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Diffstat (limited to 'af')
| -rw-r--r-- | af/n.lyx | 16 | ||||
| -rw-r--r-- | af/n1.lyx | 6758 | ||||
| -rw-r--r-- | af/n1b.lyx | 6829 |
3 files changed, 6859 insertions, 6744 deletions
@@ -309,7 +309,7 @@ https://en.wikipedia.org/wiki/%CE%A3-finite_measure \end_layout \begin_layout Chapter -Espacios de Hilbert +Espacios de Banach \end_layout \begin_layout Standard @@ -323,6 +323,20 @@ filename "n1.lyx" \end_layout \begin_layout Chapter +Espacios de Hilbert +\end_layout + +\begin_layout Standard +\begin_inset CommandInset include +LatexCommand input +filename "n1b.lyx" + +\end_inset + + +\end_layout + +\begin_layout Chapter Teoría espectral \end_layout @@ -82,31 +82,6 @@ \begin_body \begin_layout Standard -David Hilbert (1862–1943) fue un influyente matemático alemán que formuló - la teoría de los espacios de Hilbert. - En 1900 publicó una lista de 23 problemas que marcarían en buena medida - el progreso matemático en el siglo XX, y presentó 10 de ellos en el -\emph on -\lang english -International Congress of Mathematicians -\emph default -\lang spanish - de París de 1900. - Fue editor jefe de -\emph on -\lang ngerman -Mathematische Annalen -\emph default -\lang spanish -, una revista matemática muy prestigiosa por casi 150 años, y tuvo discípulos - como -\lang ngerman -Alfréd Haar, Erhard Schmidt, Hugo Steihaus, Hermann Weyl o Ernst Zermelo -\lang spanish -. -\end_layout - -\begin_layout Standard Salvo que se indique lo contrario, al hablar de espacios vectoriales entenderemo s que lo son sobre \begin_inset Formula $\mathbb{R}$ @@ -4506,4960 +4481,35 @@ nproof \end_layout \begin_layout Section -Espacios de Hilbert -\end_layout - -\begin_layout Standard -Dado un -\begin_inset Formula $\mathbb{K}$ -\end_inset - --espacio vectorial -\begin_inset Formula $H$ -\end_inset - -, -\begin_inset Formula $\langle\cdot,\cdot\rangle:H\times H\to\mathbb{K}$ -\end_inset - - es una -\series bold -forma hermitiana -\series default - si para -\begin_inset Formula $a,b\in\mathbb{K}$ -\end_inset - - y -\begin_inset Formula $x,y,z\in H$ -\end_inset - - se tiene -\begin_inset Formula $\langle ax+by,z\rangle=a\langle x,z\rangle+b\langle y,z\rangle$ -\end_inset - - y -\begin_inset Formula $\langle x,y\rangle=\overline{\langle y,x\rangle}$ -\end_inset - -, y es -\series bold -definida positiva -\series default - si para -\begin_inset Formula $x\in H\setminus0$ -\end_inset - - es -\begin_inset Formula $\langle x,x\rangle\in\mathbb{R}^{+}$ -\end_inset - -. - Un -\series bold -producto escalar -\series default - es una forma hermitiana definida positiva, y un -\series bold -espacio prehilbertiano -\series default - es par formado por un espacio vectorial y un producto escalar sobre este. -\end_layout - -\begin_layout Standard -Dado un espacio prehilbertiano -\begin_inset Formula $(H,\langle\cdot,\cdot\rangle)$ -\end_inset - -: -\end_layout - -\begin_layout Enumerate - -\series bold -Desigualdad de Cauchy-Schwartz: -\series default - -\begin_inset Formula $\forall x,y\in H,|\langle x,y\rangle|^{2}\leq\langle x,x\rangle\langle y,y\rangle$ -\end_inset - -, con igualdad si y sólo si -\begin_inset Formula $x$ -\end_inset - - e -\begin_inset Formula $y$ -\end_inset - - son linealmente dependientes. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Enumerate -\begin_inset Formula $H$ -\end_inset - - es un espacio normado con la norma -\begin_inset Formula $\Vert x\Vert\coloneqq\sqrt{\langle x,x\rangle}$ -\end_inset - -, y para -\begin_inset Formula $x,y\in H$ -\end_inset - -, -\begin_inset Formula $\Vert x+y\Vert=\Vert x\Vert+\Vert y\Vert\iff x=0\lor y=0\lor\exists a>0:x=ay$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Enumerate -Para -\begin_inset Formula $a,b\in\mathbb{K}$ -\end_inset - - y -\begin_inset Formula $x,y,z\in H$ -\end_inset - -, -\begin_inset Formula $\langle x,ay+bz\rangle=\overline{a}\langle x,y\rangle+\overline{b}\langle x,z\rangle$ -\end_inset - -. -\end_layout - -\begin_layout Enumerate -Para -\begin_inset Formula $x,y\in H$ -\end_inset - -, -\begin_inset Formula $\Vert x+y\Vert^{2}=\Vert x\Vert^{2}+\Vert y\Vert^{2}+2\text{Re}\langle x,y\rangle$ -\end_inset - -. -\end_layout - -\begin_deeper -\begin_layout Standard -\begin_inset Formula $\Vert x+y\Vert^{2}=\langle x+y,x+y\rangle=\langle x,x\rangle+\langle x,y\rangle+\overline{\langle x,y\rangle}+\langle y,y\rangle$ -\end_inset - -. -\end_layout - -\end_deeper -\begin_layout Standard - -\series bold -Identidades de polarización: -\series default - Si -\begin_inset Formula $H$ -\end_inset - - es un espacio prehilbertiano y -\begin_inset Formula $x,y\in H$ -\end_inset - -: -\end_layout - -\begin_layout Enumerate -\begin_inset Formula $\langle x,y\rangle=\frac{1}{4}(\Vert x+y\Vert^{2}-\Vert x-y\Vert^{2}+\text{i}\Vert x+\text{i}y\Vert^{2}-\text{i}\Vert x-\text{i}y\Vert^{2})$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Enumerate -Si -\begin_inset Formula $H$ -\end_inset - - se define sobre -\begin_inset Formula $\mathbb{R}$ -\end_inset - -, -\begin_inset Formula $\langle x,y\rangle=\frac{1}{4}(\Vert x+y\Vert^{2}-\Vert x-y\Vert^{2})$ -\end_inset - -. -\end_layout - -\begin_layout Standard - -\series bold -Teorema de von Neumann: -\series default - Un espacio normado -\begin_inset Formula $(X,\Vert\cdot\Vert)$ -\end_inset - - admite un producto escalar -\begin_inset Formula $\langle\cdot,\cdot\rangle$ -\end_inset - - en -\begin_inset Formula $X$ -\end_inset - - con -\begin_inset Formula $\langle x,x\rangle\equiv\Vert x\Vert^{2}$ -\end_inset - - si y sólo si -\begin_inset Formula $\Vert\cdot\Vert$ -\end_inset - - verifica la -\series bold -ley del paralelogramo: -\series default - -\begin_inset Formula -\[ -\forall x,y\in H,\Vert x+y\Vert^{2}+\Vert x-y\Vert^{2}=2(\Vert x\Vert^{2}+\Vert y\Vert^{2}). -\] - -\end_inset - - -\end_layout - -\begin_layout Itemize -\begin_inset Argument item:1 -status open - -\begin_layout Plain Layout -\begin_inset Formula $\implies]$ -\end_inset - - -\end_layout - -\end_inset - -En general -\begin_inset Formula $\langle x,y+z\rangle=\overline{\langle y+z,x\rangle}=\overline{\langle y,x\rangle}+\overline{\langle z,x\rangle}=\langle x,y\rangle+\langle x,z\rangle$ -\end_inset - -, de donde -\begin_inset Formula -\begin{multline*} -\Vert x+y\Vert^{2}+\Vert x-y\Vert^{2}=\langle x+y,x+y\rangle+\langle x-y,x-y\rangle=\\ -=\langle x,x\rangle+\langle x,y\rangle+\langle y,x\rangle+\langle y,y\rangle+\langle x,x\rangle-\langle x,y\rangle-\langle y,x\rangle+\langle y,y\rangle=2(\Vert x\Vert^{2}+\Vert y\Vert^{2}). -\end{multline*} - -\end_inset - - -\end_layout - -\begin_layout Itemize -\begin_inset Argument item:1 -status open - -\begin_layout Plain Layout -\begin_inset Formula $\impliedby]$ -\end_inset - - -\end_layout - -\end_inset - -Definimos -\begin_inset Formula $\langle\cdot,\cdot\rangle$ -\end_inset - - según la identidad de polarización, y queremos ver que es un producto escalar - cuya norma es la inicial. - Se tiene -\begin_inset Formula -\begin{align*} -\langle x,x\rangle & =\frac{1}{4}\left(\Vert2x\Vert^{2}-\Vert x-x\Vert^{2}+\text{i}\Vert x+\text{i}x\Vert^{2}-\text{i}\Vert x-\text{i}x\Vert^{2}\right)=\\ - & =\frac{1}{4}\left(4\Vert x\Vert^{2}+\text{i}|1+\text{i}|^{2}\Vert x\Vert^{2}-\text{i}|1-\text{i}|^{2}\Vert x\Vert^{2}\right)=\Vert x\Vert^{2}, -\end{align*} - -\end_inset - -y -\begin_inset Formula -\begin{align*} -4\langle x,y\rangle & =\Vert x+y\Vert^{2}-\Vert x-y\Vert^{2}+\text{i}\Vert x+\text{i}y\Vert^{2}-\text{i}\Vert x-\text{i}y\Vert^{2}\\ - & =\Vert y+x\Vert^{2}-\Vert y-x\Vert^{2}+\text{i}\Vert y-\text{i}x\Vert-\text{i}\Vert y+\text{i}x\Vert^{2}=4\overline{\langle y,x\rangle}\\ - & =\Vert-x-y\Vert^{2}-\Vert-x+y\Vert^{2}+\text{i}\Vert-x-\text{i}y\Vert^{2}-\text{i}\Vert-x+\text{i}y\Vert^{2}=-4\langle-x,y\rangle\\ - & =\Vert\text{i}x+\text{i}y\Vert^{2}-\Vert\text{i}x-\text{i}y\Vert^{2}+\text{i}\Vert\text{i}x-y\Vert^{2}-\text{i}\Vert\text{i}x+y\Vert^{2}=4\frac{\langle\text{i}x,y\rangle}{\text{i}}. -\end{align*} - -\end_inset - -Para ver que -\begin_inset Formula $\langle x+z,y\rangle=\langle x,y\rangle+\langle z,y\rangle$ -\end_inset - -, -\begin_inset Formula -\begin{multline*} -\Vert x+z+y\Vert^{2}-\Vert x+z-y\Vert^{2}=\left\Vert \left(x+\frac{y}{2}\right)+\left(z+\frac{y}{2}\right)\right\Vert ^{2}-\left\Vert \left(x+\frac{y}{2}\right)-\left(z+\frac{y}{2}\right)\right\Vert ^{2}=\\ -=2\left\Vert x+\frac{y}{2}\right\Vert ^{2}+2\left\Vert z+\frac{y}{2}\right\Vert ^{2}\cancel{-\Vert x-z\Vert^{2}}-2\left\Vert x-\frac{y}{2}\right\Vert ^{2}-2\left\Vert z-\frac{y}{2}\right\Vert ^{2}\cancel{+\Vert x-z\Vert^{2}}, -\end{multline*} - -\end_inset - -de donde -\begin_inset Formula -\begin{eqnarray*} -4\langle x+z,y\rangle & = & \Vert x+z+y\Vert^{2}-\Vert x+z-y\Vert^{2}+\text{i}\Vert x+z+\text{i}y\Vert^{2}-\text{i}\Vert x+z-\text{i}y\Vert^{2}\\ - & = & 2\left(\left\Vert x+\frac{y}{2}\right\Vert ^{2}+\left\Vert z+\frac{y}{2}\right\Vert ^{2}-\left\Vert x-\frac{y}{2}\right\Vert ^{2}-\left\Vert z-\frac{y}{2}\right\Vert \right)\\ - & & +2\text{i}\left(\left\Vert x+\text{i}\frac{y}{2}\right\Vert ^{2}+\left\Vert z+\text{i}\frac{z}{2}\right\Vert ^{2}-\left\Vert x-\text{i}\frac{y}{2}\right\Vert ^{2}-\left\Vert z-\text{i}\frac{y}{2}\right\Vert ^{2}\right)\\ - & = & 8\left\langle x,\frac{y}{2}\right\rangle +8\left\langle z,\frac{y}{2}\right\rangle , -\end{eqnarray*} - -\end_inset - -y por tanto -\begin_inset Formula -\[ -\langle x+z,y\rangle=2\left\langle x,\frac{y}{2}\right\rangle +2\left\langle z,\frac{y}{2}\right\rangle =\langle x,y\rangle+\langle z,y\rangle, -\] - -\end_inset - -donde en la segunda igualdad hemos usado la primera igualdad con -\begin_inset Formula $z=0$ -\end_inset - - o -\begin_inset Formula $x=0$ -\end_inset - -. - Usando esto y que -\begin_inset Formula $\langle-x,y\rangle$ -\end_inset - - es fácil ver que -\begin_inset Formula $\langle ax,y\rangle=a\langle x,y\rangle$ -\end_inset - - para -\begin_inset Formula $a\in\mathbb{Q}$ -\end_inset - -; para -\begin_inset Formula $a\in\mathbb{R}$ -\end_inset - - se usa la continuidad de la norma y por tanto del producto escalar, y para - -\begin_inset Formula $a\in\mathbb{C}$ -\end_inset - - se usa -\begin_inset Formula $\langle\text{i}x,y\rangle=\text{i}\langle x,y\rangle$ -\end_inset - -. -\end_layout - -\begin_layout Standard -\begin_inset Formula $(\ell^{\infty},\Vert\cdot\Vert_{\infty})$ -\end_inset - - y -\begin_inset Formula $({\cal C}([a,b]),\Vert\cdot\Vert_{1})$ -\end_inset - - son espacios normados no prehilbertianos. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Dos espacios prehilbertianos -\begin_inset Formula $(H_{1},\langle\cdot,\cdot\rangle_{1})$ -\end_inset - - y -\begin_inset Formula $(H_{2},\langle\cdot,\cdot\rangle_{2})$ -\end_inset - - son -\series bold -equivalentes -\series default - si existe un isomorfismo algebraico -\begin_inset Formula $T:H_{1}\to H_{2}$ -\end_inset - - con -\begin_inset Formula $\langle x,y\rangle_{1}=\langle T(x),T(y)\rangle_{2}$ -\end_inset - - para todo -\begin_inset Formula $x,y\in H_{1}$ -\end_inset - -, si y sólo si existe un isomorfismo isométrico entre los espacios normados. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Si -\begin_inset Formula $H$ -\end_inset - - es un espacio prehilbertiano, -\begin_inset Formula $x,y\in H$ -\end_inset - - son -\series bold -ortogonales -\series default -, -\begin_inset Formula $x\bot y$ -\end_inset - -, si -\begin_inset Formula $\langle x,y\rangle=0$ -\end_inset - -. - Decimos que -\begin_inset Formula $x\in H$ -\end_inset - - es -\series bold -ortogonal -\series default - a -\begin_inset Formula $M\subseteq H$ -\end_inset - -, -\begin_inset Formula $x\bot M$ -\end_inset - -, si -\begin_inset Formula $\forall y\in M,x\bot y$ -\end_inset - -, y llamamos -\begin_inset Formula $M^{\bot}\coloneqq\{x\in H:x\bot M\}$ -\end_inset - -. - Una familia -\begin_inset Formula $\{x_{i}\}_{i\in I}\subseteq H$ -\end_inset - - es -\series bold -ortogonal -\series default - si -\begin_inset Formula $\forall i,j\in I,(i\neq j\implies x_{i}\bot x_{j})$ -\end_inset - -, y es -\series bold -ortonormal -\series default - si además -\begin_inset Formula $\forall i,\Vert x_{i}\Vert=1$ -\end_inset - -. - Entonces: -\end_layout - -\begin_layout Enumerate - -\series bold -Teorema de Pitágoras: -\series default - Si -\begin_inset Formula $x\bot y$ -\end_inset - -, -\begin_inset Formula $\Vert x+y\Vert^{2}=\Vert x\Vert^{2}+\Vert y\Vert^{2}$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Enumerate -Si -\begin_inset Formula $(x_{i})_{i\in I}$ -\end_inset - - es una familia ortogonal de elementos no nulos, es una familia linealmente - independiente. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Enumerate -Si -\begin_inset Formula $M\subseteq H$ -\end_inset - -, -\begin_inset Formula $M^{\bot}$ -\end_inset - - es un subespacio cerrado de -\begin_inset Formula $H$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard - -\series bold -Lema de Gram-Schmidt: -\series default - Sean -\begin_inset Formula $H$ -\end_inset - - prehilbertiano, -\begin_inset Formula $\{x_{n}\}_{n}\subseteq H$ -\end_inset - - una familia contable linealmente independiente y -\begin_inset Formula $(u_{n})_{n}$ -\end_inset - - e -\begin_inset Formula $(y_{n})_{n}$ -\end_inset - - dadas por -\begin_inset Formula $u_{n}\coloneqq\frac{y_{n}}{\Vert y_{n}\Vert}$ -\end_inset - -, -\begin_inset Formula $y_{0}\coloneqq x_{0}$ -\end_inset - - y para -\begin_inset Formula $n\geq1$ -\end_inset - -, -\begin_inset Formula -\[ -y_{n}\coloneqq x_{n}-\sum_{j<n}\langle x_{n},u_{j}\rangle u_{j}, -\] - -\end_inset - - -\begin_inset Formula $(u_{n})_{n}$ -\end_inset - - es una sucesión ortonormal en -\begin_inset Formula $H$ -\end_inset - - y, para cada -\begin_inset Formula $n$ -\end_inset - -, -\begin_inset Formula $\text{span}\{u_{1},\dots,u_{n}\}=\text{span}\{x_{1},\dots,x_{n}\}$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Si -\begin_inset Formula $M$ -\end_inset - - es un subespacio de dimensión finita del espacio prehilbertiano -\begin_inset Formula $H$ -\end_inset - -: -\end_layout - -\begin_layout Enumerate -\begin_inset Formula $M$ -\end_inset - - tiene una base algebraica formada por vectores ortonormales. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Enumerate -\begin_inset Formula $M$ -\end_inset - - es equivalente a -\begin_inset Formula $\mathbb{K}^{\dim M}$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - +Redes \end_layout \begin_layout Standard Un \series bold -espacio de Hilbert +conjunto preordenado \series default - es un espacio prehilbertiano completo. - Dado un espacio de medida -\begin_inset Formula $(\Omega,\Sigma,\mu)$ -\end_inset - -, -\begin_inset Formula $L^{2}(\Omega,\Sigma,\mu)$ -\end_inset - - es un espacio de Hilbert con -\begin_inset Formula -\[ -\langle f,g\rangle\coloneqq\int_{\Omega}f\overline{g}\dif\mu, -\] - -\end_inset - -y en particular lo son -\begin_inset Formula $\ell^{2}$ -\end_inset - - con -\begin_inset Formula $\langle x,y\rangle\coloneqq\sum_{n}x_{n}\overline{y_{n}}$ -\end_inset - - y -\begin_inset Formula $\ell_{n}^{2}$ -\end_inset - - con -\begin_inset Formula $\langle x,y\rangle\coloneqq\sum_{i}x_{i}\overline{y_{i}}$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Son espacios prehilbertianos no completos: -\end_layout - -\begin_layout Enumerate -\begin_inset Formula $c_{00}$ -\end_inset - - con el producto escalar de -\begin_inset Formula $\ell^{2}$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Enumerate -\begin_inset Formula $C([a,b])$ -\end_inset - - con el producto escalar de -\begin_inset Formula $L^{2}([a,b])$ -\end_inset - - con la medida de Lebesgue, y entonces -\begin_inset Formula $C([a,b])$ -\end_inset - - es denso en -\begin_inset Formula $L^{2}([a,b])$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Section -Mejor aproximación -\end_layout - -\begin_layout Standard -Si -\begin_inset Formula $X$ -\end_inset - - es un espacio vectorial, -\begin_inset Formula $A\subseteq X$ -\end_inset - - es -\series bold -convexo -\series default - si -\begin_inset Formula $\forall\lambda\in[0,1]$ -\end_inset - -, -\begin_inset Formula $\lambda A+(1-\lambda)A\subseteq A$ -\end_inset - -. - Si -\begin_inset Formula $X$ -\end_inset - - es normado, -\begin_inset Formula $S\subseteq X$ -\end_inset - - no vacío y -\begin_inset Formula $x\in X$ -\end_inset - -, un -\begin_inset Formula $y\in S$ -\end_inset - - es un -\series bold -vector de mejor aproximación -\series default - de -\begin_inset Formula $x$ -\end_inset - - a -\begin_inset Formula $S$ -\end_inset - - si -\begin_inset Formula $\Vert x-y\Vert=\min_{z\in S}\Vert x-z\Vert$ -\end_inset - -. -\end_layout - -\begin_layout Standard - -\series bold -Teorema de mejor aproximación: -\series default - Si -\begin_inset Formula $H$ -\end_inset - - es un espacio prehilbertiano y -\begin_inset Formula $C\subseteq H$ -\end_inset - - es no vacío, convexo y completo, para cada -\begin_inset Formula $x\in H$ -\end_inset - - existe una mejor aproximación de -\begin_inset Formula $x$ + es un par +\begin_inset Formula $(C,\apprle)$ \end_inset - a + formado por un conjunto \begin_inset Formula $C$ \end_inset -. - -\series bold -Demostración: -\series default - Podemos suponer por traslación que -\begin_inset Formula $x=0$ -\end_inset - -, y llamamos -\begin_inset Formula $\alpha\coloneqq\inf_{z\in C}\Vert z\Vert$ -\end_inset - -. - Para la existencia tomamos una sucesión -\begin_inset Formula $\{y_{n}\}_{n}\subseteq C$ -\end_inset - - con -\begin_inset Formula $\lim_{n}\Vert y_{n}\Vert=\alpha$ -\end_inset - - y probamos que es de Cauchy, pues entonces por completitud existe -\begin_inset Formula $y\coloneqq\lim_{n}y_{n}\in C$ -\end_inset - - y por continuidad de la norma es -\begin_inset Formula $\Vert y\Vert=\alpha$ -\end_inset - -. - Para -\begin_inset Formula $\varepsilon>0$ -\end_inset - - existe -\begin_inset Formula $n_{0}$ -\end_inset - - tal que si -\begin_inset Formula $n\geq n_{0}$ -\end_inset - - es -\begin_inset Formula $\Vert y_{n}\Vert^{2}<\alpha^{2}+\varepsilon$ -\end_inset - -, y por la ley del paralelogramo es -\begin_inset Formula -\[ -\left\Vert \frac{y_{n}-y_{m}}{2}\right\Vert ^{2}=\frac{1}{2}(\Vert y_{n}\Vert^{2}+\Vert y_{m}\Vert^{2})-\left\Vert \frac{y_{n}+y_{m}}{2}\right\Vert ^{2}\leq\frac{1}{2}(\alpha^{2}+\varepsilon+\alpha^{2}+\varepsilon)-\alpha^{2}=\varepsilon, -\] - -\end_inset - -pues por convexidad -\begin_inset Formula $\frac{y_{n}+y_{m}}{2}\in S$ -\end_inset - - y por tanto su norma es mayor o igual a -\begin_inset Formula $\alpha$ -\end_inset - -. - Para la unicidad, si -\begin_inset Formula $y,z\in C$ -\end_inset - - cumplen -\begin_inset Formula $\Vert y\Vert=\Vert z\Vert=\alpha$ -\end_inset - -, por un argumento como el anterior, -\begin_inset Formula -\[ -\left\Vert \frac{y-z}{2}\right\Vert ^{2}=\frac{1}{2}(\Vert y\Vert^{2}+\Vert z\Vert^{2})-\left\Vert \frac{y+z}{2}\right\Vert ^{2}\leq\frac{1}{2}(\alpha^{2}+\alpha^{2})-\alpha^{2}=0. -\] - -\end_inset - - -\end_layout - -\begin_layout Standard -Como -\series bold -teorema -\series default -, si -\begin_inset Formula $Y$ -\end_inset - - es un subespacio de un espacio prehilbertiano -\begin_inset Formula $H$ -\end_inset - - y -\begin_inset Formula $x\in H$ -\end_inset - -: -\end_layout - -\begin_layout Enumerate -\begin_inset Formula $y\in Y$ -\end_inset - - es de mejor aproximación de -\begin_inset Formula $x$ -\end_inset - - a -\begin_inset Formula $Y$ -\end_inset - - si y sólo si -\begin_inset Formula $x-y\bot Y$ -\end_inset - -. -\end_layout - -\begin_deeper -\begin_layout Enumerate -\begin_inset Argument item:1 -status open - -\begin_layout Plain Layout -\begin_inset Formula $\implies]$ -\end_inset - - -\end_layout - -\end_inset - -Para -\begin_inset Formula $z\in Y$ -\end_inset - - y -\begin_inset Formula $a\in\mathbb{K}$ -\end_inset - -, como -\begin_inset Formula $y-az\in Y$ -\end_inset - -, -\begin_inset Formula -\[ -\Vert x-y\Vert^{2}\leq\Vert x-y+az\Vert^{2}=\Vert x-y\Vert^{2}+2\text{Re}(a\langle z,x-y\rangle)+|a|^{2}\Vert z\Vert^{2}, -\] - -\end_inset - -luego -\begin_inset Formula $0\leq2\text{Re}(a\langle z,x-y\rangle)+|a|^{2}\Vert z\Vert^{2}$ -\end_inset - - y, haciendo -\begin_inset Formula $a=t\langle x-y,z\rangle$ -\end_inset - - con -\begin_inset Formula $t\in\mathbb{R}$ -\end_inset - -, -\begin_inset Formula $0\leq2t|\langle x-y,z\rangle|^{2}+t^{2}|\langle x-y,z\rangle|^{2}\Vert z\Vert^{2}$ -\end_inset - -. - Si hubiera -\begin_inset Formula $z\in Y$ -\end_inset - - con -\begin_inset Formula $\langle x-y,z\rangle\neq0$ -\end_inset - -, -\begin_inset Formula $0\leq2t+t^{2}\Vert z\Vert^{2}$ -\end_inset - - para todo -\begin_inset Formula $t\in\mathbb{R}$ -\end_inset - -, pero si -\begin_inset Formula $\Vert z\Vert^{2}=0$ -\end_inset - -, esto es negativo cuando -\begin_inset Formula $t<0$ -\end_inset - -, y si -\begin_inset Formula $\Vert z\Vert^{2}>0$ -\end_inset - -, es negativo al menos cuando -\begin_inset Formula $t=-\frac{1}{\Vert z\Vert^{2}}\#$ -\end_inset - -, luego -\begin_inset Formula $x-y\bot z$ -\end_inset - - y -\begin_inset Formula $x-y\bot Y$ -\end_inset - -. -\end_layout - -\begin_layout Enumerate -\begin_inset Argument item:1 -status open - -\begin_layout Plain Layout -\begin_inset Formula $\impliedby]$ -\end_inset - - -\end_layout - -\end_inset - -Para -\begin_inset Formula $z\in Y$ -\end_inset - -, por el teorema de Pitágoras, -\begin_inset Formula -\[ -\Vert x-z\Vert^{2}=\Vert x-y+y-z\Vert^{2}=\Vert x-y\Vert^{2}+\Vert y-z\Vert^{2}\geq\Vert x-y\Vert^{2}. -\] - -\end_inset - - -\end_layout - -\end_deeper -\begin_layout Enumerate -Si existe una mejor aproximación de -\begin_inset Formula $x$ -\end_inset - - a -\begin_inset Formula $Y$ -\end_inset - -, es única. -\end_layout - -\begin_deeper -\begin_layout Standard -Sean -\begin_inset Formula $y,z\in Y$ -\end_inset - - de mejor aproximación, como -\begin_inset Formula $x-y,x-z\in Y^{\bot}$ -\end_inset - -, su diferencia -\begin_inset Formula $y-z\in Y^{\bot}\cap Y$ -\end_inset - -, luego -\begin_inset Formula $\langle y-z,y-z\rangle=0$ -\end_inset - - e -\begin_inset Formula $y=z$ -\end_inset - -. -\end_layout - -\end_deeper -\begin_layout Enumerate -Si -\begin_inset Formula $Y$ -\end_inset - - es completo, hay vector de mejor aproximación. -\end_layout - -\begin_deeper -\begin_layout Standard -Por el teorema anterior (los subespacios son convexos). -\end_layout - -\end_deeper -\begin_layout Section -Determinante de Gram -\end_layout - -\begin_layout Standard -Sean -\begin_inset Formula $H$ -\end_inset - - prehilbertiano y -\begin_inset Formula $M\leq H$ -\end_inset - - de dimensión finita con base ortonormal -\begin_inset Formula $(e_{i})_{i}$ -\end_inset - -. -\end_layout - -\begin_layout Enumerate -Para -\begin_inset Formula $x\in H$ -\end_inset - - existe un único vector de aproximación de -\begin_inset Formula $x$ -\end_inset - - a -\begin_inset Formula $M$ -\end_inset - - dado por -\begin_inset Formula -\[ -\sum_{i}\langle x,e_{i}\rangle e_{i}. -\] - -\end_inset - - -\end_layout - -\begin_layout Enumerate -\begin_inset Formula $d(x,M)^{2}=\Vert x\Vert^{2}-\sum_{i}|\langle x,e_{i}\rangle|^{2}$ -\end_inset - -. -\end_layout - -\begin_layout Standard -Llamamos -\series bold -determinante de Gram -\series default - de -\begin_inset Formula $(x_{i})_{i=1}^{n}$ -\end_inset - - a -\begin_inset Formula -\[ -G(x_{1},\dots,G_{n})\coloneqq\det(\langle x_{j},x_{i}\rangle)_{1\leq i\leq n}^{1\leq j\leq n}. -\] - -\end_inset - -Como -\series bold -teorema -\series default -, si -\begin_inset Formula $H$ -\end_inset - - es prehilbertiano, -\begin_inset Formula $M\leq H$ -\end_inset - - de dimensión finita con base -\begin_inset Formula $(b_{i})_{i}$ -\end_inset - - y -\begin_inset Formula $x\in H$ -\end_inset - -, el vector de mejor aproximación de -\begin_inset Formula $x$ -\end_inset - - a -\begin_inset Formula $M$ -\end_inset - - es -\begin_inset Formula -\[ -\frac{-1}{G(b_{1},\dots,b_{n})}\begin{vmatrix}\langle x_{1},x_{1}\rangle & \langle x_{2},x_{1}\rangle & \cdots & \langle x_{n},x_{1}\rangle & \langle x,x_{1}\rangle\\ -\langle x_{1},x_{2}\rangle & \langle x_{2},x_{2}\rangle & \cdots & \langle x_{n},x_{2}\rangle & \langle x,x_{2}\rangle\\ -\vdots & \vdots & \ddots & \vdots & \vdots\\ -\langle x_{1},x_{n}\rangle & \langle x_{2},x_{n}\rangle & \cdots & \langle x_{n},x_{n}\rangle & \langle x,x_{n}\rangle\\ -x_{1} & x_{2} & \cdots & x_{n} & 0 -\end{vmatrix}, -\] - -\end_inset - -y -\begin_inset Formula -\[ -d(x,M)=\sqrt{\frac{G(x_{1},\dots,x_{n},x)}{G(x_{1},\dots,x_{n})}}. -\] - -\end_inset - - -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Algunas aplicaciones: -\end_layout - -\begin_layout Enumerate - -\series bold -Resolución de sistemas sobre-dimensionados por mínimos cuadrados. - -\series default - Tenemos un fenómeno experimental que se puede modelar como una función - lineal -\begin_inset Formula $y(x)=a_{1}x_{1}+\dots+a_{n}x_{n}$ -\end_inset - -, pero no conocemos los -\begin_inset Formula $a_{i}$ -\end_inset - -. - Hacemos -\begin_inset Formula $m$ -\end_inset - - experimentos fijando un -\begin_inset Formula $x_{i}$ -\end_inset - - en cada uno y midiendo -\begin_inset Formula $y_{i}\coloneqq y(x_{i})$ -\end_inset - - para plantear un sistema de -\begin_inset Formula $m$ -\end_inset - - ecuaciones. - Solo hacen falta -\begin_inset Formula $n$ -\end_inset - - experimentos cuidando que los -\begin_inset Formula $x_{i}$ -\end_inset - - sean linealmente independientes, pero en general conviene hacer más, -\begin_inset Formula $m>n$ -\end_inset - -. - Como las mediciones son aproximadas, el sistema puede ser incompatible, - por lo que se eligen los -\begin_inset Formula $a_{i}\in\mathbb{R}$ -\end_inset - - de forma que se minimice -\begin_inset Formula -\[ -\sum_{i\in\mathbb{N}_{m}}\left(y_{i}-\sum_{j\in\mathbb{N}_{n}}a_{j}x_{ij}\right)^{2}=\left\Vert y-\sum_{j\in\mathbb{N}_{n}}a_{j}X_{j}\right\Vert ^{2}, -\] - -\end_inset - -donde -\begin_inset Formula $X_{j}\coloneqq(x_{1j},\dots,x_{mj})$ -\end_inset - -. - Si -\begin_inset Formula $X_{1},\dots,X_{n}$ -\end_inset - - son linealmente independientes, sea -\begin_inset Formula $M\coloneqq\text{span}\{X_{1},\dots,X_{n}\}<\mathbb{R}^{m}$ -\end_inset - -, buscamos el vector -\begin_inset Formula $Z\in M$ -\end_inset - - de mejor aproximación de -\begin_inset Formula $y$ -\end_inset - - en -\begin_inset Formula $M$ -\end_inset - - que, expresado respecto de la base -\begin_inset Formula $(X_{1},\dots,X_{n})$ -\end_inset - -, nos dará el vector -\begin_inset Formula $(a_{1},\dots,a_{n})$ -\end_inset - - buscado. -\end_layout - -\begin_layout Enumerate - -\series bold -Ajustes polinómicos por mínimos cuadrados. - -\series default - Queremos modelar un fenómeno experimental como una función polinómica -\begin_inset Formula $f:[a,b]\to\mathbb{R}$ -\end_inset - -, y tenemos -\begin_inset Formula $k$ -\end_inset - - observaciones de la forma -\begin_inset Formula $f(t_{i})=y_{i}$ -\end_inset - - con -\begin_inset Formula $t_{1}<\dots<t_{k}$ -\end_inset - -. - Existe un polinomio de grado máximo -\begin_inset Formula $k-1$ -\end_inset - - que cumple esto, pero muchas veces -\begin_inset Formula $k$ -\end_inset - - es muy grande y esto complica los cálculos y puede llevar al -\emph on -\lang english -overfitting -\emph default -\lang spanish - o fenómeno de Runge. - Entonces buscamos un polinomio -\begin_inset Formula $f$ -\end_inset - - de grado máximo -\begin_inset Formula $n$ -\end_inset - - bastante menor que -\begin_inset Formula $k-1$ -\end_inset - - que minimice -\begin_inset Formula -\[ -\sum_{i\in\mathbb{N}_{k}}|y_{i}-f(t_{i})|^{2}=\left\Vert y-\sum_{j=0}^{n}f_{j}t^{j}\right\Vert ^{2}, -\] - -\end_inset - -donde -\begin_inset Formula $t^{j}\coloneqq(t_{1}^{j},\dots,t_{k}^{j})$ -\end_inset - -. - Para ello, como para -\begin_inset Formula $k\geq2$ -\end_inset - - los -\begin_inset Formula $t^{j}$ -\end_inset - - son linealmente independientes, consideramos -\begin_inset Formula $M\coloneqq\text{span}\{1,t,t^{2},\dots,t^{n}\}<\mathbb{R}^{n+1}$ -\end_inset - - y buscamos la mejor aproximación de -\begin_inset Formula $y$ -\end_inset - - a -\begin_inset Formula $M$ -\end_inset - -. -\end_layout - -\begin_layout Section -Teorema de la proyección -\end_layout - -\begin_layout Standard - -\series bold -Teorema de la proyección: -\series default - Si -\begin_inset Formula $H$ -\end_inset - - es un espacio de Hilbert con un subespacio cerrado -\begin_inset Formula $M$ -\end_inset - - y -\begin_inset Formula $P_{M}:H\to M$ -\end_inset - - la -\series bold -proyección ortogonal -\series default - de -\begin_inset Formula $H$ -\end_inset - - sobre -\begin_inset Formula $M$ -\end_inset - - que asigna a cada -\begin_inset Formula $x\in H$ -\end_inset - - la mejor aproximación de -\begin_inset Formula $x$ -\end_inset - - a -\begin_inset Formula $M$ -\end_inset - -: -\end_layout - -\begin_layout Enumerate -\begin_inset Formula $H$ -\end_inset - - es suma directa topológica de -\begin_inset Formula $M$ -\end_inset - - y -\begin_inset Formula $M^{\bot}$ -\end_inset - -, -\begin_inset Formula $P_{M}$ -\end_inset - - es la proyección canónica y, si -\begin_inset Formula $P_{M^{\bot}}:H\to M^{\bot}$ -\end_inset - - es la otra proyección canónica, si -\begin_inset Formula $M\neq0$ -\end_inset - -, -\begin_inset Formula $\Vert P_{M}\Vert=1$ -\end_inset - -, y si -\begin_inset Formula $M^{\bot}\neq0$ -\end_inset - -, -\begin_inset Formula $\Vert P_{M^{\bot}}\Vert=1$ -\end_inset - -. -\end_layout - -\begin_deeper -\begin_layout Standard -Por la definición de producto escalar, -\begin_inset Formula $M^{\bot}\leq H$ -\end_inset - -. - Claramente -\begin_inset Formula $M\cap M^{\bot}=0$ -\end_inset - -, y para -\begin_inset Formula $x\in M$ -\end_inset - -, como -\begin_inset Formula $y\coloneqq P_{M}(x)$ -\end_inset - - cumple -\begin_inset Formula $x-y\bot M$ -\end_inset - -, -\begin_inset Formula $x=y+z$ -\end_inset - - con -\begin_inset Formula $y\in M$ -\end_inset - - y -\begin_inset Formula $z\coloneqq x-y\in M^{\bot}$ -\end_inset - -, luego -\begin_inset Formula $M+M^{\bot}=H$ -\end_inset - - y -\begin_inset Formula $H$ -\end_inset - - es suma directa algebraica de -\begin_inset Formula $M$ -\end_inset - - y -\begin_inset Formula $M^{\bot}$ -\end_inset - -. - -\begin_inset Formula $P_{M}$ -\end_inset - - es la proyección canónica porque, si -\begin_inset Formula $y\in M$ -\end_inset - - y -\begin_inset Formula $z\in M^{\bot}$ -\end_inset - -, -\begin_inset Formula $(y+z)-y=z\bot M$ -\end_inset - -, y por unicidad de la mejor aproximación, -\begin_inset Formula $P_{M}(y+z)=y$ -\end_inset - -. - -\begin_inset Formula $P_{M}$ -\end_inset - - y -\begin_inset Formula $P_{M^{\bot}}$ -\end_inset - - son lineales por ser proyecciones canónicas, y para -\begin_inset Formula $x=y+z\in S_{H}$ -\end_inset - - con -\begin_inset Formula $y\in M$ -\end_inset - - y -\begin_inset Formula $z\in M^{\bot}$ -\end_inset - -, -\begin_inset Formula $\Vert x\Vert^{2}=\Vert y\Vert^{2}+\Vert z\Vert^{2}=\Vert P_{M}(x)\Vert^{2}+\Vert P_{M^{\bot}}(x)\Vert^{2}$ -\end_inset - - y -\begin_inset Formula $\Vert P_{M}(x)\Vert,\Vert P_{M^{\bot}}(x)\Vert\leq\Vert x\Vert=1$ -\end_inset - -, lo que prueba la continuidad y por tanto que -\begin_inset Formula $M$ -\end_inset - - es topológica. - Además, si -\begin_inset Formula $M\neq0$ -\end_inset - -, existe -\begin_inset Formula $y\in S_{M}$ -\end_inset - - y -\begin_inset Formula $\Vert P_{M}(y)\Vert=\Vert y\Vert=1$ -\end_inset - -, luego -\begin_inset Formula $\Vert P_{M}\Vert=1$ -\end_inset - -, y análogamente para -\begin_inset Formula $M^{\bot}$ -\end_inset - -. -\end_layout - -\end_deeper -\begin_layout Enumerate -\begin_inset Formula $P_{M}(H)=M$ -\end_inset - -, -\begin_inset Formula $\ker P_{M}=M^{\bot}$ -\end_inset - - y -\begin_inset Formula $P_{M^{\bot}}=1_{H}-P_{M}$ -\end_inset - -. -\end_layout - -\begin_layout Enumerate -Para -\begin_inset Formula $x,y\in H$ -\end_inset - -, -\begin_inset Formula $\langle P_{M}(x),y\rangle=\langle x,P_{M}(y)\rangle$ -\end_inset - - y -\begin_inset Formula $\langle P_{M^{\bot}}(x),y\rangle=\langle x,P_{M^{\bot}}(y)\rangle$ -\end_inset - -. -\end_layout - -\begin_deeper -\begin_layout Standard -Si -\begin_inset Formula $x=x_{1}+x_{2}$ -\end_inset - - e -\begin_inset Formula $y=y_{1}+y_{2}$ -\end_inset - - con -\begin_inset Formula $x_{1},y_{1}\in M$ -\end_inset - - y -\begin_inset Formula $x_{2},y_{2}\in M^{\bot}$ -\end_inset - -, -\begin_inset Formula $\langle P_{M}(x),y\rangle=\langle x_{1},y_{1}+y_{2}\rangle=\langle x_{1},y_{1}\rangle=\langle x_{1}+x_{2},y_{1}\rangle=\langle x,P_{M}(y)\rangle$ -\end_inset - -, y para -\begin_inset Formula $P_{M^{\bot}}$ -\end_inset - - es análogo. -\end_layout - -\end_deeper -\begin_layout Enumerate -\begin_inset Formula $M^{\bot\bot}=M$ -\end_inset - -. -\end_layout - -\begin_deeper -\begin_layout Standard -Si -\begin_inset Formula $x\in M$ -\end_inset - -, para -\begin_inset Formula $y\in M^{\bot}$ -\end_inset - -, -\begin_inset Formula $\langle y,x\rangle=\overline{\langle x,y\rangle}=0$ -\end_inset - -, luego -\begin_inset Formula $x\in M^{\bot\bot}$ -\end_inset - -. - Si -\begin_inset Formula $x\in M^{\bot\bot}\subseteq H$ -\end_inset - -, sean -\begin_inset Formula $y\in M$ -\end_inset - - y -\begin_inset Formula $z\in M^{\bot}$ -\end_inset - - con -\begin_inset Formula $x=y+z$ -\end_inset - -, -\begin_inset Formula $0=\langle x,z\rangle=\langle y,z\rangle+\langle z,z\rangle=\langle z,z\rangle=\Vert z\Vert^{2}$ -\end_inset - -, luego -\begin_inset Formula $z=0$ -\end_inset - - y -\begin_inset Formula $x\in M$ -\end_inset - -. -\end_layout - -\end_deeper -\begin_layout Standard -Esto no es cierto si -\begin_inset Formula $M$ -\end_inset - - no es cerrado ni si -\begin_inset Formula $H$ -\end_inset - - no es completo. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Un espacio normado es de Hilbert si y sólo si cada subespacio cerrado tiene - un complementario topológico. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Un subconjunto -\begin_inset Formula $S$ -\end_inset - - de un espacio normado -\begin_inset Formula $(X,\Vert\cdot\Vert)$ -\end_inset - - es -\series bold -total -\series default - si -\begin_inset Formula $\overline{\text{span}S}=X$ -\end_inset - -, y si -\begin_inset Formula $H$ -\end_inset - - es de Hilbert esto ocurre si y sólo si -\begin_inset Formula $S^{\bot}=0$ -\end_inset - -. -\end_layout - -\begin_layout Section -Dual de un espacio de Hilbert -\end_layout - -\begin_layout Standard - -\series bold -Teorema de Riesz-Fréchet: -\series default - Dados un espacio de Hilbert -\begin_inset Formula $H$ -\end_inset - - y un operador -\begin_inset Formula $f:H\to\mathbb{K}$ -\end_inset - -, -\begin_inset Formula $f$ -\end_inset - - es acotado si y sólo si existe -\begin_inset Formula $y\in H$ -\end_inset - - con -\begin_inset Formula $f=\langle\cdot,y\rangle$ -\end_inset - -, en cuyo caso -\begin_inset Formula $y$ -\end_inset - - es único y -\begin_inset Formula $\Vert f\Vert=\Vert y\Vert$ -\end_inset - -. -\end_layout - -\begin_layout Itemize -\begin_inset Argument item:1 -status open - -\begin_layout Plain Layout -\begin_inset Formula $\implies]$ -\end_inset - - -\end_layout - -\end_inset - -Para la unicidad, si -\begin_inset Formula $f(x)=\langle x,y\rangle=\langle x,z\rangle$ -\end_inset - - para todo -\begin_inset Formula $x\in H$ -\end_inset - -, -\begin_inset Formula $\langle x,y-z\rangle=0$ -\end_inset - -, luego -\begin_inset Formula $y-z\bot H$ -\end_inset - - y, como -\begin_inset Formula $H^{\bot}=0$ -\end_inset - -, -\begin_inset Formula $y=z$ -\end_inset - -. - Para la existencia, si -\begin_inset Formula $f=0$ -\end_inset - - tomamos -\begin_inset Formula $y=0$ -\end_inset - -, y en otro caso, -\begin_inset Formula $Y\coloneqq\ker f$ -\end_inset - - es un subespacio cerrado de -\begin_inset Formula $H$ -\end_inset - - y por tanto -\begin_inset Formula $H=Y\oplus Y^{\bot}$ -\end_inset - -, con -\begin_inset Formula $\dim Y^{\bot}=\dim\text{Im}f=1$ -\end_inset - -. - Sea entonces -\begin_inset Formula $z\in Y^{\bot}$ -\end_inset - - unitario, la proyección ortogonal de un -\begin_inset Formula $x\in H$ -\end_inset - - sobre -\begin_inset Formula $Y^{\bot}$ -\end_inset - - es -\begin_inset Formula $\langle x,z\rangle z$ -\end_inset - -, luego -\begin_inset Formula $x-\langle x,z\rangle z\in Y$ -\end_inset - - y -\begin_inset Formula -\[ -f(x)=f(x-\langle x,z\rangle z+\langle x,z\rangle z)=f(\langle x,z\rangle z)=\langle x,z\rangle f(z)=\langle x,\overline{f(z)}z\rangle\eqqcolon\langle x,y\rangle. -\] - -\end_inset - -Para -\begin_inset Formula $x\in S_{H}$ -\end_inset - -, por la desigualdad de Cauchy-Schwartz, -\begin_inset Formula $\Vert f(x)\Vert^{2}=|\langle x,y\rangle|^{2}\leq\langle x,x\rangle\langle y,y\rangle=\Vert y\Vert^{2}$ -\end_inset - -, luego -\begin_inset Formula $\Vert f\Vert\leq\Vert y\Vert$ -\end_inset - -, pero -\begin_inset Formula $f(\frac{y}{\Vert y\Vert})=\frac{f(y)}{\Vert y\Vert}=\frac{\Vert y\Vert^{2}}{\Vert y\Vert}=\Vert y\Vert$ -\end_inset - -, luego -\begin_inset Formula $\Vert f\Vert=\Vert y\Vert$ -\end_inset - -. -\end_layout - -\begin_layout Itemize -\begin_inset Argument item:1 -status open - -\begin_layout Plain Layout -\begin_inset Formula $\impliedby]$ -\end_inset - - -\end_layout - -\end_inset - - -\begin_inset Formula $f\coloneqq\langle\cdot,y\rangle$ -\end_inset - - es lineal, y es continua por el argumento anterior que prueba que -\begin_inset Formula $\Vert f\Vert=\Vert y\Vert$ -\end_inset - -. -\end_layout - -\begin_layout Standard -El teorema no es válido si -\begin_inset Formula $H$ -\end_inset - - no es completo. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Sean -\begin_inset Formula $H$ -\end_inset - - un espacio de Hilbert y -\begin_inset Formula $T:H^{*}\to H$ -\end_inset - - que a cada -\begin_inset Formula $f$ -\end_inset - - le asocia el -\begin_inset Formula $y$ -\end_inset - - con -\begin_inset Formula $f=\langle\cdot,y\rangle$ -\end_inset - -: -\end_layout - -\begin_layout Enumerate -\begin_inset Formula $T$ -\end_inset - - es biyectiva, isométrica y lineal conjugada. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Enumerate -\begin_inset Formula $H^{*}$ -\end_inset - - es un espacio de Hilbert con el producto escalar -\begin_inset Formula $\langle f,g\rangle^{*}\coloneqq\langle T(g),T(f)\rangle$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Enumerate -\begin_inset Formula $J:H\to H^{**}$ -\end_inset - - dada por -\begin_inset Formula $J(x)(f)\coloneqq f(x)$ -\end_inset - - es un isomorfismo algebraico isométrico. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Dado un un -\begin_inset Formula $\mathbb{K}$ -\end_inset - --espacio vectorial -\begin_inset Formula $X$ -\end_inset - -, -\begin_inset Formula $B:X\times X\to\mathbb{K}$ -\end_inset - - es -\series bold -bilineal -\series default - si las -\begin_inset Formula $B(\cdot,y)$ -\end_inset - - y -\begin_inset Formula $B(x,\cdot)$ -\end_inset - - son lineales, -\series bold -sesquilineal -\series default - si las -\begin_inset Formula $B(\cdot,y)$ -\end_inset - - son lineales y las -\begin_inset Formula $B(x,\cdot)$ -\end_inset - - son lineales conjugadas, -\series bold -simétrica -\series default - si -\begin_inset Formula $B(x,y)\equiv B(y,x)$ -\end_inset - - y -\series bold -positiva -\series default - si -\begin_inset Formula $\forall x\in X,B(x,x)\geq0$ -\end_inset - -. - Si además -\begin_inset Formula $X$ -\end_inset - - es normado, -\begin_inset Formula $B$ -\end_inset - - es -\series bold -acotada -\series default - si -\begin_inset Formula $\exists M>0:\forall x,y\in X,|B(x,y)|\leq M\Vert x\Vert\Vert y\Vert$ -\end_inset - -, y es -\series bold -fuertemente positiva -\series default - si -\begin_inset Formula $\exists c>0:\forall x\in X,B(x,x)\geq c\Vert x\Vert^{2}$ -\end_inset - -. - -\end_layout - -\begin_layout Standard -Si -\begin_inset Formula $B$ -\end_inset - - es bilineal o sesquilineal, es acotada si y sólo si es continua, y para - todo -\begin_inset Formula $x$ -\end_inset - - e -\begin_inset Formula $y$ -\end_inset - - es -\begin_inset Formula $2B(x,x)+2B(y,y)=B(x+y,x+y)+B(x-y,x-y)$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard - + y un \series bold -Teorema de Lax-Milgram: +preorden \series default - Sean -\begin_inset Formula $H$ -\end_inset - - un espacio de Hilbert y -\begin_inset Formula $B$ -\end_inset - - una -\begin_inset Formula $H$ -\end_inset - --forma sesquilineal acotada y fuertemente positiva, existe un único isomorfismo - de espacios de Hilbert -\begin_inset Formula $T:H\to H$ -\end_inset - - tal que -\begin_inset Formula $\forall x,y\in H,B(x,y)=\langle x,T(y)\rangle$ -\end_inset - -. -\series bold -Demostración: -\series default - Sea -\begin_inset Formula -\[ -Y\coloneqq\{y\in H\mid\exists z\in H:\langle\cdot,y\rangle=B(\cdot,z)\}, -\] - -\end_inset - - -\begin_inset Formula $0\in Y$ -\end_inset - - tomando -\begin_inset Formula $z=0$ -\end_inset - - y -\begin_inset Formula $z$ -\end_inset - - está unívocamente determinado por -\begin_inset Formula $y$ -\end_inset - -, ya que si -\begin_inset Formula $\langle\cdot,y\rangle=B(\cdot,z)=B(\cdot,z')$ -\end_inset - - entonces -\begin_inset Formula $B(\cdot,z-z')=0$ -\end_inset - - y en particular -\begin_inset Formula $0=B(z-z',z-z')\geq c\Vert z-z'\Vert^{2}$ -\end_inset - - para cierto -\begin_inset Formula $c>0$ -\end_inset - - por ser -\begin_inset Formula $B$ -\end_inset - - fuertemente positiva, luego -\begin_inset Formula $z=z'$ -\end_inset - -. - Como -\begin_inset Formula $\langle\cdot,\cdot\rangle$ -\end_inset - - y -\begin_inset Formula $B$ -\end_inset - - son sesquilineales, -\begin_inset Formula $Y$ -\end_inset - - es un espacio vectorial y -\begin_inset Formula $S:Y\to H$ -\end_inset - - que a cada -\begin_inset Formula $y$ -\end_inset - - le asocia el -\begin_inset Formula $z$ -\end_inset - - con -\begin_inset Formula $\langle\cdot,y\rangle=B(\cdot,z)$ -\end_inset - - es lineal. - Entonces, para -\begin_inset Formula $y\in S_{Y}$ -\end_inset - -, -\begin_inset Formula -\[ -c\Vert S(y)\Vert^{2}\leq B(S(y),S(y))=\langle S(y),y\rangle\in\mathbb{R}^{+}, -\] - -\end_inset - -pero por la desigualdad de Cauchy-Schwartz, -\begin_inset Formula $\langle S(y),y\rangle^{2}=|\langle S(y),y\rangle|^{2}\leq\Vert S(y)\Vert^{2}\Vert y\Vert^{2}$ -\end_inset - -, luego -\begin_inset Formula $c\Vert S(y)\Vert^{2}\leq\langle S(y),y\rangle\leq\Vert S(y)\Vert\Vert y\Vert=\Vert S(y)\Vert$ -\end_inset - - y -\begin_inset Formula $\Vert S(y)\Vert\leq\frac{1}{c}$ -\end_inset - -, con lo que -\begin_inset Formula $S$ -\end_inset - - es continua. - Entonces, si -\begin_inset Formula $\{y_{n}\}_{n}\subseteq Y$ -\end_inset - - y existe -\begin_inset Formula $\lim_{n}y_{n}\eqqcolon y\in H$ -\end_inset - -, por continuidad de -\begin_inset Formula $S$ -\end_inset - - y de -\begin_inset Formula $B$ -\end_inset - -, -\begin_inset Formula -\[ -\langle x,y\rangle=\lim_{n}\langle x,y_{n}\rangle=\lim_{n}B(x,S(y_{n}))=B(x,S(y)), -\] - -\end_inset - -luego -\begin_inset Formula $y\in Y$ -\end_inset - - e -\begin_inset Formula $Y$ -\end_inset - - es cerrado. - Entonces, si -\begin_inset Formula $z\in Y^{\bot}$ -\end_inset - -, como -\begin_inset Formula $B(\cdot,z):H\to\mathbb{K}$ -\end_inset - - es continua, por el teorema de Riesz-Fréchet existe -\begin_inset Formula $w\in H$ -\end_inset - - con -\begin_inset Formula $B(\cdot,z)=\langle\cdot,w\rangle$ -\end_inset - -, luego -\begin_inset Formula $w\in Y$ -\end_inset - -, pero entonces -\begin_inset Formula $B(z,z)=\langle z,w\rangle=0$ -\end_inset - - y, por ser -\begin_inset Formula $B$ -\end_inset - - fuertemente positiva, -\begin_inset Formula $z=0$ -\end_inset - -, luego -\begin_inset Formula $Y^{\bot}=0$ -\end_inset - - e -\begin_inset Formula $Y=H$ -\end_inset - -. - Para -\begin_inset Formula $z\in H$ -\end_inset - -, como -\begin_inset Formula $B(\cdot,z)$ -\end_inset - - es continua, existe -\begin_inset Formula $w\in H$ -\end_inset - - con -\begin_inset Formula $B(\cdot z)=\langle\cdot,w\rangle$ -\end_inset - - y por tanto -\begin_inset Formula $z=S(w)$ -\end_inset - -, luego -\begin_inset Formula $S$ -\end_inset - - es suprayectiva. - Si -\begin_inset Formula $S(y)=0$ -\end_inset - -, para -\begin_inset Formula $x\in H$ -\end_inset - -, -\begin_inset Formula $\langle x,y\rangle=B(x,S(y))=0$ -\end_inset - - y por tanto -\begin_inset Formula $y=0$ -\end_inset - -, luego -\begin_inset Formula $S$ -\end_inset - - es inyectiva. - Por tanto -\begin_inset Formula $S$ -\end_inset - - es biyectiva y -\begin_inset Formula $T\coloneqq S^{-1}$ -\end_inset - - cumple -\begin_inset Formula $\langle x,T(y)\rangle=B(x,y)$ -\end_inset - -. - Además, para -\begin_inset Formula $y\in S_{H}$ -\end_inset - -, -\begin_inset Formula $\Vert T(y)\Vert^{2}=\langle T(y),T(y)\rangle=B(T(y),y)\leq M\Vert T(y)\Vert\Vert y\Vert=M\Vert T(y)\Vert$ -\end_inset - -, siendo -\begin_inset Formula $M$ -\end_inset - - una cota de -\begin_inset Formula $B$ -\end_inset - -, de donde -\begin_inset Formula $\Vert T\Vert\leq M$ -\end_inset - - y, como -\begin_inset Formula $\Vert T^{-1}\Vert=\Vert S\Vert\leq\frac{1}{c}$ -\end_inset - -, -\begin_inset Formula $T$ -\end_inset - - es un isomorfismo topológico isométrico. -\end_layout - -\begin_layout Standard -En particular, dado un espacio vectorial -\begin_inset Formula $H$ -\end_inset - - con dos productos escalares -\begin_inset Formula $\langle\cdot,\cdot\rangle_{1}$ -\end_inset - - y -\begin_inset Formula $\langle\cdot,\cdot\rangle_{2}$ -\end_inset - - equivalentes que hacen a -\begin_inset Formula $H$ -\end_inset - - completo, existe un isomorfismo -\begin_inset Formula $T:H\to H$ -\end_inset - - de espacios de Hilbert con -\begin_inset Formula $\langle x,y\rangle_{1}=\langle x,T(y)\rangle_{2}$ -\end_inset - -. -\end_layout - -\begin_layout Standard -Dado un espacio medible -\begin_inset Formula $(\Omega,\Sigma)$ -\end_inset - - con medidas -\begin_inset Formula $\mu$ -\end_inset - - y -\begin_inset Formula $\nu$ -\end_inset - -, -\begin_inset Formula $\nu$ -\end_inset - - es -\series bold -absolutamente continua -\series default - respecto de -\begin_inset Formula $\mu$ -\end_inset - - si -\begin_inset Formula $\forall A\in\Sigma,(\mu(A)=0\implies\nu(A)=0)$ -\end_inset - -, y es -\series bold -finita -\series default - si -\begin_inset Formula $\nu(\Omega)<\infty$ -\end_inset - -. - -\series bold -Teorema de Radon-Nicodym: -\series default - Si -\begin_inset Formula $(\Omega,\Sigma)$ -\end_inset - - es un espacio medible con medidas finitas -\begin_inset Formula $\mu$ -\end_inset - - y -\begin_inset Formula $\nu$ -\end_inset - - siendo -\begin_inset Formula $\nu$ -\end_inset - - absolutamente continua respecto de -\begin_inset Formula $\mu$ -\end_inset - -, existe -\begin_inset Formula $g:\Omega\to[0,+\infty]$ -\end_inset - - -\begin_inset Formula $\mu$ -\end_inset - --integrable tal que -\begin_inset Formula -\[ -\forall A\in\Sigma,\nu(A)=\int_{A}g\dif\mu. -\] - -\end_inset - - -\series bold -Demostración: -\series default - -\begin_inset Formula $\sigma\coloneqq\mu+\nu$ -\end_inset - - es una medida finita en -\begin_inset Formula $X$ -\end_inset - - tal que -\begin_inset Formula $\forall A\in\Sigma,(\sigma(A)=0\iff\mu(A)=0)$ -\end_inset - -, y la función lineal entre espacios de Hilbert -\begin_inset Formula $T:L^{2}(\Omega,\Sigma,\sigma)\to\mathbb{R}$ -\end_inset - - dada por -\begin_inset Formula -\[ -Tu\coloneqq\int_{\Omega}u\dif\mu -\] - -\end_inset - -está bien definida y es continua porque, si -\begin_inset Formula $\Vert u\Vert_{L^{2}(\Omega,\Sigma,\sigma)}=1$ -\end_inset - -, -\begin_inset Formula -\begin{align*} -|Tu| & =\left|\int_{\Omega}u\dif\mu\right|\leq\int_{\Omega}|u|\dif\mu\leq\sqrt{\int_{\Omega}|u|^{2}\dif\mu}+\sqrt{\int_{\Omega}\dif\mu}\leq\\ - & \leq\sqrt{\int_{\Omega}|u|^{2}\dif\mu+\int_{\Omega}|u|^{2}\dif\nu}+\sqrt{\int_{\Omega}\dif\mu+\int_{\Omega}\dif\nu}=1+\sqrt{\sigma(X)}. -\end{align*} - -\end_inset - -Por el teorema de representación de Riesz, existe -\begin_inset Formula $f\in L^{2}(\Omega,\Sigma,\sigma)$ -\end_inset - - tal que, para -\begin_inset Formula $u\in L^{2}(\Omega,\Sigma,\sigma)$ -\end_inset - -, -\begin_inset Formula -\[ -Tu=\int_{\Omega}u\dif\mu=\int_{\Omega}uf\dif\sigma, -\] - -\end_inset - -pero esta igualdad se da para cuando -\begin_inset Formula $u=\chi_{A}$ -\end_inset - - para cualquier -\begin_inset Formula $A\in{\cal F}$ -\end_inset - - y por linealidad para cualquier función -\begin_inset Formula $\Sigma$ -\end_inset - --medible simple, y por el teorema de convergencia dominada también se da - para cualquier función -\begin_inset Formula $\Sigma$ -\end_inset - --medible no negativa en casi todo punto. - Además, para -\begin_inset Formula $A\in\Sigma$ -\end_inset - -, -\begin_inset Formula -\[ -\mu(A)=\int_{\Omega}\chi_{A}f\dif\sigma=\int_{A}f\dif\sigma, -\] - -\end_inset - -de modo que -\begin_inset Formula $f$ -\end_inset - - es -\begin_inset Formula $\Sigma$ -\end_inset - --medible y, haciendo -\begin_inset Formula $A=\{x\mid f(x)\leq0\}$ -\end_inset - - o -\begin_inset Formula $A=\{x\mid f(x)>1\}$ -\end_inset - -, vemos que -\begin_inset Formula $f(\omega)\in(0,1]$ -\end_inset - - para casi todo -\begin_inset Formula $\omega\in\Omega$ -\end_inset - -, de modo que -\begin_inset Formula $\frac{1}{g}$ -\end_inset - - es -\begin_inset Formula $\Sigma$ -\end_inset - --medible no negativa en casi todo punto y, en casi todo punto, -\begin_inset Formula $\frac{1}{f}f=1$ -\end_inset - -, con lo que para -\begin_inset Formula $A\in\Sigma$ -\end_inset - -, -\begin_inset Formula -\[ -\int_{A}\frac{1}{f}\dif\mu=\int_{A}\dif\sigma\implies\nu(A)=\sigma(A)-\mu(A)=\int_{A}\left(\frac{1}{f}-1\right)\dif\mu\eqqcolon\int_{A}g\dif\mu. -\] - -\end_inset - - -\end_layout - -\begin_layout Section -Problemas variacionales cuadráticos -\end_layout - -\begin_layout Standard - -\series bold -Teorema principal de los problemas variacionales cuadráticos: -\series default - Sean -\begin_inset Formula $H$ -\end_inset - - un -\begin_inset Formula $\mathbb{R}$ -\end_inset - --espacio de Hilbert, -\begin_inset Formula $B$ -\end_inset - - una -\begin_inset Formula $H$ -\end_inset - --forma bilineal simétrica, acotada y fuertemente positiva, -\begin_inset Formula $b$ -\end_inset - - una -\begin_inset Formula $H$ -\end_inset - --forma lineal continua y -\begin_inset Formula $F:H\to\mathbb{R}$ -\end_inset - - dada por -\begin_inset Formula -\[ -F(x)\coloneqq\frac{1}{2}B(x,x)-b(x), -\] - -\end_inset - -entonces: -\end_layout - -\begin_layout Enumerate -Para -\begin_inset Formula $w\in H$ -\end_inset - -, -\begin_inset Formula $F$ -\end_inset - - alcanza su mínimo en -\begin_inset Formula $w$ -\end_inset - - si y sólo si -\begin_inset Formula $\forall y\in H,B(w,y)=b(y)$ -\end_inset - -. -\end_layout - -\begin_deeper -\begin_layout Enumerate -\begin_inset Argument item:1 -status open - -\begin_layout Plain Layout -\begin_inset Formula $\implies]$ -\end_inset - - -\end_layout - -\end_inset - -Fijado -\begin_inset Formula $y\in H$ -\end_inset - -, para -\begin_inset Formula $t\in\mathbb{R}$ -\end_inset - - -\begin_inset Formula -\begin{align*} -F(w+ty) & =\frac{1}{2}B(w+ty,w+ty)-b(w+ty)=\\ - & =\frac{1}{2}(B(w,w)+2tB(w,y)+t^{2}B(y,y))-b(w)-tb(y)=\\ - & =F(w)+t(B(w,y)-b(y))+\frac{1}{2}t^{2}B(y,y), -\end{align*} - -\end_inset - -pero por hipótesis -\begin_inset Formula $F(w)\leq F(w+ty)$ -\end_inset - - para todo -\begin_inset Formula $t\in\mathbb{R}$ -\end_inset - -, luego -\begin_inset Formula $\varphi:\mathbb{R}\to\mathbb{R}$ -\end_inset - - dada por -\begin_inset Formula $\varphi(t)\coloneqq F(w+ty)$ -\end_inset - - tiene un mínimo en -\begin_inset Formula $t=0$ -\end_inset - - y -\begin_inset Formula $0=\varphi'(0)=B(w,y)-b(y)$ -\end_inset - -. -\end_layout - -\begin_layout Enumerate -\begin_inset Argument item:1 -status open - -\begin_layout Plain Layout -\begin_inset Formula $\impliedby]$ -\end_inset - - -\end_layout - -\end_inset - -Para -\begin_inset Formula $y\in H$ -\end_inset - - y -\begin_inset Formula $t\in\mathbb{R}$ -\end_inset - -, -\begin_inset Formula -\[ -F(w+ty)=F(w)+\cancel{t(B(w,y)-b(y))}^{=0}+\frac{1}{2}t^{2}B(y,y)\geq F(w). -\] - -\end_inset - - -\end_layout - -\end_deeper -\begin_layout Enumerate -Existe un único -\begin_inset Formula $w\in H$ -\end_inset - - en el que -\begin_inset Formula $F$ -\end_inset - - alcanza su mínimo. -\end_layout - -\begin_deeper -\begin_layout Standard -Como -\begin_inset Formula $B$ -\end_inset - - es bilineal, simétrica y fuertemente positiva, es un producto escalar sobre - -\begin_inset Formula $H$ -\end_inset - -, y como existen -\begin_inset Formula $c,M>0$ -\end_inset - - con -\begin_inset Formula $c\Vert x\Vert^{2}\leq B(x,x)\leq M\Vert x\Vert^{2}$ -\end_inset - -, el producto escalar -\begin_inset Formula $B$ -\end_inset - - es equivalente al de -\begin_inset Formula $H$ -\end_inset - -, luego -\begin_inset Formula $b$ -\end_inset - - es continua con el producto escalar -\begin_inset Formula $B$ -\end_inset - - y por el teorema de Riesz-Fréchet existe un único -\begin_inset Formula $w\in H$ -\end_inset - - con -\begin_inset Formula $b=B(\cdot,w)=B(w,\cdot)$ -\end_inset - -, que es la condición del primer apartado. -\end_layout - -\end_deeper -\begin_layout Section -Convolución y aproximación de funciones -\end_layout - -\begin_layout Standard -Dado un abierto -\begin_inset Formula $\Omega\subseteq\mathbb{R}^{n}$ -\end_inset - -, -\begin_inset Formula $f:\mathbb{R}^{n}\to\mathbb{R}$ -\end_inset - - es -\series bold -localmente integrable -\series default - si -\begin_inset Formula $|f|$ -\end_inset - - es integrable en todo compacto -\begin_inset Formula $K\subseteq\Omega$ -\end_inset - -. - Dadas dos funciones localmente integrables -\begin_inset Formula $f,g:\mathbb{R}^{n}\to\mathbb{R}$ -\end_inset - -, definimos su -\series bold -producto de convolución -\series default - como -\begin_inset Formula $(f*g):D\to\mathbb{R}$ -\end_inset - - dada por -\begin_inset Formula -\[ -(f*g)(a)\coloneqq\int_{\mathbb{R}^{n}}f(x)g(a-x)\dif x, -\] - -\end_inset - -donde -\begin_inset Formula $D\coloneqq\{a\in\mathbb{R}^{n}\mid x\mapsto f(x)g(a-x)\text{ integrable}\}$ -\end_inset - -. - Si -\begin_inset Formula $f,g\in L^{2}(\mathbb{R}^{n})$ -\end_inset - -, -\begin_inset Formula $f*g$ -\end_inset - - está definida en todo -\begin_inset Formula $\mathbb{R}^{n}$ -\end_inset - - y es continua y uniformemente acotada con -\begin_inset Formula -\[ -\Vert f*g\Vert_{\infty}\leq\Vert f\Vert_{2}\Vert g\Vert_{2}. -\] - -\end_inset - - -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - -El producto de convolución es conmutativo, y si -\begin_inset Formula $f*g$ -\end_inset - - está definida en casi todo punto, -\begin_inset Formula $\text{sop}(f*g)\subseteq\overline{\text{sop}(f)+\text{sop}(g)}$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Una -\series bold -sucesión de Dirac -\series default - es una sucesión -\begin_inset Formula $(K_{m}:\mathbb{R}^{n}\to\mathbb{R}^{\geq0})_{m}$ -\end_inset - - de funciones continuas con -\begin_inset Formula -\[ -\int_{\mathbb{R}^{n}}K_{n}=1 -\] - -\end_inset - -y tal que -\begin_inset Formula -\[ -\forall\varepsilon,\delta>0,\exists n_{0}:\forall n\geq n_{0},\int_{\mathbb{R}^{n}\setminus B(0,\delta)}K_{n}(x)\dif x<\varepsilon. -\] - -\end_inset - -Por ejemplo, si -\begin_inset Formula $K:\mathbb{R}^{n}\to\mathbb{R}$ -\end_inset - - es continua, no negativa, con soporte compacto e integral 1, entonces -\begin_inset Formula $(x\mapsto m^{n}K(mx))_{m\geq1}$ -\end_inset - - es una sucesión de Dirac. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Las sucesiones de Dirac aproximan la -\series bold -delta de Dirac -\series default -, una -\begin_inset Quotes cld -\end_inset - -función extendida -\begin_inset Quotes crd -\end_inset - - con integral 1 que vale 0 en todo punto salvo en el origen en que el valor - es infinito. -\end_layout - -\begin_layout Standard -Como -\series bold -teorema -\series default -, si -\begin_inset Formula $f:\mathbb{R}^{n}\to\mathbb{R}$ -\end_inset - - es continua y acotada, la sucesión -\begin_inset Formula $(f*K_{m})_{m}$ -\end_inset - - tiende uniformemente a -\begin_inset Formula $f$ -\end_inset - - sobre subconjuntos compactos de -\begin_inset Formula $\mathbb{R}^{n}$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Si -\begin_inset Formula $f:\mathbb{R}^{n}\to\mathbb{R}$ -\end_inset - - es localmente integrable y -\begin_inset Formula $g\in{\cal D}^{k}(\mathbb{R}^{n})$ -\end_inset - -, -\begin_inset Formula $f*g\in{\cal C}^{k}(\mathbb{R}^{n})$ -\end_inset - - y para -\begin_inset Formula $\alpha\in\mathbb{N}^{n}$ -\end_inset - - con -\begin_inset Formula $\sum_{i}\alpha_{i}\leq k$ -\end_inset - - es -\begin_inset Formula -\[ -\frac{\partial^{|\alpha|}(f*g)}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{n}}}=f*\left(\frac{\partial^{|\alpha|}g}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{n}}}\right), -\] - -\end_inset - -con lo que -\begin_inset Formula $f*g$ -\end_inset - - es una regularización de -\begin_inset Formula $f$ -\end_inset - - a través de una función suave -\begin_inset Formula $g$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Como -\series bold -teorema -\series default -, dado un abierto -\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ -\end_inset - -, -\begin_inset Formula ${\cal D}(G)$ -\end_inset - - es denso en -\begin_inset Formula $(C_{c}(G),\Vert\cdot\Vert_{\infty})$ -\end_inset - - y en -\begin_inset Formula $L^{p}(G)$ -\end_inset - - para todo -\begin_inset Formula $p\in[1,\infty)$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Para -\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ -\end_inset - - abierto y -\begin_inset Formula $f\in L^{2}(G)$ -\end_inset - -, si para todo -\begin_inset Formula $\psi\in{\cal D}(G)$ -\end_inset - - es -\begin_inset Formula -\[ -\int_{G}f\psi=0 -\] - -\end_inset - -entonces -\begin_inset Formula $f=0$ -\end_inset - - en casi todo punto, y en particular, si -\begin_inset Formula $f$ -\end_inset - - es continua, -\begin_inset Formula $f=0$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Section -Principio de Dirichlet -\end_layout - -\begin_layout Standard -Dado un abierto -\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ -\end_inset - -, -\begin_inset Formula $u\in{\cal D}^{2}(G)$ -\end_inset - - es -\series bold -armónica -\series default - en -\begin_inset Formula $G$ -\end_inset - - si -\begin_inset Formula $\triangle u\coloneqq\nabla^{2}u=0$ -\end_inset - - en todo punto de -\begin_inset Formula $G$ -\end_inset - -. - Dada -\begin_inset Formula $g\in{\cal C}(S_{\mathbb{C}})$ -\end_inset - -, el -\series bold -problema de Dirichlet -\series default - consiste en encontrar -\begin_inset Formula $u\in{\cal D}^{2}(\overline{B_{X}})$ -\end_inset - - armónica con -\begin_inset Formula $u|_{S_{\mathbb{C}}}=g$ -\end_inset - -. - Para un abierto -\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ -\end_inset - -, llamamos -\begin_inset Formula ${\cal C}^{m}(\overline{G})$ -\end_inset - - al conjunto de funciones -\begin_inset Formula $u:\overline{G}\to\mathbb{R}$ -\end_inset - - con -\begin_inset Formula $u|_{G}\in{\cal C}^{m}(G)$ -\end_inset - - para las que las derivadas parciales de orden -\begin_inset Formula $m$ -\end_inset - - de -\begin_inset Formula $u$ -\end_inset - - en -\begin_inset Formula $G$ -\end_inset - - admiten prolongación continua a -\begin_inset Formula $\overline{G}$ -\end_inset - -. - Escribimos -\begin_inset Formula $\partial_{j}u\coloneqq\frac{\partial u}{\partial j}$ -\end_inset - -. -\end_layout - -\begin_layout Standard -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -begin{samepage} -\end_layout - -\end_inset - -Dados un abierto -\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ -\end_inset - - acotado y no vacío, -\begin_inset Formula $f:G\to\mathbb{R}$ -\end_inset - - y -\begin_inset Formula $g:\partial G\to\mathbb{R}$ -\end_inset - -, el -\series bold -problema de valores frontera para la ecuación de Poisson -\series default - consiste en encontrar -\begin_inset Formula $u:\overline{G}\to\mathbb{R}$ -\end_inset - - tal que -\begin_inset Formula $-\triangle u|_{G}=f$ -\end_inset - - y -\begin_inset Formula $u|_{\partial G}=g$ -\end_inset - -, y el -\series bold -problema generalizado de valores frontera -\series default - consiste en encontrar -\begin_inset Formula $u:\overline{G}\to\mathbb{R}$ -\end_inset - - con -\begin_inset Formula $u|_{\partial G}=g$ -\end_inset - - y -\begin_inset Formula -\[ -\forall v\in{\cal D}(G),\int_{G}\sum_{j=1}^{n}\frac{\partial u}{\partial x_{j}}\frac{\partial v}{\partial x_{j}}\dif x\int_{G}fv. -\] - -\end_inset - - -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{samepage} -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Si -\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ -\end_inset - - es un abierto acotado no vacío, -\begin_inset Formula $f\in{\cal C}(\overline{G})$ -\end_inset - - y -\begin_inset Formula $g\in{\cal C}(\partial G)$ -\end_inset - -: -\end_layout - -\begin_layout Enumerate -Una -\begin_inset Formula $w\in{\cal C}^{2}(\overline{G})$ -\end_inset - - es solución del problema de valores frontera para la ecuación de Poisson - y sólo si lo es del problema generalizado de valores frontera. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Enumerate -Si -\begin_inset Formula $w\in{\cal C}^{2}(\overline{G})$ -\end_inset - - es solución del problema variacional consistente en encontrar el mínimo - de -\begin_inset Formula $F:\{u\in{\cal C}^{2}(\overline{G})\mid u|_{\partial G}=g\}\to\mathbb{R}$ -\end_inset - - dada por -\begin_inset Formula -\[ -F(u)\coloneqq\frac{1}{2}\int_{G}\sum_{j=1}^{n}(\partial_{j}u(x))^{2}\dif x-\int_{G}fu, -\] - -\end_inset - -entonces es solución de los dos problemas anteriores. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -El -\series bold -teorema de integración por partes en varias variables -\series default - afirma que, si -\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ -\end_inset - - es un abierto, -\begin_inset Formula $u\in{\cal C}^{1}(G)$ -\end_inset - - y -\begin_inset Formula $v\in{\cal D}(G)$ -\end_inset - -, -\begin_inset Formula -\[ -\int_{G}u\partial_{j}v=-\int_{G}(\partial_{j}u)v. -\] - -\end_inset - - -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Si -\begin_inset Formula $G$ -\end_inset - - es un abierto de -\begin_inset Formula $\mathbb{R}^{n}$ -\end_inset - - y -\begin_inset Formula $u,w\in L^{2}(G)$ -\end_inset - -, -\begin_inset Formula $w$ -\end_inset - - es la -\series bold -derivada generalizada -\begin_inset Formula $j$ -\end_inset - --ésima -\series default - de -\begin_inset Formula $u$ -\end_inset - -, -\begin_inset Formula $w=\partial_{j}u$ -\end_inset - -, si -\begin_inset Formula -\[ -\forall v\in{\cal D}(G),\int_{G}u\partial_{j}v=-\int_{G}wv, -\] - -\end_inset - -y para -\begin_inset Formula $\alpha\in\mathbb{N}^{n}$ -\end_inset - - llamamos -\begin_inset Formula $D^{\alpha}u\coloneqq\partial_{1}^{\alpha_{1}}\cdots\partial_{n}^{\alpha_{n}}u$ -\end_inset - -. - -\end_layout - -\begin_layout Standard -Para -\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ -\end_inset - - abierto, -\begin_inset Formula $k\in\mathbb{N}$ -\end_inset - - y -\begin_inset Formula $p\in[1,\infty)$ -\end_inset - -, llamamos -\series bold -espacio de Sobolev -\series default - a -\begin_inset Formula -\[ -W^{k,p}(G)\coloneqq\{u\in L^{p}(G)\mid\forall\alpha\in\mathbb{N}^{n},(|\alpha|\leq k\implies\exists D^{\alpha}f\in L^{p}(G))\}. -\] - -\end_inset - -Escribimos -\begin_inset Formula $W^{k}(G)\coloneqq W^{k,2}(G)$ -\end_inset - -, y generalmente consideramos el espacio de Sobolev -\begin_inset Formula $W^{1}(G)$ -\end_inset - -. -\end_layout - -\begin_layout Standard -Si -\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ -\end_inset - - es abierto, definimos la relación de equivalencia en -\begin_inset Formula $G\to\mathbb{R}$ -\end_inset - - como -\begin_inset Formula $f\sim g\iff\{x\in G\mid f(x)\neq g(x)\}\text{ es de medida nula}$ -\end_inset - -, y -\begin_inset Formula $\langle\cdot,\cdot\rangle_{1,2}:W^{1}(G)/\sim\to\mathbb{R}$ -\end_inset - - dada por -\begin_inset Formula -\[ -\langle\overline{u},\overline{v}\rangle_{1,2}\coloneqq\int_{G}\left(uv+\sum_{j}(\partial_{j}u)(\partial_{j}v)\right) -\] - -\end_inset - -es un producto escalar en -\begin_inset Formula $W^{1}(G)/\sim$ -\end_inset - - que lo convierte en un espacio de Hilbert. - Identificamos -\begin_inset Formula $W^{1}(G)$ -\end_inset - - con -\begin_inset Formula $W^{1}(G)/\sim$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Llamamos -\begin_inset Formula $H_{0}^{1}(G)$ -\end_inset - - al espacio de Hilbert obtenido como la clausura de -\begin_inset Formula ${\cal D}(G)$ +\begin_inset Formula $\apprle$ \end_inset en -\begin_inset Formula $W^{1}(G)$ -\end_inset - -, que en general es un subespacio propio de -\begin_inset Formula $W^{1}(G)$ -\end_inset - - pero es igual a -\begin_inset Formula $W^{1}(G)$ -\end_inset - - si -\begin_inset Formula $G=\mathbb{R}^{n}$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Si -\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ -\end_inset - - es un abierto acotado no vacío y -\begin_inset Formula $u\in W^{1}(G)$ -\end_inset - -, -\series bold - -\begin_inset Formula $u$ -\end_inset - - se anula en la frontera de -\begin_inset Formula $G$ -\end_inset - - en sentido generalizado -\series default -, -\begin_inset Formula $u=0$ -\end_inset - - en -\begin_inset Formula $\partial G$ -\end_inset - -, si -\begin_inset Formula $u\in H_{0}^{1}(G)$ -\end_inset - -, y para -\begin_inset Formula $f,g\in W^{1}(G)$ -\end_inset - -, -\series bold - -\begin_inset Formula $f=g$ -\end_inset - - en -\begin_inset Formula $\partial G$ -\end_inset - - en sentido generalizado -\series default - si -\begin_inset Formula $f-g\in H_{0}^{1}(G)$ -\end_inset - -. -\end_layout - -\begin_layout Standard - -\series bold -Desigualdad de Poincaré-Friedrichs: -\series default - Si -\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ -\end_inset - - es un abierto acotado no vacío, existe -\begin_inset Formula $C>0$ -\end_inset - - tal que para -\begin_inset Formula $u\in H_{0}^{1}(G)$ -\end_inset - -, -\begin_inset Formula -\[ -C\int_{G}u^{2}\leq\int_{G}\sum_{j=1}^{n}(\partial_{j}u)^{2}. -\] - -\end_inset - - -\series bold -Demostración: -\series default - Sean -\begin_inset Formula $R\coloneqq\prod_{i}[a_{i},b_{i}]$ -\end_inset - - con -\begin_inset Formula $G\subseteq R$ -\end_inset - - y -\begin_inset Formula $u\in{\cal D}(G)$ -\end_inset - -, y vemos -\begin_inset Formula $u$ -\end_inset - - como una función en -\begin_inset Formula $R$ -\end_inset - - que se anula fuera de -\begin_inset Formula $G$ -\end_inset - - y con valor indefinido en -\begin_inset Formula $\partial G$ -\end_inset - -, para -\begin_inset Formula $x\in R$ -\end_inset - -, por la desigualdad de Cauchy-Schwartz, -\begin_inset Formula -\begin{align*} -(u(x))^{2} & =\left(\int_{a_{n}}^{x_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)\dif t\right)^{2}\leq\left(\int_{a_{n}}^{x_{n}}\dif t\right)\left(\int_{a_{n}}^{x_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)^{2}\dif t\right)\leq\\ - & \leq(b_{n}-a_{n})\int_{a_{n}}^{b_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)^{2}\dif t, -\end{align*} - -\end_inset - -luego -\begin_inset Formula -\begin{align*} -\int_{G}u^{2} & =\int_{R}u^{2}\leq\int_{a_{1}}^{b_{1}}\cdots\int_{a_{n}}^{b_{n}}(b_{n}-a_{n})\int_{a_{n}}^{b_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)^{2}\dif t\dif x_{n}\cdots\dif x_{1}=\\ - & =(b_{n}-a_{n})^{2}\int_{a_{1}}^{b_{1}}\cdots\int_{a_{n}}^{b_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)^{2}\dif t\dif x_{n-1}\cdots\dif x_{1}=\\ - & =(b_{n}-a_{n})^{2}\int_{R}(\partial_{n}u)^{2}\dif x\leq(b_{n}-a_{n})^{2}\int_{R}\sum_{j}(\partial_{j}u)^{2}\dif x=(b_{n}-a_{n})^{2}\int_{G}\sum_{j}(\partial_{j}u)^{2}\dif x. -\end{align*} - -\end_inset - -Para -\begin_inset Formula $u\in H_{0}^{1}(G)$ -\end_inset - -,existe una sucesión -\begin_inset Formula $\{u_{m}\}_{m}\subseteq{\cal D}(G)$ -\end_inset - - con -\begin_inset Formula $\lim_{m}\Vert u-u_{m}\Vert_{1,2}=0$ -\end_inset - - y por tanto -\begin_inset Formula $\lim_{m}\Vert u-u_{m}\Vert_{2}=\lim_{m}\Vert\partial_{j}u-\partial_{j}u_{m}\Vert_{2}=0$ -\end_inset - -, y tomando límites y usando que la norma -\begin_inset Formula $\Vert\cdot\Vert_{2}\leq\Vert\cdot\Vert_{1,2}$ -\end_inset - - y por tanto es continua en -\begin_inset Formula $W^{1}(G)$ -\end_inset - -, -\begin_inset Formula -\[ -C\int_{G}u^{2}-\int_{G}\sum_{j}(\partial_{j}u)^{2}=C\Vert u\Vert_{2}^{2}-\sum_{j}\Vert\partial_{j}u\Vert_{2}^{2}=\lim_{m}\left(C\Vert u_{m}\Vert_{2}^{2}-\sum_{j}\Vert\partial_{j}u_{m}\Vert_{2}^{2}\right)\leq0. -\] - -\end_inset - - -\end_layout - -\begin_layout Standard - -\series bold -Principio de Dirichlet: -\series default - Sean -\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ -\end_inset - - un abierto acotado no vacío, -\begin_inset Formula $f\in L^{2}(G)$ -\end_inset - - y -\begin_inset Formula $g\in W^{1}(G)$ -\end_inset - -, -\begin_inset Formula $F:\{u\in W^{1}(G)\mid u-g\in H_{0}^{1}(G)\}\to\mathbb{R}$ -\end_inset - - dada por -\begin_inset Formula -\[ -F(u)\coloneqq\frac{1}{2}\int_{G}\sum_{j=1}^{n}(\partial_{j}u)^{2}-\int_{G}fu -\] - -\end_inset - -alcanza su mínimo en un único punto, que es el único -\begin_inset Formula $u\in\text{Dom}f$ -\end_inset - - tal que -\begin_inset Formula -\[ -\forall v\in H_{0}^{1}(G),\int_{G}\sum_{j=1}^{n}(\partial_{j}u)(\partial_{j}v)=\int_{G}fv -\] - -\end_inset - -y la única solución en -\begin_inset Formula $\text{Dom}f$ -\end_inset - - del problema de valores frontera para la ecuación de Poisson -\begin_inset Formula $-\nabla^{2}u=f$ -\end_inset - -. -\end_layout - -\begin_layout Standard - -\series bold -Demostración: -\series default - Para -\begin_inset Formula $u,v\in W^{1}(G)$ -\end_inset - - definimos -\begin_inset Formula -\begin{align*} -B(u,v) & \coloneqq\int_{G}\sum_{j}(\partial_{j}u)(\partial_{j}v), & b_{0}(v) & \coloneqq\int_{G}fv, & b(v) & \coloneqq b_{0}(v)-B(v,g). -\end{align*} - -\end_inset - - -\begin_inset Formula $B$ -\end_inset - - es bilineal y simétrica, y es acotada porque -\begin_inset Formula -\[ -|B(u,v)|=\left|\sum_{j}\int_{G}(\partial_{j}u)(\partial_{j}v)\right|\leq\sum_{j}\left|\int_{G}(\partial_{j}u)(\partial_{j}v)\right|\leq\sum_{j}\Vert\partial_{j}u\Vert_{2}\Vert\partial_{j}v\Vert_{2}\leq n\Vert u\Vert_{1,2}\Vert v\Vert_{1,2}. -\] - -\end_inset - -Por la desigualdad de Poincaré-Friedrichs, existe -\begin_inset Formula $C>0$ -\end_inset - - tal que, para todo -\begin_inset Formula $v\in H$ -\end_inset - -, -\begin_inset Formula -\[ -C\int_{G}v^{2}\leq\int_{G}\sum_{j}(\partial_{j}v)^{2}, -\] - -\end_inset - -luego -\begin_inset Formula -\[ -C\Vert v\Vert_{1,2}^{2}=C\left(\int_{G}v^{2}+\sum_{j}(\partial_{j}v)^{2}\right)\leq(1+C)\int_{G}\sum_{j}(\partial_{j}v)^{2}=(1+C)B(v,v) -\] - -\end_inset - -y -\begin_inset Formula $B$ -\end_inset - - es fuertemente positiva. - Además, -\begin_inset Formula $b_{0}$ -\end_inset - - es lineal y es acotada por la desigualdad de Cauchy-Schwartz, y como además - -\begin_inset Formula $B$ -\end_inset - - es bilineal y acotada, -\begin_inset Formula $b_{0}$ -\end_inset - - es lineal acotada y se dan las condiciones del teorema principal de los - problemas variacionales cuadráticos. - Ahora bien, si -\begin_inset Formula $w\coloneqq u-g\in H_{0}^{1}(G)$ -\end_inset - -, -\begin_inset Formula -\begin{multline*} -\frac{1}{2}B(w,w)-b(w)=\frac{1}{2}\int_{G}\sum_{j}(\partial_{j}(u-g))^{2}-\int_{G}f(u-g)+\int_{G}\sum_{j}(\partial_{j}(u-g))(\partial_{j}(g))=\\ -=\frac{1}{2}\int_{G}\sum_{j}(\partial_{j}(u-g))(\partial_{j}(u+g))-\int_{G}f(u-g)=\\ -=\frac{1}{2}\int_{G}\sum_{j}(\partial_{j}u)^{2}-\int_{G}fu+\frac{1}{2}\int_{G}\sum_{j}(\partial_{j}g)^{2}+\int_{G}fg, -\end{multline*} - -\end_inset - -luego minimizar -\begin_inset Formula $F$ -\end_inset - - equivale a minimizar -\begin_inset Formula $\frac{1}{2}B(w,w)-b(w)$ -\end_inset - -, y además -\begin_inset Formula -\begin{multline*} -B(w,v)=b(v)\iff B(u,v)-B(g,v)=b_{0}(v)-B(v,g)\iff B(u,v)=b_{0}(v)\iff\\ -\iff\int_{G}\sum_{j}(\partial_{j}u)(\partial_{j}v)=\int_{G}fv. -\end{multline*} - -\end_inset - -Para la última parte, si -\begin_inset Formula $u_{0}$ -\end_inset - - cumple esta última fórmula para todo -\begin_inset Formula $v\in H_{0}^{1}(G)$ -\end_inset - -, por integración por partes, -\begin_inset Formula -\[ -0=\int_{G}\sum_{j}(\partial_{j}u_{0})(\partial_{j}v)-\int_{G}fv=-\int_{G}\sum_{j}(\partial_{j}\partial_{j}u_{0})v-\int_{G}fv=-\int_{G}(\nabla^{2}u_{0}+f)v, -\] - -\end_inset - -con lo que -\begin_inset Formula $(\nabla^{2}u_{0}+f)\bot H_{0}^{1}(G)$ -\end_inset - - y, como -\begin_inset Formula ${\cal D}(G)\subseteq H_{0}^{1}(G)$ -\end_inset - - es denso en -\begin_inset Formula $L^{2}(G)$ -\end_inset - -, -\begin_inset Formula $\nabla^{2}u_{0}+f=0$ -\end_inset - -. -\end_layout - -\begin_layout Section -Soluciones débiles -\end_layout - -\begin_layout Standard -Dados -\begin_inset Formula $k,n\in\mathbb{N}$ -\end_inset - - y -\begin_inset Formula $a_{\alpha}\in\mathbb{K}^{n}$ -\end_inset - - para cada -\begin_inset Formula $\alpha\in\mathbb{N}^{n}$ -\end_inset - - con -\begin_inset Formula $|\alpha|<k$ -\end_inset - -, un -\series bold -operador diferencial lineal de coeficientes constantes -\series default - es uno de la forma -\begin_inset Formula -\[ -L\coloneqq\sum_{|\alpha|\leq k}a_{\alpha}\left(\frac{\partial}{\partial x}\right)^{\alpha}\coloneqq\sum_{|\alpha|\leq k}a_{\alpha}\frac{\partial^{|\alpha|}}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{n}}}, -\] - -\end_inset - -y su -\series bold -operador adjunto -\series default - es -\begin_inset Formula -\[ -L^{*}\coloneqq\sum_{|\alpha|\leq k}(-1)^{|\alpha|}\overline{a_{\alpha}}\left(\frac{\partial}{\partial x}\right)^{\alpha}. -\] - -\end_inset - -Si -\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ -\end_inset - - es abierto, -\begin_inset Formula $\varphi,\psi\in L^{2}(G)$ -\end_inset - - son de clase -\begin_inset Formula ${\cal C}^{k}$ -\end_inset - - y una de las dos tiene soporte compacto, entonces -\begin_inset Formula $\langle L\psi,\varphi\rangle=\langle\psi,L^{*}\varphi\rangle$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Así, si -\begin_inset Formula $G$ -\end_inset - - es un abierto en -\begin_inset Formula $\mathbb{R}^{n}$ -\end_inset - -, -\begin_inset Formula $f,u\in L^{2}(G)$ -\end_inset - - son de clase -\begin_inset Formula ${\cal C}^{k}$ -\end_inset - - y -\begin_inset Formula $Lu=f$ -\end_inset - -, entonces -\begin_inset Formula $\langle f,\psi\rangle=\langle u,L^{*}\psi\rangle$ -\end_inset - - para todo -\begin_inset Formula $\psi\in{\cal D}(G)$ -\end_inset - -. - Para -\begin_inset Formula $f\in L^{2}(G)$ -\end_inset - -, -\begin_inset Formula $u\in L^{2}(G)$ -\end_inset - - es -\series bold -solución débil -\series default - de la ecuación en derivadas parciales -\begin_inset Formula $Lu=f$ -\end_inset - - si para todo -\begin_inset Formula $\psi\in{\cal D}(G)$ -\end_inset - - es -\begin_inset Formula $\langle f,\psi\rangle=\langle u,L^{*}\psi\rangle$ -\end_inset - -. -\end_layout - -\begin_layout Standard -Si -\begin_inset Formula $L=\od{}{x}$ -\end_inset - - y -\begin_inset Formula $u,f\in L^{2}((0,1))$ -\end_inset - -, -\begin_inset Formula $Lu=f$ -\end_inset - - en sentido débil si y sólo si existe -\begin_inset Formula $F:(0,1)\to\mathbb{R}$ -\end_inset - - absolutamente continua con -\begin_inset Formula $F=u$ -\end_inset - - y -\begin_inset Formula $F'=f$ -\end_inset - - para casi todo -\begin_inset Formula $x\in(0,1)$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -La ecuación de ondas en una dimensión, -\begin_inset Formula -\[ -\left\{ \begin{array}{rlrl} -\frac{\partial^{2}u}{\partial x^{2}}-\frac{\partial^{2}u}{\partial t^{2}} & =0, & t & \in[0,+\infty),\\ -u(x,0) & \equiv f(x), & x & \in[0,\pi],\\ -\frac{\partial u}{\partial t}(x,0) & \equiv0, -\end{array}\right. -\] - -\end_inset - -siendo -\begin_inset Formula $f:[0,\pi]\to\mathbb{R}$ -\end_inset - - una función lineal a trozos, admite soluciones débiles que no son soluciones - ordinarias. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard - -\series bold -Teorema de Malgrange-Ehrenpreis: -\series default - Sean -\begin_inset Formula $G$ -\end_inset - - un abierto acotado de -\begin_inset Formula $\mathbb{R}^{n}$ -\end_inset - - y -\begin_inset Formula $L$ -\end_inset - - un operador en derivadas parciales lineal con coeficientes constantes, - existe un operador lineal continuo -\begin_inset Formula $K:L^{2}(G)\to L^{2}(G)$ -\end_inset - - tal que para todo -\begin_inset Formula $f\in L^{2}(G)$ -\end_inset - -, -\begin_inset Formula $u\coloneqq K(f)$ -\end_inset - - es solución débil de -\begin_inset Formula $Lu=f$ -\end_inset - -. -\end_layout - -\begin_layout Standard - -\series bold -Demostración: -\series default - Definimos -\begin_inset Formula $\langle\varphi,\psi\rangle_{L}\coloneqq\langle L^{*}\varphi,L^{*}\psi\rangle_{2}$ -\end_inset - -, y para ver que es un producto escalar sobre -\begin_inset Formula ${\cal D}(G)$ -\end_inset - - vemos que existe -\begin_inset Formula $C>0$ -\end_inset - - tal que, para -\begin_inset Formula $\psi\in{\cal D}(G)$ -\end_inset - -, -\begin_inset Formula $\Vert\psi\Vert_{2}\leq C\Vert L^{*}\psi\Vert_{2}$ -\end_inset - -. - Si -\begin_inset Formula $L^{*}=\frac{\partial}{\partial x_{1}}$ -\end_inset - -, llamando -\begin_inset Formula $\psi(x)\coloneqq0$ -\end_inset - - para -\begin_inset Formula $x\notin G$ -\end_inset - -, para -\begin_inset Formula $x\in G$ -\end_inset - -, como -\begin_inset Formula $\text{sop}\psi\subseteq G$ -\end_inset - - es compacto, sea -\begin_inset Formula $m\coloneqq\inf_{x\in G}x_{1}$ -\end_inset - -, -\begin_inset Formula -\begin{align*} -\psi(x)^{2} & =\left(\int_{m}^{x_{1}}\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\dif t\right)^{2}\leq\left(\int_{m}^{x_{1}}\left|\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\right|\cdot1\dif t\right)\leq\\ - & \leq\int_{m}^{x_{1}}\dif t\int_{m}^{x_{1}}\left|\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\right|^{2}\dif t\leq d\int_{m}^{x_{1}}\left|\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\right|^{2}, -\end{align*} - -\end_inset - -donde -\begin_inset Formula $d$ -\end_inset - - es el diámetro de -\begin_inset Formula $G$ -\end_inset - -, e integrando de nuevo, -\begin_inset Formula -\begin{align*} -\Vert\psi\Vert_{2}^{2} & =\int_{G}\psi(x)^{2}\dif x\leq d\int_{m}^{x_{1}}\int_{-\infty}^{x_{2}}\cdots\int_{-\infty}^{x_{n}}\int_{m}^{x_{1}}\left|\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\right|^{2}\dif t\dif x_{n}\cdots\dif x_{1}\leq\\ - & \leq d^{2}\int_{G}\left|\frac{\partial\psi}{\partial x_{1}}(x)\right|^{2}\dif x=d^{2}\Vert L^{*}\psi\Vert_{2}^{2}. -\end{align*} - -\end_inset - -Si -\begin_inset Formula $L^{*}=\frac{\partial}{\partial x_{i}}$ -\end_inset - - para otro -\begin_inset Formula $i$ -\end_inset - -, es análogo, y si -\begin_inset Formula $L^{*}=\left(\frac{\partial}{\partial x}\right)^{|\alpha|}$ -\end_inset - -, por inducción, -\begin_inset Formula $\Vert\psi\Vert_{2}\leq d^{|\alpha|}\Vert L^{*}\psi\Vert_{2}$ -\end_inset - -. - Para -\begin_inset Formula $L$ -\end_inset - - arbitrario basta hacer combinaciones lineales. - Visto esto, sean -\begin_inset Formula $H_{0}\coloneqq({\cal D}(G),\langle\cdot,\cdot\rangle_{L})$ -\end_inset - - y -\begin_inset Formula $H$ -\end_inset - - su compleción, -\begin_inset Formula $L^{*}:H_{0}\to L^{2}(G)$ -\end_inset - - es lineal y continuo y por tanto admite una extensión lineal y continua - -\begin_inset Formula $\hat{L}^{*}:H\to L^{2}(G)$ -\end_inset - -. - Sea ahora -\begin_inset Formula $f\in L^{2}(G)$ -\end_inset - - y -\begin_inset Formula $l_{0}:H_{0}\to\mathbb{K}$ -\end_inset - - dada por -\begin_inset Formula $l_{0}(\psi)\coloneqq\langle\psi,f\rangle_{2}$ -\end_inset - -, -\begin_inset Formula -\[ -|l_{0}(\psi)|=|\langle\psi,f\rangle_{2}|\leq\Vert\psi\Vert_{2}\Vert f\Vert_{2}\leq C\Vert f\Vert_{2}\Vert L^{*}\psi\Vert_{2}, -\] - -\end_inset - -donde -\begin_inset Formula $C$ -\end_inset - - es tal que -\begin_inset Formula $\Vert\psi\Vert_{2}\leq C\Vert L^{*}\psi\Vert_{2}$ -\end_inset - - para todo \begin_inset Formula $C$ \end_inset -, de modo que -\begin_inset Formula $l_{0}$ -\end_inset - - es lineal continua por la cota -\begin_inset Formula $C\Vert f\Vert_{2}$ -\end_inset - - y se puede extender a una forma lineal y continua -\begin_inset Formula $l:H\to\mathbb{K}$ -\end_inset - - con -\begin_inset Formula $\Vert l\Vert\leq C\Vert f\Vert_{2}$ -\end_inset - -. - Por el teorema de Riesz, existe un único -\begin_inset Formula $\hat{u}\in H$ -\end_inset - - con -\begin_inset Formula $l(h)\equiv\langle h,\hat{u}\rangle_{L}$ -\end_inset - - para -\begin_inset Formula $h\in H$ -\end_inset - - y además -\begin_inset Formula $\Vert\hat{u}\Vert_{H}=\Vert l\Vert_{H}$ -\end_inset - -, y tomando -\begin_inset Formula $u\coloneqq\hat{L}^{*}\hat{u}$ -\end_inset - -, -\begin_inset Formula $l(h)=\langle\hat{L}^{*}h,\hat{L}^{*}\hat{u}\rangle=\langle\hat{L}^{*}h,u\rangle_{2}$ -\end_inset - -, pero para -\begin_inset Formula $\psi\in{\cal D}(G)$ -\end_inset - -, -\begin_inset Formula $l(\psi)=\langle\psi,f\rangle_{2}$ -\end_inset - - y -\begin_inset Formula $\hat{L}^{*}(\psi)=L^{*}\psi$ -\end_inset - -, con lo que -\begin_inset Formula $\langle L^{*}\psi,u\rangle_{2}=l(\psi)=\langle\psi,f\rangle_{2}$ -\end_inset - -, y basta llamar -\begin_inset Formula $K(f)\coloneqq u$ -\end_inset - -. - Para la continuidad de -\begin_inset Formula $K$ -\end_inset - -, -\begin_inset Formula -\[ -\Vert K(f)\Vert_{2}=\Vert u\Vert_{2}=\Vert\hat{L}^{*}\hat{u}\Vert_{2}=\Vert\hat{u}\Vert_{H}=\Vert l\Vert_{H}=\sup_{\Vert\psi\Vert_{H}=\Vert L^{*}\psi\Vert_{2}=1}|l(\psi)|\leq C\Vert f\Vert_{2}. -\] - -\end_inset - - -\end_layout - -\begin_layout Section -Método de Galerkin -\end_layout - -\begin_layout Standard -Sean -\begin_inset Formula $M_{1}\subseteq M_{2}\subseteq\dots\subseteq M_{n}\subseteq\dots$ -\end_inset - - una sucesión de subespacios cerrados de un espacio de Hilbert -\begin_inset Formula $H$ -\end_inset - - con unión densa en -\begin_inset Formula $H$ -\end_inset - -, -\begin_inset Formula $a:H\times H\to\mathbb{R}$ -\end_inset - - bilineal, simétrica, continua y fuertemente positiva, -\begin_inset Formula $b:H\to\mathbb{R}$ -\end_inset - - lineal continua, -\begin_inset Formula -\[ -J(x)\coloneqq\frac{1}{2}a(x,x)-b(x) -\] - -\end_inset - -para -\begin_inset Formula $x\in H$ -\end_inset - -, -\begin_inset Formula $u\in H$ -\end_inset - - con -\begin_inset Formula $J(u)$ -\end_inset - - mínimo y, para -\begin_inset Formula $n\in\mathbb{N}$ -\end_inset - -, -\begin_inset Formula $u_{n}\in M_{n}$ -\end_inset - - con -\begin_inset Formula $J(u_{n})$ -\end_inset - - mínimo, de modo que -\begin_inset Formula $a(x,u_{n})=b(x)$ -\end_inset - - para todo -\begin_inset Formula $x\in M_{n}$ -\end_inset - - y -\begin_inset Formula $a(x,u)=b(x)$ -\end_inset - - para todo -\begin_inset Formula $x\in H$ -\end_inset - -: -\end_layout - -\begin_layout Enumerate - -\series bold -Teorema de Galerkin-Ritz: -\series default - -\begin_inset Formula $\lim_{n}u_{n}=u$ -\end_inset - -. -\end_layout - -\begin_deeper -\begin_layout Standard -Para -\begin_inset Formula $x\in M_{n}$ -\end_inset - -, -\begin_inset Formula $a(x,u_{n})=b(x)$ -\end_inset - -, y para -\begin_inset Formula $x\in H$ -\end_inset - -, -\begin_inset Formula $a(x,u)=f(x)$ -\end_inset - -, luego -\begin_inset Formula $a(x,u-u_{n})=b(x)-b(x)=0$ -\end_inset - - para -\begin_inset Formula $x\in M_{n}$ -\end_inset - -. - Pero -\begin_inset Formula $a$ -\end_inset - - es un producto escalar equivalente al de -\begin_inset Formula $H$ -\end_inset - -, luego -\begin_inset Formula $u-u_{n}\bot M_{n}$ -\end_inset - - y, si -\begin_inset Formula $P_{n}:H\to M_{n}$ -\end_inset - - es la proyección ortogonal, -\begin_inset Formula $P_{n}(u)=u_{n}$ -\end_inset - -. - Por el teorema de la proyección, -\begin_inset Formula $\Vert u-u_{n}\Vert=\Vert u-P_{n}(u)\Vert=d(u,M_{n})$ -\end_inset - -, pero por la densidad es -\begin_inset Formula $d(u,\bigcup_{n}M_{n})=0$ -\end_inset - -, y para -\begin_inset Formula $\varepsilon>0$ -\end_inset - - existen -\begin_inset Formula $n_{0}\in\mathbb{N}$ -\end_inset - - e -\begin_inset Formula $y\in M_{n_{0}}$ -\end_inset - - con -\begin_inset Formula $\Vert u-y\Vert<\varepsilon$ -\end_inset - -, y como la sucesión es creciente, para -\begin_inset Formula $n\geq n_{0}$ -\end_inset - -, -\begin_inset Formula $\Vert u-u_{n}\Vert=d(u,M_{n})\leq d(u,M_{n_{0}})\leq\Vert u-y\Vert<\varepsilon$ -\end_inset - -, con lo que -\begin_inset Formula $\lim_{n}u_{n}=u$ -\end_inset - -. -\end_layout - -\end_deeper -\begin_layout Enumerate -Dados -\begin_inset Formula $c,d>0$ -\end_inset - - con -\begin_inset Formula $a(x,y)\leq d\Vert x\Vert\Vert y\Vert$ -\end_inset - - y -\begin_inset Formula $c\Vert x\Vert^{2}\leq a(x,x)$ -\end_inset - - para todo -\begin_inset Formula $x,y\in H$ -\end_inset - -, -\begin_inset Formula $c\Vert u\Vert\leq\Vert b\Vert$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Enumerate - -\series bold -Razón de convergencia: -\series default - -\begin_inset Formula $\Vert u-u_{n}\Vert\leq\frac{d}{c}d(u,M_{n})$ -\end_inset - -. -\end_layout - -\begin_layout Enumerate - -\series bold -Estimación del error: -\series default - Si -\begin_inset Formula $\beta\leq J(x)$ -\end_inset - - para todo -\begin_inset Formula $x\in H$ -\end_inset - -, para -\begin_inset Formula $n\in\mathbb{N}$ -\end_inset - - es -\begin_inset Formula $\frac{c}{2}\Vert u-u_{n}\Vert^{2}\leq J(u_{n})-\beta$ -\end_inset - -. -\end_layout - -\begin_layout Standard -El -\series bold -método de Galerkin -\series default - para resolver un problema de esta forma consiste en tomar en el teorema - anterior los -\begin_inset Formula $M_{n}$ -\end_inset - - de dimensión finita y resolver los sistemas de ecuaciones lineales resultantes, - con matriz de coeficientes simétrica y definida positiva de tamaño -\begin_inset Formula $\dim M_{n}$ -\end_inset - -. - Tomando adecuadamente las bases de los -\begin_inset Formula $M_{n}$ -\end_inset - - se puede conseguir que las matrices tengan muchas entradas nulas. -\end_layout - -\begin_layout Section -Redes +, una relación reflexiva y transitiva. \end_layout \begin_layout Standard @@ -9467,19 +4517,11 @@ Un \series bold conjunto dirigido \series default - es un par -\begin_inset Formula $(D,\geq)$ + es un conjunto preordenado +\begin_inset Formula $(D,\leq)$ \end_inset - formado por un conjunto -\begin_inset Formula $D$ -\end_inset - - y una relación -\begin_inset Formula $\geq$ -\end_inset - - transitiva reflexiva y tal que + tal que \begin_inset Formula $\forall i,j\in D,\exists k\in D:k\geq i,j$ \end_inset @@ -9497,7 +4539,7 @@ red \end_inset donde -\begin_inset Formula $(D,\geq)$ +\begin_inset Formula $(D,\leq)$ \end_inset es un conjunto dirigido, que escribimos como @@ -9510,7 +4552,7 @@ red . Todo conjunto totalmente ordenado es dirigido, y en particular -\begin_inset Formula $(\mathbb{N},\geq)$ +\begin_inset Formula $(\mathbb{N},\leq)$ \end_inset lo es y así las sucesiones son redes. @@ -9956,7 +4998,7 @@ Si \end_inset , que es un conjunto dirigido por -\begin_inset Formula $\supseteq$ +\begin_inset Formula $\subseteq$ \end_inset . @@ -10349,1775 +5391,5 @@ nproof \end_layout -\begin_layout Section -Bases hilbertiana -\end_layout - -\begin_layout Standard -Sean -\begin_inset Formula $(H_{i})_{i\in I}$ -\end_inset - - una familia de -\begin_inset Formula $\mathbb{K}$ -\end_inset - --espacios de Hilbert, -\begin_inset Formula $H_{0}\coloneqq\prod_{i\in I}H_{i}$ -\end_inset - - y -\begin_inset Formula $\langle\cdot,\cdot\rangle:H_{0}\times H_{0}\to[0,+\infty]$ -\end_inset - - dada por -\begin_inset Formula -\[ -\langle x,y\rangle\coloneqq\sum_{i\in I}\langle x_{i},y_{i}\rangle_{H_{i}}, -\] - -\end_inset - -llamamos -\series bold -suma directa hilbertiana -\series default - o -\series bold -suma -\begin_inset Formula $\ell^{2}$ -\end_inset - - -\series default - de -\begin_inset Formula $\{H_{i}\}_{i\in I}$ -\end_inset - - al espacio de Hilbert -\begin_inset Formula -\[ -\bigoplus_{i\in I}H_{i}\coloneqq\ell^{2}((H_{i})_{i\in I})\coloneqq(\{x\in H_{0}\mid\langle x,x\rangle<\infty\},\langle\cdot,\cdot\rangle). -\] - -\end_inset - - -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Cada -\begin_inset Formula $H_{i}$ -\end_inset - - es isométricamente isomorfo al subespacio de -\begin_inset Formula $H$ -\end_inset - - de los vectores con todas las coordenadas nulas salvo la -\begin_inset Formula $i$ -\end_inset - -, los -\begin_inset Formula $H_{i}$ -\end_inset - - son mutuamente ortogonales en -\begin_inset Formula $H$ -\end_inset - -, -\begin_inset Formula $H$ -\end_inset - - es la clausura lineal cerrada de los -\begin_inset Formula $H_{i}$ -\end_inset - - y cada -\begin_inset Formula $x\in H$ -\end_inset - - se puede expresar de forma única como -\begin_inset Formula $\sum_{i\in I}x_{i}$ -\end_inset - - con cada -\begin_inset Formula $x_{i}\in H_{i}$ -\end_inset - -. -\end_layout - -\begin_layout Standard -Si -\begin_inset Formula $H$ -\end_inset - - es un -\begin_inset Formula $\mathbb{K}$ -\end_inset - --espacio de Hilbert y -\begin_inset Formula $(H_{i})_{i\in I}$ -\end_inset - - es una familia de subespacios cerrados de -\begin_inset Formula $H$ -\end_inset - - mutuamente ortogonales con -\begin_inset Formula $H=\overline{\text{span}\{H_{i}\}_{i\in I}}$ -\end_inset - -, entonces -\begin_inset Formula $H$ -\end_inset - - es isométricamente isomorfo a -\begin_inset Formula $\bigoplus_{i\in I}H_{i}$ -\end_inset - -, e identificamos -\begin_inset Formula $H$ -\end_inset - - con -\begin_inset Formula $\bigoplus_{i\in I}H_{i}$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard - -\series bold -Desigualdad de Bessel: -\series default - Sean -\begin_inset Formula $H$ -\end_inset - - un espacio prehilbertiano y -\begin_inset Formula $\{e_{i}\}_{i\in I}\subseteq H$ -\end_inset - - una familia ortonormal, para -\begin_inset Formula $x\in H$ -\end_inset - -, -\begin_inset Formula -\[ -\sum_{i\in I}|\langle x,e_{i}\rangle|^{2}\leq\Vert x\Vert^{2}. -\] - -\end_inset - - -\end_layout - -\begin_layout Standard -Para un conjunto -\begin_inset Formula $I$ -\end_inset - - arbitrario, llamamos -\begin_inset Formula $\ell^{2}(I)\coloneqq\bigoplus_{i\in I}\mathbb{K}$ -\end_inset - -. -\end_layout - -\begin_layout Standard - -\series bold -Teorema de la base hilbertiana: -\series default - Sean -\begin_inset Formula $H$ -\end_inset - - un espacio de Hilbert y -\begin_inset Formula $\{e_{i}\}_{i\in I}\subseteq H$ -\end_inset - - una familia ortonormal, -\begin_inset Formula $\{e_{i}\}_{i\in I}$ -\end_inset - - es ortonormal maximal (por inclusión) si y sólo si -\begin_inset Formula $\forall x\in H,(\forall i\in I,\langle x,e_{i}\rangle=0\implies x=0)$ -\end_inset - -, si y sólo si es un conjunto total, si y sólo si -\begin_inset Formula $\hat{}:H\to\ell^{2}(I)$ -\end_inset - - dada por -\begin_inset Formula $\hat{x}\coloneqq(\langle x,e_{i}\rangle)_{i\in I}$ -\end_inset - - es inyectiva, si y sólo si todo -\begin_inset Formula $x\in H$ -\end_inset - - admite un -\series bold -desarrollo de Fourier -\series default - -\begin_inset Formula $x=\sum_{i\in I}\langle x,e_{i}\rangle e_{i}$ -\end_inset - -, si y sólo si -\begin_inset Formula $\forall x,y\in H,\langle x,y\rangle=\sum_{i\in I}\langle x,e_{i}\rangle\overline{\langle y,e_{i}\rangle}$ -\end_inset - -, si y sólo si todo -\begin_inset Formula $x\in H$ -\end_inset - - cumple la -\series bold -identidad de Parseval -\series default -, -\begin_inset Formula $\Vert x\Vert^{2}=\sum_{i\in I}|\langle x,e_{i}\rangle|^{2}$ -\end_inset - -, y entonces decimos que -\begin_inset Formula $(e_{i})_{i\in I}$ -\end_inset - - es una -\series bold -base hilbertiana -\series default - de -\begin_inset Formula $H$ -\end_inset - - o un -\series bold -sistema ortonormal completo -\series default -. -\end_layout - -\begin_layout Description -\begin_inset Formula $1\implies2]$ -\end_inset - - Entonces -\begin_inset Formula $x\bot\{e_{i}\}_{i\in I}$ -\end_inset - -, por lo que si -\begin_inset Formula $x\neq0$ -\end_inset - -, -\begin_inset Formula $\{e_{i}\}_{i\in I}\cup\{x\}$ -\end_inset - - sería ortogonal. -\begin_inset Formula $\#$ -\end_inset - - -\end_layout - -\begin_layout Description -\begin_inset Formula $2\iff3]$ -\end_inset - - Sabemos que un -\begin_inset Formula $S\subseteq H$ -\end_inset - - es total si y sólo si -\begin_inset Formula $S^{\bot}=0$ -\end_inset - -. -\end_layout - -\begin_layout Description -\begin_inset Formula $2\iff4]$ -\end_inset - - Por ser -\begin_inset Formula $\hat{}$ -\end_inset - - lineal. -\end_layout - -\begin_layout Description -\begin_inset Formula $4\implies5]$ -\end_inset - - -\begin_inset Formula $\widehat{\sum_{i}\langle x,e_{i}\rangle e_{i}}=\sum_{i}\langle x,e_{i}\rangle\hat{e}_{i}=\sum_{i}\langle x,e_{i}\rangle e_{i}=\hat{x}$ -\end_inset - -, y por inyectividad -\begin_inset Formula $x=\sum_{i\in I}\langle x,e_{i}\rangle e_{i}$ -\end_inset - -. -\end_layout - -\begin_layout Description -\begin_inset Formula $5\implies6]$ -\end_inset - - -\begin_inset Formula $\langle x,y\rangle=\sum_{i,j\in I}\langle\langle x,e_{i}\rangle e_{i},\langle y,e_{j}\rangle e_{j}\rangle=\sum_{i,j\in I}\langle x,e_{i}\rangle\overline{\langle y,e_{j}\rangle}\langle e_{i},e_{j}\rangle=\sum_{i\in I}\langle x,e_{i}\rangle\overline{\langle y,e_{j}\rangle}$ -\end_inset - -. -\end_layout - -\begin_layout Description -\begin_inset Formula $6\implies7]$ -\end_inset - - Basta tomar -\begin_inset Formula $x=y$ -\end_inset - -. -\end_layout - -\begin_layout Description -\begin_inset Formula $7\implies1]$ -\end_inset - - Si fuera -\begin_inset Formula $\{e_{i}\}_{i}\subsetneq M\subseteq H$ -\end_inset - - con -\begin_inset Formula $M$ -\end_inset - - ortonormal, para -\begin_inset Formula $x\in M\setminus\{e_{i}\}_{i}$ -\end_inset - -, -\begin_inset Formula $1=\Vert x\Vert^{2}=\sum_{i\in I}|\langle x,e_{i}\rangle|^{2}=0\#$ -\end_inset - -. -\end_layout - -\begin_layout Standard - -\series bold -Primer teorema de Riesz-Fischer: -\series default - Si -\begin_inset Formula $H$ -\end_inset - - es un espacio prehilbertiano con una familia ortonormal -\begin_inset Formula $\{e_{i}\}_{i\in I}$ -\end_inset - - y -\begin_inset Formula $\hat{}:H\to\mathbb{K}^{I}$ -\end_inset - - viene dada por -\begin_inset Formula $\hat{x}\coloneqq(\langle x,e_{i}\rangle)_{i\in I}$ -\end_inset - -, -\begin_inset Formula $\hat{}$ -\end_inset - - es lineal y continua con imagen contenida en -\begin_inset Formula $\ell^{2}(I)$ -\end_inset - - e igual a -\begin_inset Formula $\ell^{2}(I)$ -\end_inset - - si -\begin_inset Formula $H$ -\end_inset - - es de Hilbert. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Si -\begin_inset Formula $H$ -\end_inset - - es un espacio de Hilbert, todo espacio ortonormal de vectores en -\begin_inset Formula $H$ -\end_inset - - se puede completar a una base hilbertiana de -\begin_inset Formula $H$ -\end_inset - -, y en particular todo espacio de Hilbert posee una base hilbertiana y es - isométricamente isomorfo a un -\begin_inset Formula $\ell^{2}(I)$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Los espacios de Hilbert -\begin_inset Formula $\ell^{2}(I)$ -\end_inset - - y -\begin_inset Formula $\ell^{2}(J)$ -\end_inset - - son topológicamente isomorfos si y sólo si -\begin_inset Formula $|I|=|J|$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Llamamos -\series bold -dimensión hilbertiana -\series default - de un espacio de Hilbert al cardinal de cualquier base hilbertiana. - -\series bold -Segundo teorema de Riesz-Fischer: -\series default - Si -\begin_inset Formula $H$ -\end_inset - - es de dimensión infinita, -\begin_inset Formula $\dim H=\aleph_{0}\coloneqq|\mathbb{N}|$ -\end_inset - - si y sólo si -\begin_inset Formula $H\cong\ell^{2}$ -\end_inset - -, si y sólo si -\begin_inset Formula $H$ -\end_inset - - es separable. -\end_layout - -\begin_layout Description -\begin_inset Formula $1\iff2]$ -\end_inset - - Por lo anterior. -\end_layout - -\begin_layout Description -\begin_inset Formula $2\implies3]$ -\end_inset - - Visto. -\end_layout - -\begin_layout Description -\begin_inset Formula $3\implies2]$ -\end_inset - - Dado -\begin_inset Formula $\{x_{n}\}_{n\in\mathbb{N}}\subseteq H$ -\end_inset - - denso, como -\begin_inset Formula $H$ -\end_inset - - es de dimensión infinita, existe una subsucesión -\begin_inset Formula $(x_{n_{k}})_{k}$ -\end_inset - - linealmente independiente de -\begin_inset Formula $(x_{n})_{n}$ -\end_inset - - con -\begin_inset Formula $\text{span}\{x_{n}\}_{n}=\text{span}\{x_{n_{k}}\}_{k}$ -\end_inset - -, luego -\begin_inset Formula $\overline{\text{span}\{x_{n_{k}}\}_{k}}=H$ -\end_inset - - y el proceso de ortonormalización de Gram-Schmidt nos da una base hilbertiana - numerable de -\begin_inset Formula $H$ -\end_inset - -. -\end_layout - -\begin_layout Standard -Así, si -\begin_inset Formula $Z\leq_{\mathbb{K}}\ell^{2}$ -\end_inset - - es cerrado de dimensión infinita, -\begin_inset Formula $Z\cong\ell^{2}$ -\end_inset - -. -\end_layout - -\begin_layout Section -Aproximaciones por polinomios -\end_layout - -\begin_layout Standard -Si -\begin_inset Formula $I\subseteq\mathbb{R}$ -\end_inset - - es un intervalo cerrado, llamamos -\begin_inset Formula ${\cal C}(I)$ -\end_inset - - al conjunto de funciones -\begin_inset Formula $I\to\mathbb{R}$ -\end_inset - - continuas en el interior de -\begin_inset Formula $I$ -\end_inset - -. -\end_layout - -\begin_layout Standard - -\series bold -Teorema de Korovkin: -\series default - Sean -\begin_inset Formula $p_{0},p_{1},p_{2}:[a,b]\subseteq\mathbb{R}\to\mathbb{R}$ -\end_inset - - dadas por -\begin_inset Formula $p_{k}(t)\coloneqq t^{k}$ -\end_inset - - y -\begin_inset Formula $(P_{n}:{\cal C}([a,b])\to{\cal C}([a,b]))_{n}$ -\end_inset - - una sucesión de funciones lineales positivas ( -\begin_inset Formula $\forall f\in{\cal C}([a,b]),(f\geq0\implies P_{n}(f)\geq0)$ -\end_inset - -) con -\begin_inset Formula $\lim_{n}\Vert P_{n}(p_{k})-p_{k}\Vert_{\infty}=0$ -\end_inset - - para -\begin_inset Formula $k\in\{0,1,2\}$ -\end_inset - -, entonces, para -\begin_inset Formula $f\in{\cal C}([a,b])$ -\end_inset - -, -\begin_inset Formula $\lim_{n}\Vert P_{n}(f)-f\Vert_{\infty}=0$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard - -\series bold -Teorema de Weierstrass: -\series default - El conjunto de polinomios en una variable es denso -\begin_inset Formula $({\cal C}([a,b]),\Vert\cdot\Vert_{\infty})$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Así, para -\begin_inset Formula $f\in{\cal C}([a,b])$ -\end_inset - -, se puede encontrar una sucesión de polinomios que converja uniformemente - a -\begin_inset Formula $f$ -\end_inset - -. - Hacerlo con polinomios de interpolación por nodos prefijados no es una - buena estrategia ya que para toda secuencia de nodos de interpolación en - -\begin_inset Formula $[a,b]$ -\end_inset - -, existe -\begin_inset Formula $f\in{\cal C}([a,b])$ -\end_inset - - para la que los polinomios de interpolación en dichos nodos no converge - uniformemente a -\begin_inset Formula $f$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - Si se hace con nodos equidistantes se da el fenómeno de Runge. -\end_layout - -\begin_layout Standard - -\series bold -Teorema de Čebyšev: -\series default - Para -\begin_inset Formula $f\in{\cal C}([a,b])$ -\end_inset - - y -\begin_inset Formula $n\in\mathbb{N}$ -\end_inset - -, si -\begin_inset Formula $K_{n}\subseteq\mathbb{K}[X]$ -\end_inset - - es el conjunto de polinomio de grado máximo -\begin_inset Formula $n$ -\end_inset - -, -\begin_inset Formula $p:K_{n}\mapsto\Vert f-p\Vert_{\infty}$ -\end_inset - - tiene un único mínimo -\begin_inset Formula $p_{n}$ -\end_inset - -, y -\begin_inset Formula $(p_{n})_{n}$ -\end_inset - - converge uniformemente a -\begin_inset Formula $f$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Un -\series bold -polinomio trigonométrico real -\series default - es una función -\begin_inset Formula $p:\mathbb{R}\to\mathbb{R}$ -\end_inset - - de la forma -\begin_inset Formula -\[ -p(x)\coloneqq\sum_{n=0}^{m}(a_{n}\cos(nx)+b_{n}\sin(nx)) -\] - -\end_inset - -para ciertos -\begin_inset Formula $a_{n},b_{n}\in\mathbb{R}$ -\end_inset - -. - -\series bold -Teorema de Weierstrass: -\series default - Si -\begin_inset Formula $f:[-\pi,\pi]\to\mathbb{R}$ -\end_inset - - es continua con -\begin_inset Formula $f(-\pi)=f(\pi)$ -\end_inset - -, para cada -\begin_inset Formula $\varepsilon>0$ -\end_inset - - existe un polinomio trigonométrico real -\begin_inset Formula $p$ -\end_inset - - con -\begin_inset Formula $\Vert f-p\Vert_{\infty}<\varepsilon$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Para -\begin_inset Formula $f:[-\pi,\pi]\to\mathbb{C}$ -\end_inset - - integrable y -\begin_inset Formula $r\in\mathbb{Z}$ -\end_inset - -, llamamos -\series bold - -\begin_inset Formula $r$ -\end_inset - --ésimo coeficiente de Fourier -\series default - de -\begin_inset Formula $f$ -\end_inset - - a -\begin_inset Formula -\[ -\hat{f}(r)\coloneqq\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)\text{e}^{-\text{i}rt}\dif t, -\] - -\end_inset - -y -\series bold -serie de Fourier -\series default - de -\begin_inset Formula $f$ -\end_inset - - a la serie formal -\begin_inset Formula -\[ -\sum_{r\in\mathbb{Z}}\hat{f}(r)\text{e}^{-\text{i}rt}. -\] - -\end_inset - - -\end_layout - -\begin_layout Standard -Para -\begin_inset Formula $f:[-\pi,\pi]\to\mathbb{R}$ -\end_inset - - integrable y -\begin_inset Formula $n\in\mathbb{N}^{*}$ -\end_inset - -, llamando -\begin_inset Formula -\begin{align*} -a_{0} & \coloneqq\frac{1}{2\pi}\int_{-\pi}^{\pi}f, & a_{n} & \coloneqq\frac{1}{\pi}\int_{-\pi}^{\pi}f(t)\cos(nt)\dif t, & b_{n} & \coloneqq\frac{1}{\pi}\int_{-\pi}^{\pi}f(t)\sin(nt)\dif t, -\end{align*} - -\end_inset - -la -\series bold -serie de Fourier real -\series default - de -\begin_inset Formula $f$ -\end_inset - - es -\begin_inset Formula -\[ -\sum_{n=0}^{\infty}a_{n}\cos(nt)+\sum_{n=1}^{\infty}b_{n}\sin(nt). -\] - -\end_inset - - -\end_layout - -\begin_layout Standard -Como -\series bold -teorema -\series default -, sean -\begin_inset Formula $([-\pi,\pi],\Sigma,\mu)$ -\end_inset - - es el espacio de medida usual en -\begin_inset Formula $[-\pi,\pi]$ -\end_inset - -, -\begin_inset Formula $M_{\mathbb{R}}\coloneqq L_{\mathbb{R}}^{2}([-\pi,\pi],\Sigma,\frac{\mu}{\pi})$ -\end_inset - - y -\begin_inset Formula $M_{\mathbb{C}}\coloneqq L_{\mathbb{C}}^{2}([-\pi,\pi],\Sigma,\frac{\mu}{2\pi})$ -\end_inset - -: -\end_layout - -\begin_layout Enumerate -El -\series bold -sistema trigonométrico -\series default - -\begin_inset Formula $(\text{e}^{\text{i}rt})_{r\in\mathbb{Z}}$ -\end_inset - - es una base hilbertiana de -\begin_inset Formula $M_{\mathbb{C}}$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Enumerate -\begin_inset Formula $(\cos(nt))_{n\in\mathbb{N}}\star(\sin(nt))_{n\in\mathbb{N}^{*}}$ -\end_inset - - es una base hilbertiana de -\begin_inset Formula $M_{\mathbb{R}}$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Enumerate -Para -\begin_inset Formula $f\in M_{\mathbb{C}}$ -\end_inset - -, -\begin_inset Formula $f$ -\end_inset - - coincide con su serie de Fourier en -\begin_inset Formula $\Vert\cdot\Vert_{2}$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Enumerate -Para -\begin_inset Formula $f\in M_{\mathbb{R}}$ -\end_inset - -, -\begin_inset Formula $f$ -\end_inset - - coincide con su serie de Fourier real en -\begin_inset Formula $\Vert\cdot\Vert_{2}$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Enumerate -\begin_inset Formula ${\cal F}:M_{\mathbb{C}}\to\ell^{2}(\mathbb{Z})$ -\end_inset - - que asigna a cada función su familia de coeficientes de Fourier -\begin_inset Formula $(\hat{f}(n))_{n\in\mathbb{Z}}$ -\end_inset - - es un isomorfismo de espacios de Hilbert. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Un -\series bold -peso -\series default - en un intervalo cerrado -\begin_inset Formula $I\subseteq\mathbb{R}$ -\end_inset - - es una -\begin_inset Formula $p\in{\cal C}(I)$ -\end_inset - - estrictamente positiva tal que -\begin_inset Formula -\[ -\forall n\in\mathbb{N},\int_{I}|t|^{n}p(t)\dif t<\infty. -\] - -\end_inset - -Entonces -\begin_inset Formula $\langle\cdot,\cdot\rangle:{\cal C}(I)\times{\cal C}(I)\to[-\infty,+\infty]$ -\end_inset - - dada por -\begin_inset Formula -\[ -\langle f,g\rangle\coloneqq\int_{I}f\overline{g}p -\] - -\end_inset - -es un producto escalar en -\begin_inset Formula $H_{p}\coloneqq\{f\in{\cal C}(I)\mid\langle f,f\rangle<\infty\}$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Llamamos -\series bold -sucesión de polinomios ortonormales -\series default - asociada a -\begin_inset Formula $\langle\cdot,\cdot\rangle$ -\end_inset - - o al peso -\begin_inset Formula $p$ -\end_inset - - en -\begin_inset Formula $I$ -\end_inset - - a una sucesión -\begin_inset Formula $\{P_{n}\}_{n\in\mathbb{N}}\subseteq H_{p}$ -\end_inset - - de polinomios con -\begin_inset Formula $\text{span}\{1,t,\dots,t^{n}\}=\text{span}\{P_{0},P_{1},\dots,P_{n}\}$ -\end_inset - - para cada -\begin_inset Formula $n\in\mathbb{N}$ -\end_inset - -, y entonces, para -\begin_inset Formula $n\in\mathbb{N}$ -\end_inset - -: -\end_layout - -\begin_layout Enumerate -\begin_inset Formula $P_{n}$ -\end_inset - - es un polinomio de grado -\begin_inset Formula $n$ -\end_inset - - con coeficientes reales. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Enumerate -\begin_inset Formula $P_{n}$ -\end_inset - - es ortogonal en -\begin_inset Formula $H_{p}$ -\end_inset - - al subespacio de polinomios de grado menor que -\begin_inset Formula $n$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Enumerate -\begin_inset Formula $P_{n}$ -\end_inset - - tiene -\begin_inset Formula $n$ -\end_inset - - raíces distintas en -\begin_inset Formula $(a,b)$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Ejemplos: -\end_layout - -\begin_layout Enumerate - -\series bold -Polinomios de Legendre. -\series default - -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\begin_inset Formula -\begin{align*} -I & =[-1,1], & p(t) & =1, & P_{n}(t) & =\frac{\sqrt{\frac{2n+1}{2}}}{2^{n}n!}\od[n]{(t^{2}-1)^{n}}{t}. -\end{align*} - -\end_inset - - -\end_layout - -\begin_layout Enumerate - -\series bold -Polinomios de Laguerre. -\series default - -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\series bold - -\begin_inset Formula -\begin{align*} -I & =[0,\infty), & p(t) & =\text{e}^{-t}, & P_{n}(t) & =\frac{\text{e}^{t}}{n!}\od[n]{\text{e}^{-t}t^{n}}{t}. -\end{align*} - -\end_inset - - -\end_layout - -\begin_layout Enumerate - -\series bold -Polinomios de Hermite. -\series default - -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\begin_inset Formula -\begin{align*} -I & =(-\infty,\infty), & p(t) & =\text{e}^{-t^{2}}, & P_{n}(t) & =\frac{\text{e}^{t^{2}}}{\sqrt[4]{\pi}\sqrt{2^{n}n!}}\od[n]{\text{e}^{-t^{2}}}{t}. -\end{align*} - -\end_inset - - -\end_layout - -\begin_layout Enumerate - -\series bold -Polinomios de Čebyšev. -\series default - -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\begin_inset Formula -\begin{align*} -I & =[-1,1], & p(t) & =\frac{1}{\sqrt{1-t^{2}}}, & P_{n}(t) & =\cos(n\arccos t), -\end{align*} - -\end_inset - -siendo -\begin_inset Formula $\arccos:[-1,1]\to[0,\pi]$ -\end_inset - -. -\end_layout - -\begin_layout Standard -Una sucesión de polinomios ortonormales asociada a un peso -\begin_inset Formula $p$ -\end_inset - - en un intervalo compacto es total en -\begin_inset Formula $H_{p}$ -\end_inset - -, y en particular los polinomios de Legendre forman una base hilbertiana - en -\begin_inset Formula $L^{2}([-1,1]).$ -\end_inset - - -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Si -\begin_inset Formula $p$ -\end_inset - - es un peso en -\begin_inset Formula $[a,b]$ -\end_inset - - y -\begin_inset Formula $a\leq t_{1}<\dots<t_{n}\leq b$ -\end_inset - -, se tiene una -\series bold -fórmula de cuadratura gaussiana -\series default -, -\begin_inset Formula -\[ -\int_{a}^{b}fp\approx\sum_{k=1}^{n}A_{k}f(t_{k}) -\] - -\end_inset - -para ciertos -\begin_inset Formula $A_{1},\dots,A_{n}\in\mathbb{R}$ -\end_inset - -, y se alcanza la igualdad si -\begin_inset Formula $f$ -\end_inset - - es un polinomio de grado menor que -\begin_inset Formula $n$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard - -\series bold -Teorema de Gauss: -\series default - Dados un peso -\begin_inset Formula $p$ -\end_inset - - en -\begin_inset Formula $[a,b]$ -\end_inset - - con una sucesión de polinomios ortonormales -\begin_inset Formula $(P_{n})_{n}$ -\end_inset - -, -\begin_inset Formula $n\in\mathbb{N}^{*}$ -\end_inset - -, -\begin_inset Formula $a<t_{1}<\dots<t_{n}<b$ -\end_inset - - y -\begin_inset Formula $A_{1},\dots,A_{n}\in\mathbb{R}$ -\end_inset - -, si -\begin_inset Formula -\[ -\int_{a}^{b}fp=\sum_{k=1}^{n}A_{k}f(t_{k}) -\] - -\end_inset - -para todo polinomio -\begin_inset Formula $f$ -\end_inset - - de grado menor que -\begin_inset Formula $n$ -\end_inset - -, esta fórmula se para polinomios de grado menor que -\begin_inset Formula $2n$ -\end_inset - - si y sólo si -\begin_inset Formula $t_{1},\dots,t_{n}$ -\end_inset - - son los ceros de -\begin_inset Formula $P_{n}$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard - -\series bold -Teorema de Stieltjes: -\series default - Sean -\begin_inset Formula $p$ -\end_inset - - un peso en -\begin_inset Formula $[a,b]$ -\end_inset - - con una sucesión de polinomios ortonormales -\begin_inset Formula $(P_{n})_{n}$ -\end_inset - - y, para -\begin_inset Formula $n\in\mathbb{N}$ -\end_inset - -, -\begin_inset Formula $t_{n1}<\dots<t_{nn}$ -\end_inset - - los ceros de -\begin_inset Formula $P_{n}$ -\end_inset - - y -\begin_inset Formula $A_{n1},\dots,A_{nn}\in\mathbb{R}$ -\end_inset - - los correspondientes coeficientes en la fórmula de cuadratura gaussiana, - para -\begin_inset Formula $f\in{\cal C}([a,b])$ -\end_inset - -, -\begin_inset Formula -\[ -\int_{a}^{b}fp=\lim_{n}\sum_{k=1}^{n}A_{nk}f(t_{nk}). -\] - -\end_inset - - -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Section -El espacio de Bergman -\end_layout - -\begin_layout Standard -Llamamos -\begin_inset Formula $D(a,r)\coloneqq B(a,r)\subseteq\mathbb{C}$ -\end_inset - -. - Si -\begin_inset Formula $\Omega\subseteq\mathbb{C}$ -\end_inset - - es abierto, -\begin_inset Formula ${\cal H}(\Omega)$ -\end_inset - - es el conjunto de las funciones holomorfas en -\begin_inset Formula $\Omega$ -\end_inset - -, y para -\begin_inset Formula $f\in{\cal H}(\Omega)$ -\end_inset - - y -\begin_inset Formula $\overline{D(a,r)}\subseteq\Omega$ -\end_inset - -, la serie -\begin_inset Formula $\sum_{n\in\mathbb{N}}a_{n}(z-a)^{n}$ -\end_inset - - con -\begin_inset Formula $z\in D(a,r)$ -\end_inset - - converge uniformemente a -\begin_inset Formula $f$ -\end_inset - - en compactos de -\begin_inset Formula $D(a,r)$ -\end_inset - - para ciertos -\begin_inset Formula $a_{n}\in\mathbb{C}$ -\end_inset - -. -\end_layout - -\begin_layout Standard -Si -\begin_inset Formula $\Omega\subseteq\mathbb{C}$ -\end_inset - - es abierto, llamamos -\begin_inset Formula ${\cal T}_{\text{K}}$ -\end_inset - - a la topología en -\begin_inset Formula ${\cal H}(\Omega)$ -\end_inset - - de convergencia uniforme sobre compactos, y -\series bold -espacio de Bergman -\series default - en el abierto -\begin_inset Formula $\Omega\subseteq\mathbb{C}$ -\end_inset - - a -\begin_inset Formula -\[ -A^{2}(\Omega)\coloneqq\left\{ f\in{\cal H}(\Omega)\;\middle|\;\int_{\Omega}|f|^{2}<\infty\right\} , -\] - -\end_inset - -un subespacio cerrado y separable de -\begin_inset Formula $L^{2}(\Omega)$ -\end_inset - - que es pues un espacio de Hilbert numerable con -\begin_inset Formula $\langle\cdot,\cdot\rangle_{2}$ -\end_inset - -, y en el que la topología inducida por -\begin_inset Formula $L^{2}(\Omega)$ -\end_inset - - es más fina que la inducida por -\begin_inset Formula ${\cal T}_{\text{K}}$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Si -\begin_inset Formula $\Omega\subseteq\mathbb{C}$ -\end_inset - - es abierto, -\begin_inset Formula $(\omega_{n})_{n}$ -\end_inset - - es base hilbertiana de -\begin_inset Formula $A^{2}(\Omega)$ -\end_inset - - y -\begin_inset Formula $f\in A^{2}(\Omega)$ -\end_inset - -, el desarrollo en serie de Fourier de -\begin_inset Formula $f$ -\end_inset - -, -\begin_inset Formula $\sum_{n}\langle f,\omega_{n}\rangle\omega_{n}$ -\end_inset - -, converge uniformemente a -\begin_inset Formula $f$ -\end_inset - - en compactos de -\begin_inset Formula $\Omega$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Si -\begin_inset Formula $\psi_{n}(z)\coloneqq(z-a)^{n}$ -\end_inset - -, -\begin_inset Formula $(\frac{\psi_{n}}{\Vert\psi_{n}\Vert})_{n}$ -\end_inset - - es una base hilbertiana de -\begin_inset Formula $A^{2}(D(a,r))$ -\end_inset - -, y el desarrollo en serie de potencias es el desarrollo en serie de Fourier - sobre esta base. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -Como -\series bold -teorema -\series default -, si -\begin_inset Formula $\Omega\subsetneq\mathbb{C}$ -\end_inset - - es un abierto simplemente conexo y -\begin_inset Formula $f:\Omega\to D(0,1)$ -\end_inset - - es un isomorfismo, -\begin_inset Formula -\[ -\left(z\mapsto\sqrt{\frac{n}{\pi}}(f(z))^{n-1}\dot{f}(z)\right)_{n} -\] - -\end_inset - -es base hilbertiana de -\begin_inset Formula $A^{2}(\Omega)$ -\end_inset - -, y en particular para -\begin_inset Formula $R>0$ -\end_inset - -, -\begin_inset Formula -\[ -\left(z\mapsto\sqrt{\frac{n}{\pi}}R^{-n}z^{n-1}\right)_{n} -\] - -\end_inset - - es base hilbertiana de -\begin_inset Formula $A^{2}(D(0,R))$ -\end_inset - -. -\begin_inset Note Note -status open - -\begin_layout Plain Layout -nproof -\end_layout - -\end_inset - - -\end_layout - \end_body \end_document diff --git a/af/n1b.lyx b/af/n1b.lyx new file mode 100644 index 0000000..8338697 --- /dev/null +++ b/af/n1b.lyx @@ -0,0 +1,6829 @@ +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass book +\begin_preamble +\input{../defs} +\usepackage{commath} +\end_preamble +\use_default_options true +\maintain_unincluded_children false +\language spanish +\language_package default +\inputencoding auto +\fontencoding global +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\spacing single +\use_hyperref false +\papersize default +\use_geometry false +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 1 +\use_minted 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style french +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Standard +David Hilbert (1862–1943) fue un influyente matemático alemán que formuló + la teoría de los espacios de Hilbert. + En 1900 publicó una lista de 23 problemas que marcarían en buena medida + el progreso matemático en el siglo XX, y presentó 10 de ellos en el +\emph on +\lang english +International Congress of Mathematicians +\emph default +\lang spanish + de París de 1900. + Fue editor jefe de +\emph on +\lang ngerman +Mathematische Annalen +\emph default +\lang spanish +, una revista matemática muy prestigiosa por casi 150 años, y tuvo discípulos + como +\lang ngerman +Alfréd Haar, Erhard Schmidt, Hugo Steihaus, Hermann Weyl o Ernst Zermelo +\lang spanish +. +\end_layout + +\begin_layout Standard +Dado un +\begin_inset Formula $\mathbb{K}$ +\end_inset + +-espacio vectorial +\begin_inset Formula $H$ +\end_inset + +, +\begin_inset Formula $\langle\cdot,\cdot\rangle:H\times H\to\mathbb{K}$ +\end_inset + + es una +\series bold +forma hermitiana +\series default + si para +\begin_inset Formula $a,b\in\mathbb{K}$ +\end_inset + + y +\begin_inset Formula $x,y,z\in H$ +\end_inset + + se tiene +\begin_inset Formula $\langle ax+by,z\rangle=a\langle x,z\rangle+b\langle y,z\rangle$ +\end_inset + + y +\begin_inset Formula $\langle x,y\rangle=\overline{\langle y,x\rangle}$ +\end_inset + +, y es +\series bold +definida positiva +\series default + si para +\begin_inset Formula $x\in H\setminus0$ +\end_inset + + es +\begin_inset Formula $\langle x,x\rangle\in\mathbb{R}^{+}$ +\end_inset + +. + Un +\series bold +producto escalar +\series default + es una forma hermitiana definida positiva, y un +\series bold +espacio prehilbertiano +\series default + es par formado por un espacio vectorial y un producto escalar sobre este. +\end_layout + +\begin_layout Standard +Dado un espacio prehilbertiano +\begin_inset Formula $(H,\langle\cdot,\cdot\rangle)$ +\end_inset + +: +\end_layout + +\begin_layout Enumerate + +\series bold +Desigualdad de Cauchy-Schwartz: +\series default + +\begin_inset Formula $\forall x,y\in H,|\langle x,y\rangle|^{2}\leq\langle x,x\rangle\langle y,y\rangle$ +\end_inset + +, con igualdad si y sólo si +\begin_inset Formula $x$ +\end_inset + + e +\begin_inset Formula $y$ +\end_inset + + son linealmente dependientes. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $H$ +\end_inset + + es un espacio normado con la norma +\begin_inset Formula $\Vert x\Vert\coloneqq\sqrt{\langle x,x\rangle}$ +\end_inset + +, y para +\begin_inset Formula $x,y\in H$ +\end_inset + +, +\begin_inset Formula $\Vert x+y\Vert=\Vert x\Vert+\Vert y\Vert\iff x=0\lor y=0\lor\exists a>0:x=ay$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +Para +\begin_inset Formula $a,b\in\mathbb{K}$ +\end_inset + + y +\begin_inset Formula $x,y,z\in H$ +\end_inset + +, +\begin_inset Formula $\langle x,ay+bz\rangle=\overline{a}\langle x,y\rangle+\overline{b}\langle x,z\rangle$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Para +\begin_inset Formula $x,y\in H$ +\end_inset + +, +\begin_inset Formula $\Vert x+y\Vert^{2}=\Vert x\Vert^{2}+\Vert y\Vert^{2}+2\text{Re}\langle x,y\rangle$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +\begin_inset Formula $\Vert x+y\Vert^{2}=\langle x+y,x+y\rangle=\langle x,x\rangle+\langle x,y\rangle+\overline{\langle x,y\rangle}+\langle y,y\rangle$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Standard + +\series bold +Identidades de polarización: +\series default + Si +\begin_inset Formula $H$ +\end_inset + + es un espacio prehilbertiano y +\begin_inset Formula $x,y\in H$ +\end_inset + +: +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $\langle x,y\rangle=\frac{1}{4}(\Vert x+y\Vert^{2}-\Vert x-y\Vert^{2}+\text{i}\Vert x+\text{i}y\Vert^{2}-\text{i}\Vert x-\text{i}y\Vert^{2})$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +Si +\begin_inset Formula $H$ +\end_inset + + se define sobre +\begin_inset Formula $\mathbb{R}$ +\end_inset + +, +\begin_inset Formula $\langle x,y\rangle=\frac{1}{4}(\Vert x+y\Vert^{2}-\Vert x-y\Vert^{2})$ +\end_inset + +. +\end_layout + +\begin_layout Standard + +\series bold +Teorema de von Neumann: +\series default + Un espacio normado +\begin_inset Formula $(X,\Vert\cdot\Vert)$ +\end_inset + + admite un producto escalar +\begin_inset Formula $\langle\cdot,\cdot\rangle$ +\end_inset + + en +\begin_inset Formula $X$ +\end_inset + + con +\begin_inset Formula $\langle x,x\rangle\equiv\Vert x\Vert^{2}$ +\end_inset + + si y sólo si +\begin_inset Formula $\Vert\cdot\Vert$ +\end_inset + + verifica la +\series bold +ley del paralelogramo: +\series default + +\begin_inset Formula +\[ +\forall x,y\in H,\Vert x+y\Vert^{2}+\Vert x-y\Vert^{2}=2(\Vert x\Vert^{2}+\Vert y\Vert^{2}). +\] + +\end_inset + + +\end_layout + +\begin_layout Itemize +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +\begin_inset Formula $\implies]$ +\end_inset + + +\end_layout + +\end_inset + +En general +\begin_inset Formula $\langle x,y+z\rangle=\overline{\langle y+z,x\rangle}=\overline{\langle y,x\rangle}+\overline{\langle z,x\rangle}=\langle x,y\rangle+\langle x,z\rangle$ +\end_inset + +, de donde +\begin_inset Formula +\begin{multline*} +\Vert x+y\Vert^{2}+\Vert x-y\Vert^{2}=\langle x+y,x+y\rangle+\langle x-y,x-y\rangle=\\ +=\langle x,x\rangle+\langle x,y\rangle+\langle y,x\rangle+\langle y,y\rangle+\langle x,x\rangle-\langle x,y\rangle-\langle y,x\rangle+\langle y,y\rangle=2(\Vert x\Vert^{2}+\Vert y\Vert^{2}). +\end{multline*} + +\end_inset + + +\end_layout + +\begin_layout Itemize +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +\begin_inset Formula $\impliedby]$ +\end_inset + + +\end_layout + +\end_inset + +Definimos +\begin_inset Formula $\langle\cdot,\cdot\rangle$ +\end_inset + + según la identidad de polarización, y queremos ver que es un producto escalar + cuya norma es la inicial. + Se tiene +\begin_inset Formula +\begin{align*} +\langle x,x\rangle & =\frac{1}{4}\left(\Vert2x\Vert^{2}-\Vert x-x\Vert^{2}+\text{i}\Vert x+\text{i}x\Vert^{2}-\text{i}\Vert x-\text{i}x\Vert^{2}\right)=\\ + & =\frac{1}{4}\left(4\Vert x\Vert^{2}+\text{i}|1+\text{i}|^{2}\Vert x\Vert^{2}-\text{i}|1-\text{i}|^{2}\Vert x\Vert^{2}\right)=\Vert x\Vert^{2}, +\end{align*} + +\end_inset + +y +\begin_inset Formula +\begin{align*} +4\langle x,y\rangle & =\Vert x+y\Vert^{2}-\Vert x-y\Vert^{2}+\text{i}\Vert x+\text{i}y\Vert^{2}-\text{i}\Vert x-\text{i}y\Vert^{2}\\ + & =\Vert y+x\Vert^{2}-\Vert y-x\Vert^{2}+\text{i}\Vert y-\text{i}x\Vert-\text{i}\Vert y+\text{i}x\Vert^{2}=4\overline{\langle y,x\rangle}\\ + & =\Vert-x-y\Vert^{2}-\Vert-x+y\Vert^{2}+\text{i}\Vert-x-\text{i}y\Vert^{2}-\text{i}\Vert-x+\text{i}y\Vert^{2}=-4\langle-x,y\rangle\\ + & =\Vert\text{i}x+\text{i}y\Vert^{2}-\Vert\text{i}x-\text{i}y\Vert^{2}+\text{i}\Vert\text{i}x-y\Vert^{2}-\text{i}\Vert\text{i}x+y\Vert^{2}=4\frac{\langle\text{i}x,y\rangle}{\text{i}}. +\end{align*} + +\end_inset + +Para ver que +\begin_inset Formula $\langle x+z,y\rangle=\langle x,y\rangle+\langle z,y\rangle$ +\end_inset + +, +\begin_inset Formula +\begin{multline*} +\Vert x+z+y\Vert^{2}-\Vert x+z-y\Vert^{2}=\left\Vert \left(x+\frac{y}{2}\right)+\left(z+\frac{y}{2}\right)\right\Vert ^{2}-\left\Vert \left(x+\frac{y}{2}\right)-\left(z+\frac{y}{2}\right)\right\Vert ^{2}=\\ +=2\left\Vert x+\frac{y}{2}\right\Vert ^{2}+2\left\Vert z+\frac{y}{2}\right\Vert ^{2}\cancel{-\Vert x-z\Vert^{2}}-2\left\Vert x-\frac{y}{2}\right\Vert ^{2}-2\left\Vert z-\frac{y}{2}\right\Vert ^{2}\cancel{+\Vert x-z\Vert^{2}}, +\end{multline*} + +\end_inset + +de donde +\begin_inset Formula +\begin{eqnarray*} +4\langle x+z,y\rangle & = & \Vert x+z+y\Vert^{2}-\Vert x+z-y\Vert^{2}+\text{i}\Vert x+z+\text{i}y\Vert^{2}-\text{i}\Vert x+z-\text{i}y\Vert^{2}\\ + & = & 2\left(\left\Vert x+\frac{y}{2}\right\Vert ^{2}+\left\Vert z+\frac{y}{2}\right\Vert ^{2}-\left\Vert x-\frac{y}{2}\right\Vert ^{2}-\left\Vert z-\frac{y}{2}\right\Vert \right)\\ + & & +2\text{i}\left(\left\Vert x+\text{i}\frac{y}{2}\right\Vert ^{2}+\left\Vert z+\text{i}\frac{z}{2}\right\Vert ^{2}-\left\Vert x-\text{i}\frac{y}{2}\right\Vert ^{2}-\left\Vert z-\text{i}\frac{y}{2}\right\Vert ^{2}\right)\\ + & = & 8\left\langle x,\frac{y}{2}\right\rangle +8\left\langle z,\frac{y}{2}\right\rangle , +\end{eqnarray*} + +\end_inset + +y por tanto +\begin_inset Formula +\[ +\langle x+z,y\rangle=2\left\langle x,\frac{y}{2}\right\rangle +2\left\langle z,\frac{y}{2}\right\rangle =\langle x,y\rangle+\langle z,y\rangle, +\] + +\end_inset + +donde en la segunda igualdad hemos usado la primera igualdad con +\begin_inset Formula $z=0$ +\end_inset + + o +\begin_inset Formula $x=0$ +\end_inset + +. + Usando esto y que +\begin_inset Formula $\langle-x,y\rangle$ +\end_inset + + es fácil ver que +\begin_inset Formula $\langle ax,y\rangle=a\langle x,y\rangle$ +\end_inset + + para +\begin_inset Formula $a\in\mathbb{Q}$ +\end_inset + +; para +\begin_inset Formula $a\in\mathbb{R}$ +\end_inset + + se usa la continuidad de la norma y por tanto del producto escalar, y para + +\begin_inset Formula $a\in\mathbb{C}$ +\end_inset + + se usa +\begin_inset Formula $\langle\text{i}x,y\rangle=\text{i}\langle x,y\rangle$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $(\ell^{\infty},\Vert\cdot\Vert_{\infty})$ +\end_inset + + y +\begin_inset Formula $({\cal C}([a,b]),\Vert\cdot\Vert_{1})$ +\end_inset + + son espacios normados no prehilbertianos. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Dos espacios prehilbertianos +\begin_inset Formula $(H_{1},\langle\cdot,\cdot\rangle_{1})$ +\end_inset + + y +\begin_inset Formula $(H_{2},\langle\cdot,\cdot\rangle_{2})$ +\end_inset + + son +\series bold +equivalentes +\series default + si existe un isomorfismo algebraico +\begin_inset Formula $T:H_{1}\to H_{2}$ +\end_inset + + con +\begin_inset Formula $\langle x,y\rangle_{1}=\langle T(x),T(y)\rangle_{2}$ +\end_inset + + para todo +\begin_inset Formula $x,y\in H_{1}$ +\end_inset + +, si y sólo si existe un isomorfismo isométrico entre los espacios normados. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $H$ +\end_inset + + es un espacio prehilbertiano, +\begin_inset Formula $x,y\in H$ +\end_inset + + son +\series bold +ortogonales +\series default +, +\begin_inset Formula $x\bot y$ +\end_inset + +, si +\begin_inset Formula $\langle x,y\rangle=0$ +\end_inset + +. + Decimos que +\begin_inset Formula $x\in H$ +\end_inset + + es +\series bold +ortogonal +\series default + a +\begin_inset Formula $M\subseteq H$ +\end_inset + +, +\begin_inset Formula $x\bot M$ +\end_inset + +, si +\begin_inset Formula $\forall y\in M,x\bot y$ +\end_inset + +, y llamamos +\begin_inset Formula $M^{\bot}\coloneqq\{x\in H:x\bot M\}$ +\end_inset + +. + Una familia +\begin_inset Formula $\{x_{i}\}_{i\in I}\subseteq H$ +\end_inset + + es +\series bold +ortogonal +\series default + si +\begin_inset Formula $\forall i,j\in I,(i\neq j\implies x_{i}\bot x_{j})$ +\end_inset + +, y es +\series bold +ortonormal +\series default + si además +\begin_inset Formula $\forall i,\Vert x_{i}\Vert=1$ +\end_inset + +. + Entonces: +\end_layout + +\begin_layout Enumerate + +\series bold +Teorema de Pitágoras: +\series default + Si +\begin_inset Formula $x\bot y$ +\end_inset + +, +\begin_inset Formula $\Vert x+y\Vert^{2}=\Vert x\Vert^{2}+\Vert y\Vert^{2}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +Si +\begin_inset Formula $(x_{i})_{i\in I}$ +\end_inset + + es una familia ortogonal de elementos no nulos, es una familia linealmente + independiente. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +Si +\begin_inset Formula $M\subseteq H$ +\end_inset + +, +\begin_inset Formula $M^{\bot}$ +\end_inset + + es un subespacio cerrado de +\begin_inset Formula $H$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\series bold +Lema de Gram-Schmidt: +\series default + Sean +\begin_inset Formula $H$ +\end_inset + + prehilbertiano, +\begin_inset Formula $\{x_{n}\}_{n}\subseteq H$ +\end_inset + + una familia contable linealmente independiente y +\begin_inset Formula $(u_{n})_{n}$ +\end_inset + + e +\begin_inset Formula $(y_{n})_{n}$ +\end_inset + + dadas por +\begin_inset Formula $u_{n}\coloneqq\frac{y_{n}}{\Vert y_{n}\Vert}$ +\end_inset + +, +\begin_inset Formula $y_{0}\coloneqq x_{0}$ +\end_inset + + y para +\begin_inset Formula $n\geq1$ +\end_inset + +, +\begin_inset Formula +\[ +y_{n}\coloneqq x_{n}-\sum_{j<n}\langle x_{n},u_{j}\rangle u_{j}, +\] + +\end_inset + + +\begin_inset Formula $(u_{n})_{n}$ +\end_inset + + es una sucesión ortonormal en +\begin_inset Formula $H$ +\end_inset + + y, para cada +\begin_inset Formula $n$ +\end_inset + +, +\begin_inset Formula $\text{span}\{u_{1},\dots,u_{n}\}=\text{span}\{x_{1},\dots,x_{n}\}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $M$ +\end_inset + + es un subespacio de dimensión finita del espacio prehilbertiano +\begin_inset Formula $H$ +\end_inset + +: +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $M$ +\end_inset + + tiene una base algebraica formada por vectores ortonormales. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $M$ +\end_inset + + es equivalente a +\begin_inset Formula $\mathbb{K}^{\dim M}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Un +\series bold +espacio de Hilbert +\series default + es un espacio prehilbertiano completo. + Dado un espacio de medida +\begin_inset Formula $(\Omega,\Sigma,\mu)$ +\end_inset + +, +\begin_inset Formula $L^{2}(\Omega,\Sigma,\mu)$ +\end_inset + + es un espacio de Hilbert con +\begin_inset Formula +\[ +\langle f,g\rangle\coloneqq\int_{\Omega}f\overline{g}\dif\mu, +\] + +\end_inset + +y en particular lo son +\begin_inset Formula $\ell^{2}$ +\end_inset + + con +\begin_inset Formula $\langle x,y\rangle\coloneqq\sum_{n}x_{n}\overline{y_{n}}$ +\end_inset + + y +\begin_inset Formula $\ell_{n}^{2}$ +\end_inset + + con +\begin_inset Formula $\langle x,y\rangle\coloneqq\sum_{i}x_{i}\overline{y_{i}}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Son espacios prehilbertianos no completos: +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $c_{00}$ +\end_inset + + con el producto escalar de +\begin_inset Formula $\ell^{2}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $C([a,b])$ +\end_inset + + con el producto escalar de +\begin_inset Formula $L^{2}([a,b])$ +\end_inset + + con la medida de Lebesgue, y entonces +\begin_inset Formula $C([a,b])$ +\end_inset + + es denso en +\begin_inset Formula $L^{2}([a,b])$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Section +Mejor aproximación +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $X$ +\end_inset + + es un espacio vectorial, +\begin_inset Formula $A\subseteq X$ +\end_inset + + es +\series bold +convexo +\series default + si +\begin_inset Formula $\forall\lambda\in[0,1]$ +\end_inset + +, +\begin_inset Formula $\lambda A+(1-\lambda)A\subseteq A$ +\end_inset + +. + Si +\begin_inset Formula $X$ +\end_inset + + es normado, +\begin_inset Formula $S\subseteq X$ +\end_inset + + no vacío y +\begin_inset Formula $x\in X$ +\end_inset + +, un +\begin_inset Formula $y\in S$ +\end_inset + + es un +\series bold +vector de mejor aproximación +\series default + de +\begin_inset Formula $x$ +\end_inset + + a +\begin_inset Formula $S$ +\end_inset + + si +\begin_inset Formula $\Vert x-y\Vert=\min_{z\in S}\Vert x-z\Vert$ +\end_inset + +. +\end_layout + +\begin_layout Standard + +\series bold +Teorema de mejor aproximación: +\series default + Si +\begin_inset Formula $H$ +\end_inset + + es un espacio prehilbertiano y +\begin_inset Formula $C\subseteq H$ +\end_inset + + es no vacío, convexo y completo, para cada +\begin_inset Formula $x\in H$ +\end_inset + + existe una mejor aproximación de +\begin_inset Formula $x$ +\end_inset + + a +\begin_inset Formula $C$ +\end_inset + +. + +\series bold +Demostración: +\series default + Podemos suponer por traslación que +\begin_inset Formula $x=0$ +\end_inset + +, y llamamos +\begin_inset Formula $\alpha\coloneqq\inf_{z\in C}\Vert z\Vert$ +\end_inset + +. + Para la existencia tomamos una sucesión +\begin_inset Formula $\{y_{n}\}_{n}\subseteq C$ +\end_inset + + con +\begin_inset Formula $\lim_{n}\Vert y_{n}\Vert=\alpha$ +\end_inset + + y probamos que es de Cauchy, pues entonces por completitud existe +\begin_inset Formula $y\coloneqq\lim_{n}y_{n}\in C$ +\end_inset + + y por continuidad de la norma es +\begin_inset Formula $\Vert y\Vert=\alpha$ +\end_inset + +. + Para +\begin_inset Formula $\varepsilon>0$ +\end_inset + + existe +\begin_inset Formula $n_{0}$ +\end_inset + + tal que si +\begin_inset Formula $n\geq n_{0}$ +\end_inset + + es +\begin_inset Formula $\Vert y_{n}\Vert^{2}<\alpha^{2}+\varepsilon$ +\end_inset + +, y por la ley del paralelogramo es +\begin_inset Formula +\[ +\left\Vert \frac{y_{n}-y_{m}}{2}\right\Vert ^{2}=\frac{1}{2}(\Vert y_{n}\Vert^{2}+\Vert y_{m}\Vert^{2})-\left\Vert \frac{y_{n}+y_{m}}{2}\right\Vert ^{2}\leq\frac{1}{2}(\alpha^{2}+\varepsilon+\alpha^{2}+\varepsilon)-\alpha^{2}=\varepsilon, +\] + +\end_inset + +pues por convexidad +\begin_inset Formula $\frac{y_{n}+y_{m}}{2}\in S$ +\end_inset + + y por tanto su norma es mayor o igual a +\begin_inset Formula $\alpha$ +\end_inset + +. + Para la unicidad, si +\begin_inset Formula $y,z\in C$ +\end_inset + + cumplen +\begin_inset Formula $\Vert y\Vert=\Vert z\Vert=\alpha$ +\end_inset + +, por un argumento como el anterior, +\begin_inset Formula +\[ +\left\Vert \frac{y-z}{2}\right\Vert ^{2}=\frac{1}{2}(\Vert y\Vert^{2}+\Vert z\Vert^{2})-\left\Vert \frac{y+z}{2}\right\Vert ^{2}\leq\frac{1}{2}(\alpha^{2}+\alpha^{2})-\alpha^{2}=0. +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Como +\series bold +teorema +\series default +, si +\begin_inset Formula $Y$ +\end_inset + + es un subespacio de un espacio prehilbertiano +\begin_inset Formula $H$ +\end_inset + + y +\begin_inset Formula $x\in H$ +\end_inset + +: +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $y\in Y$ +\end_inset + + es de mejor aproximación de +\begin_inset Formula $x$ +\end_inset + + a +\begin_inset Formula $Y$ +\end_inset + + si y sólo si +\begin_inset Formula $x-y\bot Y$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +\begin_inset Formula $\implies]$ +\end_inset + + +\end_layout + +\end_inset + +Para +\begin_inset Formula $z\in Y$ +\end_inset + + y +\begin_inset Formula $a\in\mathbb{K}$ +\end_inset + +, como +\begin_inset Formula $y-az\in Y$ +\end_inset + +, +\begin_inset Formula +\[ +\Vert x-y\Vert^{2}\leq\Vert x-y+az\Vert^{2}=\Vert x-y\Vert^{2}+2\text{Re}(a\langle z,x-y\rangle)+|a|^{2}\Vert z\Vert^{2}, +\] + +\end_inset + +luego +\begin_inset Formula $0\leq2\text{Re}(a\langle z,x-y\rangle)+|a|^{2}\Vert z\Vert^{2}$ +\end_inset + + y, haciendo +\begin_inset Formula $a=t\langle x-y,z\rangle$ +\end_inset + + con +\begin_inset Formula $t\in\mathbb{R}$ +\end_inset + +, +\begin_inset Formula $0\leq2t|\langle x-y,z\rangle|^{2}+t^{2}|\langle x-y,z\rangle|^{2}\Vert z\Vert^{2}$ +\end_inset + +. + Si hubiera +\begin_inset Formula $z\in Y$ +\end_inset + + con +\begin_inset Formula $\langle x-y,z\rangle\neq0$ +\end_inset + +, +\begin_inset Formula $0\leq2t+t^{2}\Vert z\Vert^{2}$ +\end_inset + + para todo +\begin_inset Formula $t\in\mathbb{R}$ +\end_inset + +, pero si +\begin_inset Formula $\Vert z\Vert^{2}=0$ +\end_inset + +, esto es negativo cuando +\begin_inset Formula $t<0$ +\end_inset + +, y si +\begin_inset Formula $\Vert z\Vert^{2}>0$ +\end_inset + +, es negativo al menos cuando +\begin_inset Formula $t=-\frac{1}{\Vert z\Vert^{2}}\#$ +\end_inset + +, luego +\begin_inset Formula $x-y\bot z$ +\end_inset + + y +\begin_inset Formula $x-y\bot Y$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +\begin_inset Formula $\impliedby]$ +\end_inset + + +\end_layout + +\end_inset + +Para +\begin_inset Formula $z\in Y$ +\end_inset + +, por el teorema de Pitágoras, +\begin_inset Formula +\[ +\Vert x-z\Vert^{2}=\Vert x-y+y-z\Vert^{2}=\Vert x-y\Vert^{2}+\Vert y-z\Vert^{2}\geq\Vert x-y\Vert^{2}. +\] + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Enumerate +Si existe una mejor aproximación de +\begin_inset Formula $x$ +\end_inset + + a +\begin_inset Formula $Y$ +\end_inset + +, es única. +\end_layout + +\begin_deeper +\begin_layout Standard +Sean +\begin_inset Formula $y,z\in Y$ +\end_inset + + de mejor aproximación, como +\begin_inset Formula $x-y,x-z\in Y^{\bot}$ +\end_inset + +, su diferencia +\begin_inset Formula $y-z\in Y^{\bot}\cap Y$ +\end_inset + +, luego +\begin_inset Formula $\langle y-z,y-z\rangle=0$ +\end_inset + + e +\begin_inset Formula $y=z$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +Si +\begin_inset Formula $Y$ +\end_inset + + es completo, hay vector de mejor aproximación. +\end_layout + +\begin_deeper +\begin_layout Standard +Por el teorema anterior (los subespacios son convexos). +\end_layout + +\end_deeper +\begin_layout Section +Determinante de Gram +\end_layout + +\begin_layout Standard +Sean +\begin_inset Formula $H$ +\end_inset + + prehilbertiano y +\begin_inset Formula $M\leq H$ +\end_inset + + de dimensión finita con base ortonormal +\begin_inset Formula $(e_{i})_{i}$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Para +\begin_inset Formula $x\in H$ +\end_inset + + existe un único vector de aproximación de +\begin_inset Formula $x$ +\end_inset + + a +\begin_inset Formula $M$ +\end_inset + + dado por +\begin_inset Formula +\[ +\sum_{i}\langle x,e_{i}\rangle e_{i}. +\] + +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $d(x,M)^{2}=\Vert x\Vert^{2}-\sum_{i}|\langle x,e_{i}\rangle|^{2}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Llamamos +\series bold +determinante de Gram +\series default + de +\begin_inset Formula $(x_{i})_{i=1}^{n}$ +\end_inset + + a +\begin_inset Formula +\[ +G(x_{1},\dots,G_{n})\coloneqq\det(\langle x_{j},x_{i}\rangle)_{1\leq i\leq n}^{1\leq j\leq n}. +\] + +\end_inset + +Como +\series bold +teorema +\series default +, si +\begin_inset Formula $H$ +\end_inset + + es prehilbertiano, +\begin_inset Formula $M\leq H$ +\end_inset + + de dimensión finita con base +\begin_inset Formula $(b_{i})_{i}$ +\end_inset + + y +\begin_inset Formula $x\in H$ +\end_inset + +, el vector de mejor aproximación de +\begin_inset Formula $x$ +\end_inset + + a +\begin_inset Formula $M$ +\end_inset + + es +\begin_inset Formula +\[ +\frac{-1}{G(b_{1},\dots,b_{n})}\begin{vmatrix}\langle x_{1},x_{1}\rangle & \langle x_{2},x_{1}\rangle & \cdots & \langle x_{n},x_{1}\rangle & \langle x,x_{1}\rangle\\ +\langle x_{1},x_{2}\rangle & \langle x_{2},x_{2}\rangle & \cdots & \langle x_{n},x_{2}\rangle & \langle x,x_{2}\rangle\\ +\vdots & \vdots & \ddots & \vdots & \vdots\\ +\langle x_{1},x_{n}\rangle & \langle x_{2},x_{n}\rangle & \cdots & \langle x_{n},x_{n}\rangle & \langle x,x_{n}\rangle\\ +x_{1} & x_{2} & \cdots & x_{n} & 0 +\end{vmatrix}, +\] + +\end_inset + +y +\begin_inset Formula +\[ +d(x,M)=\sqrt{\frac{G(x_{1},\dots,x_{n},x)}{G(x_{1},\dots,x_{n})}}. +\] + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Algunas aplicaciones: +\end_layout + +\begin_layout Enumerate + +\series bold +Resolución de sistemas sobre-dimensionados por mínimos cuadrados. + +\series default + Tenemos un fenómeno experimental que se puede modelar como una función + lineal +\begin_inset Formula $y(x)=a_{1}x_{1}+\dots+a_{n}x_{n}$ +\end_inset + +, pero no conocemos los +\begin_inset Formula $a_{i}$ +\end_inset + +. + Hacemos +\begin_inset Formula $m$ +\end_inset + + experimentos fijando un +\begin_inset Formula $x_{i}$ +\end_inset + + en cada uno y midiendo +\begin_inset Formula $y_{i}\coloneqq y(x_{i})$ +\end_inset + + para plantear un sistema de +\begin_inset Formula $m$ +\end_inset + + ecuaciones. + Solo hacen falta +\begin_inset Formula $n$ +\end_inset + + experimentos cuidando que los +\begin_inset Formula $x_{i}$ +\end_inset + + sean linealmente independientes, pero en general conviene hacer más, +\begin_inset Formula $m>n$ +\end_inset + +. + Como las mediciones son aproximadas, el sistema puede ser incompatible, + por lo que se eligen los +\begin_inset Formula $a_{i}\in\mathbb{R}$ +\end_inset + + de forma que se minimice +\begin_inset Formula +\[ +\sum_{i\in\mathbb{N}_{m}}\left(y_{i}-\sum_{j\in\mathbb{N}_{n}}a_{j}x_{ij}\right)^{2}=\left\Vert y-\sum_{j\in\mathbb{N}_{n}}a_{j}X_{j}\right\Vert ^{2}, +\] + +\end_inset + +donde +\begin_inset Formula $X_{j}\coloneqq(x_{1j},\dots,x_{mj})$ +\end_inset + +. + Si +\begin_inset Formula $X_{1},\dots,X_{n}$ +\end_inset + + son linealmente independientes, sea +\begin_inset Formula $M\coloneqq\text{span}\{X_{1},\dots,X_{n}\}<\mathbb{R}^{m}$ +\end_inset + +, buscamos el vector +\begin_inset Formula $Z\in M$ +\end_inset + + de mejor aproximación de +\begin_inset Formula $y$ +\end_inset + + en +\begin_inset Formula $M$ +\end_inset + + que, expresado respecto de la base +\begin_inset Formula $(X_{1},\dots,X_{n})$ +\end_inset + +, nos dará el vector +\begin_inset Formula $(a_{1},\dots,a_{n})$ +\end_inset + + buscado. +\end_layout + +\begin_layout Enumerate + +\series bold +Ajustes polinómicos por mínimos cuadrados. + +\series default + Queremos modelar un fenómeno experimental como una función polinómica +\begin_inset Formula $f:[a,b]\to\mathbb{R}$ +\end_inset + +, y tenemos +\begin_inset Formula $k$ +\end_inset + + observaciones de la forma +\begin_inset Formula $f(t_{i})=y_{i}$ +\end_inset + + con +\begin_inset Formula $t_{1}<\dots<t_{k}$ +\end_inset + +. + Existe un polinomio de grado máximo +\begin_inset Formula $k-1$ +\end_inset + + que cumple esto, pero muchas veces +\begin_inset Formula $k$ +\end_inset + + es muy grande y esto complica los cálculos y puede llevar al +\emph on +\lang english +overfitting +\emph default +\lang spanish + o fenómeno de Runge. + Entonces buscamos un polinomio +\begin_inset Formula $f$ +\end_inset + + de grado máximo +\begin_inset Formula $n$ +\end_inset + + bastante menor que +\begin_inset Formula $k-1$ +\end_inset + + que minimice +\begin_inset Formula +\[ +\sum_{i\in\mathbb{N}_{k}}|y_{i}-f(t_{i})|^{2}=\left\Vert y-\sum_{j=0}^{n}f_{j}t^{j}\right\Vert ^{2}, +\] + +\end_inset + +donde +\begin_inset Formula $t^{j}\coloneqq(t_{1}^{j},\dots,t_{k}^{j})$ +\end_inset + +. + Para ello, como para +\begin_inset Formula $k\geq2$ +\end_inset + + los +\begin_inset Formula $t^{j}$ +\end_inset + + son linealmente independientes, consideramos +\begin_inset Formula $M\coloneqq\text{span}\{1,t,t^{2},\dots,t^{n}\}<\mathbb{R}^{n+1}$ +\end_inset + + y buscamos la mejor aproximación de +\begin_inset Formula $y$ +\end_inset + + a +\begin_inset Formula $M$ +\end_inset + +. +\end_layout + +\begin_layout Section +Teorema de la proyección +\end_layout + +\begin_layout Standard + +\series bold +Teorema de la proyección: +\series default + Si +\begin_inset Formula $H$ +\end_inset + + es un espacio de Hilbert con un subespacio cerrado +\begin_inset Formula $M$ +\end_inset + + y +\begin_inset Formula $P_{M}:H\to M$ +\end_inset + + la +\series bold +proyección ortogonal +\series default + de +\begin_inset Formula $H$ +\end_inset + + sobre +\begin_inset Formula $M$ +\end_inset + + que asigna a cada +\begin_inset Formula $x\in H$ +\end_inset + + la mejor aproximación de +\begin_inset Formula $x$ +\end_inset + + a +\begin_inset Formula $M$ +\end_inset + +: +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $H$ +\end_inset + + es suma directa topológica de +\begin_inset Formula $M$ +\end_inset + + y +\begin_inset Formula $M^{\bot}$ +\end_inset + +, +\begin_inset Formula $P_{M}$ +\end_inset + + es la proyección canónica y, si +\begin_inset Formula $P_{M^{\bot}}:H\to M^{\bot}$ +\end_inset + + es la otra proyección canónica, si +\begin_inset Formula $M\neq0$ +\end_inset + +, +\begin_inset Formula $\Vert P_{M}\Vert=1$ +\end_inset + +, y si +\begin_inset Formula $M^{\bot}\neq0$ +\end_inset + +, +\begin_inset Formula $\Vert P_{M^{\bot}}\Vert=1$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +Por la definición de producto escalar, +\begin_inset Formula $M^{\bot}\leq H$ +\end_inset + +. + Claramente +\begin_inset Formula $M\cap M^{\bot}=0$ +\end_inset + +, y para +\begin_inset Formula $x\in M$ +\end_inset + +, como +\begin_inset Formula $y\coloneqq P_{M}(x)$ +\end_inset + + cumple +\begin_inset Formula $x-y\bot M$ +\end_inset + +, +\begin_inset Formula $x=y+z$ +\end_inset + + con +\begin_inset Formula $y\in M$ +\end_inset + + y +\begin_inset Formula $z\coloneqq x-y\in M^{\bot}$ +\end_inset + +, luego +\begin_inset Formula $M+M^{\bot}=H$ +\end_inset + + y +\begin_inset Formula $H$ +\end_inset + + es suma directa algebraica de +\begin_inset Formula $M$ +\end_inset + + y +\begin_inset Formula $M^{\bot}$ +\end_inset + +. + +\begin_inset Formula $P_{M}$ +\end_inset + + es la proyección canónica porque, si +\begin_inset Formula $y\in M$ +\end_inset + + y +\begin_inset Formula $z\in M^{\bot}$ +\end_inset + +, +\begin_inset Formula $(y+z)-y=z\bot M$ +\end_inset + +, y por unicidad de la mejor aproximación, +\begin_inset Formula $P_{M}(y+z)=y$ +\end_inset + +. + +\begin_inset Formula $P_{M}$ +\end_inset + + y +\begin_inset Formula $P_{M^{\bot}}$ +\end_inset + + son lineales por ser proyecciones canónicas, y para +\begin_inset Formula $x=y+z\in S_{H}$ +\end_inset + + con +\begin_inset Formula $y\in M$ +\end_inset + + y +\begin_inset Formula $z\in M^{\bot}$ +\end_inset + +, +\begin_inset Formula $\Vert x\Vert^{2}=\Vert y\Vert^{2}+\Vert z\Vert^{2}=\Vert P_{M}(x)\Vert^{2}+\Vert P_{M^{\bot}}(x)\Vert^{2}$ +\end_inset + + y +\begin_inset Formula $\Vert P_{M}(x)\Vert,\Vert P_{M^{\bot}}(x)\Vert\leq\Vert x\Vert=1$ +\end_inset + +, lo que prueba la continuidad y por tanto que +\begin_inset Formula $M$ +\end_inset + + es topológica. + Además, si +\begin_inset Formula $M\neq0$ +\end_inset + +, existe +\begin_inset Formula $y\in S_{M}$ +\end_inset + + y +\begin_inset Formula $\Vert P_{M}(y)\Vert=\Vert y\Vert=1$ +\end_inset + +, luego +\begin_inset Formula $\Vert P_{M}\Vert=1$ +\end_inset + +, y análogamente para +\begin_inset Formula $M^{\bot}$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +\begin_inset Formula $P_{M}(H)=M$ +\end_inset + +, +\begin_inset Formula $\ker P_{M}=M^{\bot}$ +\end_inset + + y +\begin_inset Formula $P_{M^{\bot}}=1_{H}-P_{M}$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Para +\begin_inset Formula $x,y\in H$ +\end_inset + +, +\begin_inset Formula $\langle P_{M}(x),y\rangle=\langle x,P_{M}(y)\rangle$ +\end_inset + + y +\begin_inset Formula $\langle P_{M^{\bot}}(x),y\rangle=\langle x,P_{M^{\bot}}(y)\rangle$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +Si +\begin_inset Formula $x=x_{1}+x_{2}$ +\end_inset + + e +\begin_inset Formula $y=y_{1}+y_{2}$ +\end_inset + + con +\begin_inset Formula $x_{1},y_{1}\in M$ +\end_inset + + y +\begin_inset Formula $x_{2},y_{2}\in M^{\bot}$ +\end_inset + +, +\begin_inset Formula $\langle P_{M}(x),y\rangle=\langle x_{1},y_{1}+y_{2}\rangle=\langle x_{1},y_{1}\rangle=\langle x_{1}+x_{2},y_{1}\rangle=\langle x,P_{M}(y)\rangle$ +\end_inset + +, y para +\begin_inset Formula $P_{M^{\bot}}$ +\end_inset + + es análogo. +\end_layout + +\end_deeper +\begin_layout Enumerate +\begin_inset Formula $M^{\bot\bot}=M$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +Si +\begin_inset Formula $x\in M$ +\end_inset + +, para +\begin_inset Formula $y\in M^{\bot}$ +\end_inset + +, +\begin_inset Formula $\langle y,x\rangle=\overline{\langle x,y\rangle}=0$ +\end_inset + +, luego +\begin_inset Formula $x\in M^{\bot\bot}$ +\end_inset + +. + Si +\begin_inset Formula $x\in M^{\bot\bot}\subseteq H$ +\end_inset + +, sean +\begin_inset Formula $y\in M$ +\end_inset + + y +\begin_inset Formula $z\in M^{\bot}$ +\end_inset + + con +\begin_inset Formula $x=y+z$ +\end_inset + +, +\begin_inset Formula $0=\langle x,z\rangle=\langle y,z\rangle+\langle z,z\rangle=\langle z,z\rangle=\Vert z\Vert^{2}$ +\end_inset + +, luego +\begin_inset Formula $z=0$ +\end_inset + + y +\begin_inset Formula $x\in M$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Standard +Esto no es cierto si +\begin_inset Formula $M$ +\end_inset + + no es cerrado ni si +\begin_inset Formula $H$ +\end_inset + + no es completo. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Un espacio normado es de Hilbert si y sólo si cada subespacio cerrado tiene + un complementario topológico. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Un subconjunto +\begin_inset Formula $S$ +\end_inset + + de un espacio normado +\begin_inset Formula $(X,\Vert\cdot\Vert)$ +\end_inset + + es +\series bold +total +\series default + si +\begin_inset Formula $\overline{\text{span}S}=X$ +\end_inset + +, y si +\begin_inset Formula $H$ +\end_inset + + es de Hilbert esto ocurre si y sólo si +\begin_inset Formula $S^{\bot}=0$ +\end_inset + +. +\end_layout + +\begin_layout Section +Dual de un espacio de Hilbert +\end_layout + +\begin_layout Standard + +\series bold +Teorema de Riesz-Fréchet: +\series default + Dados un espacio de Hilbert +\begin_inset Formula $H$ +\end_inset + + y un operador +\begin_inset Formula $f:H\to\mathbb{K}$ +\end_inset + +, +\begin_inset Formula $f$ +\end_inset + + es acotado si y sólo si existe +\begin_inset Formula $y\in H$ +\end_inset + + con +\begin_inset Formula $f=\langle\cdot,y\rangle$ +\end_inset + +, en cuyo caso +\begin_inset Formula $y$ +\end_inset + + es único y +\begin_inset Formula $\Vert f\Vert=\Vert y\Vert$ +\end_inset + +. +\end_layout + +\begin_layout Itemize +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +\begin_inset Formula $\implies]$ +\end_inset + + +\end_layout + +\end_inset + +Para la unicidad, si +\begin_inset Formula $f(x)=\langle x,y\rangle=\langle x,z\rangle$ +\end_inset + + para todo +\begin_inset Formula $x\in H$ +\end_inset + +, +\begin_inset Formula $\langle x,y-z\rangle=0$ +\end_inset + +, luego +\begin_inset Formula $y-z\bot H$ +\end_inset + + y, como +\begin_inset Formula $H^{\bot}=0$ +\end_inset + +, +\begin_inset Formula $y=z$ +\end_inset + +. + Para la existencia, si +\begin_inset Formula $f=0$ +\end_inset + + tomamos +\begin_inset Formula $y=0$ +\end_inset + +, y en otro caso, +\begin_inset Formula $Y\coloneqq\ker f$ +\end_inset + + es un subespacio cerrado de +\begin_inset Formula $H$ +\end_inset + + y por tanto +\begin_inset Formula $H=Y\oplus Y^{\bot}$ +\end_inset + +, con +\begin_inset Formula $\dim Y^{\bot}=\dim\text{Im}f=1$ +\end_inset + +. + Sea entonces +\begin_inset Formula $z\in Y^{\bot}$ +\end_inset + + unitario, la proyección ortogonal de un +\begin_inset Formula $x\in H$ +\end_inset + + sobre +\begin_inset Formula $Y^{\bot}$ +\end_inset + + es +\begin_inset Formula $\langle x,z\rangle z$ +\end_inset + +, luego +\begin_inset Formula $x-\langle x,z\rangle z\in Y$ +\end_inset + + y +\begin_inset Formula +\[ +f(x)=f(x-\langle x,z\rangle z+\langle x,z\rangle z)=f(\langle x,z\rangle z)=\langle x,z\rangle f(z)=\langle x,\overline{f(z)}z\rangle\eqqcolon\langle x,y\rangle. +\] + +\end_inset + +Para +\begin_inset Formula $x\in S_{H}$ +\end_inset + +, por la desigualdad de Cauchy-Schwartz, +\begin_inset Formula $\Vert f(x)\Vert^{2}=|\langle x,y\rangle|^{2}\leq\langle x,x\rangle\langle y,y\rangle=\Vert y\Vert^{2}$ +\end_inset + +, luego +\begin_inset Formula $\Vert f\Vert\leq\Vert y\Vert$ +\end_inset + +, pero +\begin_inset Formula $f(\frac{y}{\Vert y\Vert})=\frac{f(y)}{\Vert y\Vert}=\frac{\Vert y\Vert^{2}}{\Vert y\Vert}=\Vert y\Vert$ +\end_inset + +, luego +\begin_inset Formula $\Vert f\Vert=\Vert y\Vert$ +\end_inset + +. +\end_layout + +\begin_layout Itemize +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +\begin_inset Formula $\impliedby]$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula $f\coloneqq\langle\cdot,y\rangle$ +\end_inset + + es lineal, y es continua por el argumento anterior que prueba que +\begin_inset Formula $\Vert f\Vert=\Vert y\Vert$ +\end_inset + +. +\end_layout + +\begin_layout Standard +El teorema no es válido si +\begin_inset Formula $H$ +\end_inset + + no es completo. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Sean +\begin_inset Formula $H$ +\end_inset + + un espacio de Hilbert y +\begin_inset Formula $T:H^{*}\to H$ +\end_inset + + que a cada +\begin_inset Formula $f$ +\end_inset + + le asocia el +\begin_inset Formula $y$ +\end_inset + + con +\begin_inset Formula $f=\langle\cdot,y\rangle$ +\end_inset + +: +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $T$ +\end_inset + + es biyectiva, isométrica y lineal conjugada. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $H^{*}$ +\end_inset + + es un espacio de Hilbert con el producto escalar +\begin_inset Formula $\langle f,g\rangle^{*}\coloneqq\langle T(g),T(f)\rangle$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $J:H\to H^{**}$ +\end_inset + + dada por +\begin_inset Formula $J(x)(f)\coloneqq f(x)$ +\end_inset + + es un isomorfismo algebraico isométrico. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Dado un un +\begin_inset Formula $\mathbb{K}$ +\end_inset + +-espacio vectorial +\begin_inset Formula $X$ +\end_inset + +, +\begin_inset Formula $B:X\times X\to\mathbb{K}$ +\end_inset + + es +\series bold +bilineal +\series default + si las +\begin_inset Formula $B(\cdot,y)$ +\end_inset + + y +\begin_inset Formula $B(x,\cdot)$ +\end_inset + + son lineales, +\series bold +sesquilineal +\series default + si las +\begin_inset Formula $B(\cdot,y)$ +\end_inset + + son lineales y las +\begin_inset Formula $B(x,\cdot)$ +\end_inset + + son lineales conjugadas, +\series bold +simétrica +\series default + si +\begin_inset Formula $B(x,y)\equiv B(y,x)$ +\end_inset + + y +\series bold +positiva +\series default + si +\begin_inset Formula $\forall x\in X,B(x,x)\geq0$ +\end_inset + +. + Si además +\begin_inset Formula $X$ +\end_inset + + es normado, +\begin_inset Formula $B$ +\end_inset + + es +\series bold +acotada +\series default + si +\begin_inset Formula $\exists M>0:\forall x,y\in X,|B(x,y)|\leq M\Vert x\Vert\Vert y\Vert$ +\end_inset + +, y es +\series bold +fuertemente positiva +\series default + si +\begin_inset Formula $\exists c>0:\forall x\in X,B(x,x)\geq c\Vert x\Vert^{2}$ +\end_inset + +. + +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $B$ +\end_inset + + es bilineal o sesquilineal, es acotada si y sólo si es continua, y para + todo +\begin_inset Formula $x$ +\end_inset + + e +\begin_inset Formula $y$ +\end_inset + + es +\begin_inset Formula $2B(x,x)+2B(y,y)=B(x+y,x+y)+B(x-y,x-y)$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\series bold +Teorema de Lax-Milgram: +\series default + Sean +\begin_inset Formula $H$ +\end_inset + + un espacio de Hilbert y +\begin_inset Formula $B$ +\end_inset + + una +\begin_inset Formula $H$ +\end_inset + +-forma sesquilineal acotada y fuertemente positiva, existe un único isomorfismo + de espacios de Hilbert +\begin_inset Formula $T:H\to H$ +\end_inset + + tal que +\begin_inset Formula $\forall x,y\in H,B(x,y)=\langle x,T(y)\rangle$ +\end_inset + +. + +\series bold +Demostración: +\series default + Sea +\begin_inset Formula +\[ +Y\coloneqq\{y\in H\mid\exists z\in H:\langle\cdot,y\rangle=B(\cdot,z)\}, +\] + +\end_inset + + +\begin_inset Formula $0\in Y$ +\end_inset + + tomando +\begin_inset Formula $z=0$ +\end_inset + + y +\begin_inset Formula $z$ +\end_inset + + está unívocamente determinado por +\begin_inset Formula $y$ +\end_inset + +, ya que si +\begin_inset Formula $\langle\cdot,y\rangle=B(\cdot,z)=B(\cdot,z')$ +\end_inset + + entonces +\begin_inset Formula $B(\cdot,z-z')=0$ +\end_inset + + y en particular +\begin_inset Formula $0=B(z-z',z-z')\geq c\Vert z-z'\Vert^{2}$ +\end_inset + + para cierto +\begin_inset Formula $c>0$ +\end_inset + + por ser +\begin_inset Formula $B$ +\end_inset + + fuertemente positiva, luego +\begin_inset Formula $z=z'$ +\end_inset + +. + Como +\begin_inset Formula $\langle\cdot,\cdot\rangle$ +\end_inset + + y +\begin_inset Formula $B$ +\end_inset + + son sesquilineales, +\begin_inset Formula $Y$ +\end_inset + + es un espacio vectorial y +\begin_inset Formula $S:Y\to H$ +\end_inset + + que a cada +\begin_inset Formula $y$ +\end_inset + + le asocia el +\begin_inset Formula $z$ +\end_inset + + con +\begin_inset Formula $\langle\cdot,y\rangle=B(\cdot,z)$ +\end_inset + + es lineal. + Entonces, para +\begin_inset Formula $y\in S_{Y}$ +\end_inset + +, +\begin_inset Formula +\[ +c\Vert S(y)\Vert^{2}\leq B(S(y),S(y))=\langle S(y),y\rangle\in\mathbb{R}^{+}, +\] + +\end_inset + +pero por la desigualdad de Cauchy-Schwartz, +\begin_inset Formula $\langle S(y),y\rangle^{2}=|\langle S(y),y\rangle|^{2}\leq\Vert S(y)\Vert^{2}\Vert y\Vert^{2}$ +\end_inset + +, luego +\begin_inset Formula $c\Vert S(y)\Vert^{2}\leq\langle S(y),y\rangle\leq\Vert S(y)\Vert\Vert y\Vert=\Vert S(y)\Vert$ +\end_inset + + y +\begin_inset Formula $\Vert S(y)\Vert\leq\frac{1}{c}$ +\end_inset + +, con lo que +\begin_inset Formula $S$ +\end_inset + + es continua. + Entonces, si +\begin_inset Formula $\{y_{n}\}_{n}\subseteq Y$ +\end_inset + + y existe +\begin_inset Formula $\lim_{n}y_{n}\eqqcolon y\in H$ +\end_inset + +, por continuidad de +\begin_inset Formula $S$ +\end_inset + + y de +\begin_inset Formula $B$ +\end_inset + +, +\begin_inset Formula +\[ +\langle x,y\rangle=\lim_{n}\langle x,y_{n}\rangle=\lim_{n}B(x,S(y_{n}))=B(x,S(y)), +\] + +\end_inset + +luego +\begin_inset Formula $y\in Y$ +\end_inset + + e +\begin_inset Formula $Y$ +\end_inset + + es cerrado. + Entonces, si +\begin_inset Formula $z\in Y^{\bot}$ +\end_inset + +, como +\begin_inset Formula $B(\cdot,z):H\to\mathbb{K}$ +\end_inset + + es continua, por el teorema de Riesz-Fréchet existe +\begin_inset Formula $w\in H$ +\end_inset + + con +\begin_inset Formula $B(\cdot,z)=\langle\cdot,w\rangle$ +\end_inset + +, luego +\begin_inset Formula $w\in Y$ +\end_inset + +, pero entonces +\begin_inset Formula $B(z,z)=\langle z,w\rangle=0$ +\end_inset + + y, por ser +\begin_inset Formula $B$ +\end_inset + + fuertemente positiva, +\begin_inset Formula $z=0$ +\end_inset + +, luego +\begin_inset Formula $Y^{\bot}=0$ +\end_inset + + e +\begin_inset Formula $Y=H$ +\end_inset + +. + Para +\begin_inset Formula $z\in H$ +\end_inset + +, como +\begin_inset Formula $B(\cdot,z)$ +\end_inset + + es continua, existe +\begin_inset Formula $w\in H$ +\end_inset + + con +\begin_inset Formula $B(\cdot z)=\langle\cdot,w\rangle$ +\end_inset + + y por tanto +\begin_inset Formula $z=S(w)$ +\end_inset + +, luego +\begin_inset Formula $S$ +\end_inset + + es suprayectiva. + Si +\begin_inset Formula $S(y)=0$ +\end_inset + +, para +\begin_inset Formula $x\in H$ +\end_inset + +, +\begin_inset Formula $\langle x,y\rangle=B(x,S(y))=0$ +\end_inset + + y por tanto +\begin_inset Formula $y=0$ +\end_inset + +, luego +\begin_inset Formula $S$ +\end_inset + + es inyectiva. + Por tanto +\begin_inset Formula $S$ +\end_inset + + es biyectiva y +\begin_inset Formula $T\coloneqq S^{-1}$ +\end_inset + + cumple +\begin_inset Formula $\langle x,T(y)\rangle=B(x,y)$ +\end_inset + +. + Además, para +\begin_inset Formula $y\in S_{H}$ +\end_inset + +, +\begin_inset Formula $\Vert T(y)\Vert^{2}=\langle T(y),T(y)\rangle=B(T(y),y)\leq M\Vert T(y)\Vert\Vert y\Vert=M\Vert T(y)\Vert$ +\end_inset + +, siendo +\begin_inset Formula $M$ +\end_inset + + una cota de +\begin_inset Formula $B$ +\end_inset + +, de donde +\begin_inset Formula $\Vert T\Vert\leq M$ +\end_inset + + y, como +\begin_inset Formula $\Vert T^{-1}\Vert=\Vert S\Vert\leq\frac{1}{c}$ +\end_inset + +, +\begin_inset Formula $T$ +\end_inset + + es un isomorfismo topológico isométrico. +\end_layout + +\begin_layout Standard +En particular, dado un espacio vectorial +\begin_inset Formula $H$ +\end_inset + + con dos productos escalares +\begin_inset Formula $\langle\cdot,\cdot\rangle_{1}$ +\end_inset + + y +\begin_inset Formula $\langle\cdot,\cdot\rangle_{2}$ +\end_inset + + equivalentes que hacen a +\begin_inset Formula $H$ +\end_inset + + completo, existe un isomorfismo +\begin_inset Formula $T:H\to H$ +\end_inset + + de espacios de Hilbert con +\begin_inset Formula $\langle x,y\rangle_{1}=\langle x,T(y)\rangle_{2}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Dado un espacio medible +\begin_inset Formula $(\Omega,\Sigma)$ +\end_inset + + con medidas +\begin_inset Formula $\mu$ +\end_inset + + y +\begin_inset Formula $\nu$ +\end_inset + +, +\begin_inset Formula $\nu$ +\end_inset + + es +\series bold +absolutamente continua +\series default + respecto de +\begin_inset Formula $\mu$ +\end_inset + + si +\begin_inset Formula $\forall A\in\Sigma,(\mu(A)=0\implies\nu(A)=0)$ +\end_inset + +, y es +\series bold +finita +\series default + si +\begin_inset Formula $\nu(\Omega)<\infty$ +\end_inset + +. + +\series bold +Teorema de Radon-Nicodym: +\series default + Si +\begin_inset Formula $(\Omega,\Sigma)$ +\end_inset + + es un espacio medible con medidas finitas +\begin_inset Formula $\mu$ +\end_inset + + y +\begin_inset Formula $\nu$ +\end_inset + + siendo +\begin_inset Formula $\nu$ +\end_inset + + absolutamente continua respecto de +\begin_inset Formula $\mu$ +\end_inset + +, existe +\begin_inset Formula $g:\Omega\to[0,+\infty]$ +\end_inset + + +\begin_inset Formula $\mu$ +\end_inset + +-integrable tal que +\begin_inset Formula +\[ +\forall A\in\Sigma,\nu(A)=\int_{A}g\dif\mu. +\] + +\end_inset + + +\series bold +Demostración: +\series default + +\begin_inset Formula $\sigma\coloneqq\mu+\nu$ +\end_inset + + es una medida finita en +\begin_inset Formula $X$ +\end_inset + + tal que +\begin_inset Formula $\forall A\in\Sigma,(\sigma(A)=0\iff\mu(A)=0)$ +\end_inset + +, y la función lineal entre espacios de Hilbert +\begin_inset Formula $T:L^{2}(\Omega,\Sigma,\sigma)\to\mathbb{R}$ +\end_inset + + dada por +\begin_inset Formula +\[ +Tu\coloneqq\int_{\Omega}u\dif\mu +\] + +\end_inset + +está bien definida y es continua porque, si +\begin_inset Formula $\Vert u\Vert_{L^{2}(\Omega,\Sigma,\sigma)}=1$ +\end_inset + +, +\begin_inset Formula +\begin{align*} +|Tu| & =\left|\int_{\Omega}u\dif\mu\right|\leq\int_{\Omega}|u|\dif\mu\leq\sqrt{\int_{\Omega}|u|^{2}\dif\mu}+\sqrt{\int_{\Omega}\dif\mu}\leq\\ + & \leq\sqrt{\int_{\Omega}|u|^{2}\dif\mu+\int_{\Omega}|u|^{2}\dif\nu}+\sqrt{\int_{\Omega}\dif\mu+\int_{\Omega}\dif\nu}=1+\sqrt{\sigma(X)}. +\end{align*} + +\end_inset + +Por el teorema de representación de Riesz, existe +\begin_inset Formula $f\in L^{2}(\Omega,\Sigma,\sigma)$ +\end_inset + + tal que, para +\begin_inset Formula $u\in L^{2}(\Omega,\Sigma,\sigma)$ +\end_inset + +, +\begin_inset Formula +\[ +Tu=\int_{\Omega}u\dif\mu=\int_{\Omega}uf\dif\sigma, +\] + +\end_inset + +pero esta igualdad se da para cuando +\begin_inset Formula $u=\chi_{A}$ +\end_inset + + para cualquier +\begin_inset Formula $A\in{\cal F}$ +\end_inset + + y por linealidad para cualquier función +\begin_inset Formula $\Sigma$ +\end_inset + +-medible simple, y por el teorema de convergencia dominada también se da + para cualquier función +\begin_inset Formula $\Sigma$ +\end_inset + +-medible no negativa en casi todo punto. + Además, para +\begin_inset Formula $A\in\Sigma$ +\end_inset + +, +\begin_inset Formula +\[ +\mu(A)=\int_{\Omega}\chi_{A}f\dif\sigma=\int_{A}f\dif\sigma, +\] + +\end_inset + +de modo que +\begin_inset Formula $f$ +\end_inset + + es +\begin_inset Formula $\Sigma$ +\end_inset + +-medible y, haciendo +\begin_inset Formula $A=\{x\mid f(x)\leq0\}$ +\end_inset + + o +\begin_inset Formula $A=\{x\mid f(x)>1\}$ +\end_inset + +, vemos que +\begin_inset Formula $f(\omega)\in(0,1]$ +\end_inset + + para casi todo +\begin_inset Formula $\omega\in\Omega$ +\end_inset + +, de modo que +\begin_inset Formula $\frac{1}{g}$ +\end_inset + + es +\begin_inset Formula $\Sigma$ +\end_inset + +-medible no negativa en casi todo punto y, en casi todo punto, +\begin_inset Formula $\frac{1}{f}f=1$ +\end_inset + +, con lo que para +\begin_inset Formula $A\in\Sigma$ +\end_inset + +, +\begin_inset Formula +\[ +\int_{A}\frac{1}{f}\dif\mu=\int_{A}\dif\sigma\implies\nu(A)=\sigma(A)-\mu(A)=\int_{A}\left(\frac{1}{f}-1\right)\dif\mu\eqqcolon\int_{A}g\dif\mu. +\] + +\end_inset + + +\end_layout + +\begin_layout Section +Problemas variacionales cuadráticos +\end_layout + +\begin_layout Standard + +\series bold +Teorema principal de los problemas variacionales cuadráticos: +\series default + Sean +\begin_inset Formula $H$ +\end_inset + + un +\begin_inset Formula $\mathbb{R}$ +\end_inset + +-espacio de Hilbert, +\begin_inset Formula $B$ +\end_inset + + una +\begin_inset Formula $H$ +\end_inset + +-forma bilineal simétrica, acotada y fuertemente positiva, +\begin_inset Formula $b$ +\end_inset + + una +\begin_inset Formula $H$ +\end_inset + +-forma lineal continua y +\begin_inset Formula $F:H\to\mathbb{R}$ +\end_inset + + dada por +\begin_inset Formula +\[ +F(x)\coloneqq\frac{1}{2}B(x,x)-b(x), +\] + +\end_inset + +entonces: +\end_layout + +\begin_layout Enumerate +Para +\begin_inset Formula $w\in H$ +\end_inset + +, +\begin_inset Formula $F$ +\end_inset + + alcanza su mínimo en +\begin_inset Formula $w$ +\end_inset + + si y sólo si +\begin_inset Formula $\forall y\in H,B(w,y)=b(y)$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +\begin_inset Formula $\implies]$ +\end_inset + + +\end_layout + +\end_inset + +Fijado +\begin_inset Formula $y\in H$ +\end_inset + +, para +\begin_inset Formula $t\in\mathbb{R}$ +\end_inset + + +\begin_inset Formula +\begin{align*} +F(w+ty) & =\frac{1}{2}B(w+ty,w+ty)-b(w+ty)=\\ + & =\frac{1}{2}(B(w,w)+2tB(w,y)+t^{2}B(y,y))-b(w)-tb(y)=\\ + & =F(w)+t(B(w,y)-b(y))+\frac{1}{2}t^{2}B(y,y), +\end{align*} + +\end_inset + +pero por hipótesis +\begin_inset Formula $F(w)\leq F(w+ty)$ +\end_inset + + para todo +\begin_inset Formula $t\in\mathbb{R}$ +\end_inset + +, luego +\begin_inset Formula $\varphi:\mathbb{R}\to\mathbb{R}$ +\end_inset + + dada por +\begin_inset Formula $\varphi(t)\coloneqq F(w+ty)$ +\end_inset + + tiene un mínimo en +\begin_inset Formula $t=0$ +\end_inset + + y +\begin_inset Formula $0=\varphi'(0)=B(w,y)-b(y)$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +\begin_inset Formula $\impliedby]$ +\end_inset + + +\end_layout + +\end_inset + +Para +\begin_inset Formula $y\in H$ +\end_inset + + y +\begin_inset Formula $t\in\mathbb{R}$ +\end_inset + +, +\begin_inset Formula +\[ +F(w+ty)=F(w)+\cancel{t(B(w,y)-b(y))}^{=0}+\frac{1}{2}t^{2}B(y,y)\geq F(w). +\] + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Enumerate +Existe un único +\begin_inset Formula $w\in H$ +\end_inset + + en el que +\begin_inset Formula $F$ +\end_inset + + alcanza su mínimo. +\end_layout + +\begin_deeper +\begin_layout Standard +Como +\begin_inset Formula $B$ +\end_inset + + es bilineal, simétrica y fuertemente positiva, es un producto escalar sobre + +\begin_inset Formula $H$ +\end_inset + +, y como existen +\begin_inset Formula $c,M>0$ +\end_inset + + con +\begin_inset Formula $c\Vert x\Vert^{2}\leq B(x,x)\leq M\Vert x\Vert^{2}$ +\end_inset + +, el producto escalar +\begin_inset Formula $B$ +\end_inset + + es equivalente al de +\begin_inset Formula $H$ +\end_inset + +, luego +\begin_inset Formula $b$ +\end_inset + + es continua con el producto escalar +\begin_inset Formula $B$ +\end_inset + + y por el teorema de Riesz-Fréchet existe un único +\begin_inset Formula $w\in H$ +\end_inset + + con +\begin_inset Formula $b=B(\cdot,w)=B(w,\cdot)$ +\end_inset + +, que es la condición del primer apartado. +\end_layout + +\end_deeper +\begin_layout Section +Convolución y aproximación de funciones +\end_layout + +\begin_layout Standard +Dado un abierto +\begin_inset Formula $\Omega\subseteq\mathbb{R}^{n}$ +\end_inset + +, +\begin_inset Formula $f:\mathbb{R}^{n}\to\mathbb{R}$ +\end_inset + + es +\series bold +localmente integrable +\series default + si +\begin_inset Formula $|f|$ +\end_inset + + es integrable en todo compacto +\begin_inset Formula $K\subseteq\Omega$ +\end_inset + +. + Dadas dos funciones localmente integrables +\begin_inset Formula $f,g:\mathbb{R}^{n}\to\mathbb{R}$ +\end_inset + +, definimos su +\series bold +producto de convolución +\series default + como +\begin_inset Formula $(f*g):D\to\mathbb{R}$ +\end_inset + + dada por +\begin_inset Formula +\[ +(f*g)(a)\coloneqq\int_{\mathbb{R}^{n}}f(x)g(a-x)\dif x, +\] + +\end_inset + +donde +\begin_inset Formula $D\coloneqq\{a\in\mathbb{R}^{n}\mid x\mapsto f(x)g(a-x)\text{ integrable}\}$ +\end_inset + +. + Si +\begin_inset Formula $f,g\in L^{2}(\mathbb{R}^{n})$ +\end_inset + +, +\begin_inset Formula $f*g$ +\end_inset + + está definida en todo +\begin_inset Formula $\mathbb{R}^{n}$ +\end_inset + + y es continua y uniformemente acotada con +\begin_inset Formula +\[ +\Vert f*g\Vert_{\infty}\leq\Vert f\Vert_{2}\Vert g\Vert_{2}. +\] + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + +El producto de convolución es conmutativo, y si +\begin_inset Formula $f*g$ +\end_inset + + está definida en casi todo punto, +\begin_inset Formula $\text{sop}(f*g)\subseteq\overline{\text{sop}(f)+\text{sop}(g)}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Una +\series bold +sucesión de Dirac +\series default + es una sucesión +\begin_inset Formula $(K_{m}:\mathbb{R}^{n}\to\mathbb{R}^{\geq0})_{m}$ +\end_inset + + de funciones continuas con +\begin_inset Formula +\[ +\int_{\mathbb{R}^{n}}K_{n}=1 +\] + +\end_inset + +y tal que +\begin_inset Formula +\[ +\forall\varepsilon,\delta>0,\exists n_{0}:\forall n\geq n_{0},\int_{\mathbb{R}^{n}\setminus B(0,\delta)}K_{n}(x)\dif x<\varepsilon. +\] + +\end_inset + +Por ejemplo, si +\begin_inset Formula $K:\mathbb{R}^{n}\to\mathbb{R}$ +\end_inset + + es continua, no negativa, con soporte compacto e integral 1, entonces +\begin_inset Formula $(x\mapsto m^{n}K(mx))_{m\geq1}$ +\end_inset + + es una sucesión de Dirac. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Las sucesiones de Dirac aproximan la +\series bold +delta de Dirac +\series default +, una +\begin_inset Quotes cld +\end_inset + +función extendida +\begin_inset Quotes crd +\end_inset + + con integral 1 que vale 0 en todo punto salvo en el origen en que el valor + es infinito. +\end_layout + +\begin_layout Standard +Como +\series bold +teorema +\series default +, si +\begin_inset Formula $f:\mathbb{R}^{n}\to\mathbb{R}$ +\end_inset + + es continua y acotada, la sucesión +\begin_inset Formula $(f*K_{m})_{m}$ +\end_inset + + tiende uniformemente a +\begin_inset Formula $f$ +\end_inset + + sobre subconjuntos compactos de +\begin_inset Formula $\mathbb{R}^{n}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $f:\mathbb{R}^{n}\to\mathbb{R}$ +\end_inset + + es localmente integrable y +\begin_inset Formula $g\in{\cal D}^{k}(\mathbb{R}^{n})$ +\end_inset + +, +\begin_inset Formula $f*g\in{\cal C}^{k}(\mathbb{R}^{n})$ +\end_inset + + y para +\begin_inset Formula $\alpha\in\mathbb{N}^{n}$ +\end_inset + + con +\begin_inset Formula $\sum_{i}\alpha_{i}\leq k$ +\end_inset + + es +\begin_inset Formula +\[ +\frac{\partial^{|\alpha|}(f*g)}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{n}}}=f*\left(\frac{\partial^{|\alpha|}g}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{n}}}\right), +\] + +\end_inset + +con lo que +\begin_inset Formula $f*g$ +\end_inset + + es una regularización de +\begin_inset Formula $f$ +\end_inset + + a través de una función suave +\begin_inset Formula $g$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Como +\series bold +teorema +\series default +, dado un abierto +\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ +\end_inset + +, +\begin_inset Formula ${\cal D}(G)$ +\end_inset + + es denso en +\begin_inset Formula $(C_{c}(G),\Vert\cdot\Vert_{\infty})$ +\end_inset + + y en +\begin_inset Formula $L^{p}(G)$ +\end_inset + + para todo +\begin_inset Formula $p\in[1,\infty)$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Para +\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ +\end_inset + + abierto y +\begin_inset Formula $f\in L^{2}(G)$ +\end_inset + +, si para todo +\begin_inset Formula $\psi\in{\cal D}(G)$ +\end_inset + + es +\begin_inset Formula +\[ +\int_{G}f\psi=0 +\] + +\end_inset + +entonces +\begin_inset Formula $f=0$ +\end_inset + + en casi todo punto, y en particular, si +\begin_inset Formula $f$ +\end_inset + + es continua, +\begin_inset Formula $f=0$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Section +Principio de Dirichlet +\end_layout + +\begin_layout Standard +Dado un abierto +\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ +\end_inset + +, +\begin_inset Formula $u\in{\cal D}^{2}(G)$ +\end_inset + + es +\series bold +armónica +\series default + en +\begin_inset Formula $G$ +\end_inset + + si +\begin_inset Formula $\triangle u\coloneqq\nabla^{2}u=0$ +\end_inset + + en todo punto de +\begin_inset Formula $G$ +\end_inset + +. + Dada +\begin_inset Formula $g\in{\cal C}(S_{\mathbb{C}})$ +\end_inset + +, el +\series bold +problema de Dirichlet +\series default + consiste en encontrar +\begin_inset Formula $u\in{\cal D}^{2}(\overline{B_{X}})$ +\end_inset + + armónica con +\begin_inset Formula $u|_{S_{\mathbb{C}}}=g$ +\end_inset + +. + Para un abierto +\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ +\end_inset + +, llamamos +\begin_inset Formula ${\cal C}^{m}(\overline{G})$ +\end_inset + + al conjunto de funciones +\begin_inset Formula $u:\overline{G}\to\mathbb{R}$ +\end_inset + + con +\begin_inset Formula $u|_{G}\in{\cal C}^{m}(G)$ +\end_inset + + para las que las derivadas parciales de orden +\begin_inset Formula $m$ +\end_inset + + de +\begin_inset Formula $u$ +\end_inset + + en +\begin_inset Formula $G$ +\end_inset + + admiten prolongación continua a +\begin_inset Formula $\overline{G}$ +\end_inset + +. + Escribimos +\begin_inset Formula $\partial_{j}u\coloneqq\frac{\partial u}{\partial j}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{samepage} +\end_layout + +\end_inset + +Dados un abierto +\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ +\end_inset + + acotado y no vacío, +\begin_inset Formula $f:G\to\mathbb{R}$ +\end_inset + + y +\begin_inset Formula $g:\partial G\to\mathbb{R}$ +\end_inset + +, el +\series bold +problema de valores frontera para la ecuación de Poisson +\series default + consiste en encontrar +\begin_inset Formula $u:\overline{G}\to\mathbb{R}$ +\end_inset + + tal que +\begin_inset Formula $-\triangle u|_{G}=f$ +\end_inset + + y +\begin_inset Formula $u|_{\partial G}=g$ +\end_inset + +, y el +\series bold +problema generalizado de valores frontera +\series default + consiste en encontrar +\begin_inset Formula $u:\overline{G}\to\mathbb{R}$ +\end_inset + + con +\begin_inset Formula $u|_{\partial G}=g$ +\end_inset + + y +\begin_inset Formula +\[ +\forall v\in{\cal D}(G),\int_{G}\sum_{j=1}^{n}\frac{\partial u}{\partial x_{j}}\frac{\partial v}{\partial x_{j}}\dif x\int_{G}fv. +\] + +\end_inset + + +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{samepage} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ +\end_inset + + es un abierto acotado no vacío, +\begin_inset Formula $f\in{\cal C}(\overline{G})$ +\end_inset + + y +\begin_inset Formula $g\in{\cal C}(\partial G)$ +\end_inset + +: +\end_layout + +\begin_layout Enumerate +Una +\begin_inset Formula $w\in{\cal C}^{2}(\overline{G})$ +\end_inset + + es solución del problema de valores frontera para la ecuación de Poisson + y sólo si lo es del problema generalizado de valores frontera. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +Si +\begin_inset Formula $w\in{\cal C}^{2}(\overline{G})$ +\end_inset + + es solución del problema variacional consistente en encontrar el mínimo + de +\begin_inset Formula $F:\{u\in{\cal C}^{2}(\overline{G})\mid u|_{\partial G}=g\}\to\mathbb{R}$ +\end_inset + + dada por +\begin_inset Formula +\[ +F(u)\coloneqq\frac{1}{2}\int_{G}\sum_{j=1}^{n}(\partial_{j}u(x))^{2}\dif x-\int_{G}fu, +\] + +\end_inset + +entonces es solución de los dos problemas anteriores. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +El +\series bold +teorema de integración por partes en varias variables +\series default + afirma que, si +\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ +\end_inset + + es un abierto, +\begin_inset Formula $u\in{\cal C}^{1}(G)$ +\end_inset + + y +\begin_inset Formula $v\in{\cal D}(G)$ +\end_inset + +, +\begin_inset Formula +\[ +\int_{G}u\partial_{j}v=-\int_{G}(\partial_{j}u)v. +\] + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $G$ +\end_inset + + es un abierto de +\begin_inset Formula $\mathbb{R}^{n}$ +\end_inset + + y +\begin_inset Formula $u,w\in L^{2}(G)$ +\end_inset + +, +\begin_inset Formula $w$ +\end_inset + + es la +\series bold +derivada generalizada +\begin_inset Formula $j$ +\end_inset + +-ésima +\series default + de +\begin_inset Formula $u$ +\end_inset + +, +\begin_inset Formula $w=\partial_{j}u$ +\end_inset + +, si +\begin_inset Formula +\[ +\forall v\in{\cal D}(G),\int_{G}u\partial_{j}v=-\int_{G}wv, +\] + +\end_inset + +y para +\begin_inset Formula $\alpha\in\mathbb{N}^{n}$ +\end_inset + + llamamos +\begin_inset Formula $D^{\alpha}u\coloneqq\partial_{1}^{\alpha_{1}}\cdots\partial_{n}^{\alpha_{n}}u$ +\end_inset + +. + +\end_layout + +\begin_layout Standard +Para +\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ +\end_inset + + abierto, +\begin_inset Formula $k\in\mathbb{N}$ +\end_inset + + y +\begin_inset Formula $p\in[1,\infty)$ +\end_inset + +, llamamos +\series bold +espacio de Sobolev +\series default + a +\begin_inset Formula +\[ +W^{k,p}(G)\coloneqq\{u\in L^{p}(G)\mid\forall\alpha\in\mathbb{N}^{n},(|\alpha|\leq k\implies\exists D^{\alpha}f\in L^{p}(G))\}. +\] + +\end_inset + +Escribimos +\begin_inset Formula $W^{k}(G)\coloneqq W^{k,2}(G)$ +\end_inset + +, y generalmente consideramos el espacio de Sobolev +\begin_inset Formula $W^{1}(G)$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ +\end_inset + + es abierto, definimos la relación de equivalencia en +\begin_inset Formula $G\to\mathbb{R}$ +\end_inset + + como +\begin_inset Formula $f\sim g\iff\{x\in G\mid f(x)\neq g(x)\}\text{ es de medida nula}$ +\end_inset + +, y +\begin_inset Formula $\langle\cdot,\cdot\rangle_{1,2}:W^{1}(G)/\sim\to\mathbb{R}$ +\end_inset + + dada por +\begin_inset Formula +\[ +\langle\overline{u},\overline{v}\rangle_{1,2}\coloneqq\int_{G}\left(uv+\sum_{j}(\partial_{j}u)(\partial_{j}v)\right) +\] + +\end_inset + +es un producto escalar en +\begin_inset Formula $W^{1}(G)/\sim$ +\end_inset + + que lo convierte en un espacio de Hilbert. + Identificamos +\begin_inset Formula $W^{1}(G)$ +\end_inset + + con +\begin_inset Formula $W^{1}(G)/\sim$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Llamamos +\begin_inset Formula $H_{0}^{1}(G)$ +\end_inset + + al espacio de Hilbert obtenido como la clausura de +\begin_inset Formula ${\cal D}(G)$ +\end_inset + + en +\begin_inset Formula $W^{1}(G)$ +\end_inset + +, que en general es un subespacio propio de +\begin_inset Formula $W^{1}(G)$ +\end_inset + + pero es igual a +\begin_inset Formula $W^{1}(G)$ +\end_inset + + si +\begin_inset Formula $G=\mathbb{R}^{n}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ +\end_inset + + es un abierto acotado no vacío y +\begin_inset Formula $u\in W^{1}(G)$ +\end_inset + +, +\series bold + +\begin_inset Formula $u$ +\end_inset + + se anula en la frontera de +\begin_inset Formula $G$ +\end_inset + + en sentido generalizado +\series default +, +\begin_inset Formula $u=0$ +\end_inset + + en +\begin_inset Formula $\partial G$ +\end_inset + +, si +\begin_inset Formula $u\in H_{0}^{1}(G)$ +\end_inset + +, y para +\begin_inset Formula $f,g\in W^{1}(G)$ +\end_inset + +, +\series bold + +\begin_inset Formula $f=g$ +\end_inset + + en +\begin_inset Formula $\partial G$ +\end_inset + + en sentido generalizado +\series default + si +\begin_inset Formula $f-g\in H_{0}^{1}(G)$ +\end_inset + +. +\end_layout + +\begin_layout Standard + +\series bold +Desigualdad de Poincaré-Friedrichs: +\series default + Si +\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ +\end_inset + + es un abierto acotado no vacío, existe +\begin_inset Formula $C>0$ +\end_inset + + tal que para +\begin_inset Formula $u\in H_{0}^{1}(G)$ +\end_inset + +, +\begin_inset Formula +\[ +C\int_{G}u^{2}\leq\int_{G}\sum_{j=1}^{n}(\partial_{j}u)^{2}. +\] + +\end_inset + + +\series bold +Demostración: +\series default + Sean +\begin_inset Formula $R\coloneqq\prod_{i}[a_{i},b_{i}]$ +\end_inset + + con +\begin_inset Formula $G\subseteq R$ +\end_inset + + y +\begin_inset Formula $u\in{\cal D}(G)$ +\end_inset + +, y vemos +\begin_inset Formula $u$ +\end_inset + + como una función en +\begin_inset Formula $R$ +\end_inset + + que se anula fuera de +\begin_inset Formula $G$ +\end_inset + + y con valor indefinido en +\begin_inset Formula $\partial G$ +\end_inset + +, para +\begin_inset Formula $x\in R$ +\end_inset + +, por la desigualdad de Cauchy-Schwartz, +\begin_inset Formula +\begin{align*} +(u(x))^{2} & =\left(\int_{a_{n}}^{x_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)\dif t\right)^{2}\leq\left(\int_{a_{n}}^{x_{n}}\dif t\right)\left(\int_{a_{n}}^{x_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)^{2}\dif t\right)\leq\\ + & \leq(b_{n}-a_{n})\int_{a_{n}}^{b_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)^{2}\dif t, +\end{align*} + +\end_inset + +luego +\begin_inset Formula +\begin{align*} +\int_{G}u^{2} & =\int_{R}u^{2}\leq\int_{a_{1}}^{b_{1}}\cdots\int_{a_{n}}^{b_{n}}(b_{n}-a_{n})\int_{a_{n}}^{b_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)^{2}\dif t\dif x_{n}\cdots\dif x_{1}=\\ + & =(b_{n}-a_{n})^{2}\int_{a_{1}}^{b_{1}}\cdots\int_{a_{n}}^{b_{n}}\partial_{n}u(x_{1},\dots,x_{n-1},t)^{2}\dif t\dif x_{n-1}\cdots\dif x_{1}=\\ + & =(b_{n}-a_{n})^{2}\int_{R}(\partial_{n}u)^{2}\dif x\leq(b_{n}-a_{n})^{2}\int_{R}\sum_{j}(\partial_{j}u)^{2}\dif x=(b_{n}-a_{n})^{2}\int_{G}\sum_{j}(\partial_{j}u)^{2}\dif x. +\end{align*} + +\end_inset + +Para +\begin_inset Formula $u\in H_{0}^{1}(G)$ +\end_inset + +,existe una sucesión +\begin_inset Formula $\{u_{m}\}_{m}\subseteq{\cal D}(G)$ +\end_inset + + con +\begin_inset Formula $\lim_{m}\Vert u-u_{m}\Vert_{1,2}=0$ +\end_inset + + y por tanto +\begin_inset Formula $\lim_{m}\Vert u-u_{m}\Vert_{2}=\lim_{m}\Vert\partial_{j}u-\partial_{j}u_{m}\Vert_{2}=0$ +\end_inset + +, y tomando límites y usando que la norma +\begin_inset Formula $\Vert\cdot\Vert_{2}\leq\Vert\cdot\Vert_{1,2}$ +\end_inset + + y por tanto es continua en +\begin_inset Formula $W^{1}(G)$ +\end_inset + +, +\begin_inset Formula +\[ +C\int_{G}u^{2}-\int_{G}\sum_{j}(\partial_{j}u)^{2}=C\Vert u\Vert_{2}^{2}-\sum_{j}\Vert\partial_{j}u\Vert_{2}^{2}=\lim_{m}\left(C\Vert u_{m}\Vert_{2}^{2}-\sum_{j}\Vert\partial_{j}u_{m}\Vert_{2}^{2}\right)\leq0. +\] + +\end_inset + + +\end_layout + +\begin_layout Standard + +\series bold +Principio de Dirichlet: +\series default + Sean +\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ +\end_inset + + un abierto acotado no vacío, +\begin_inset Formula $f\in L^{2}(G)$ +\end_inset + + y +\begin_inset Formula $g\in W^{1}(G)$ +\end_inset + +, +\begin_inset Formula $F:\{u\in W^{1}(G)\mid u-g\in H_{0}^{1}(G)\}\to\mathbb{R}$ +\end_inset + + dada por +\begin_inset Formula +\[ +F(u)\coloneqq\frac{1}{2}\int_{G}\sum_{j=1}^{n}(\partial_{j}u)^{2}-\int_{G}fu +\] + +\end_inset + +alcanza su mínimo en un único punto, que es el único +\begin_inset Formula $u\in\text{Dom}f$ +\end_inset + + tal que +\begin_inset Formula +\[ +\forall v\in H_{0}^{1}(G),\int_{G}\sum_{j=1}^{n}(\partial_{j}u)(\partial_{j}v)=\int_{G}fv +\] + +\end_inset + +y la única solución en +\begin_inset Formula $\text{Dom}f$ +\end_inset + + del problema de valores frontera para la ecuación de Poisson +\begin_inset Formula $-\nabla^{2}u=f$ +\end_inset + +. +\end_layout + +\begin_layout Standard + +\series bold +Demostración: +\series default + Para +\begin_inset Formula $u,v\in W^{1}(G)$ +\end_inset + + definimos +\begin_inset Formula +\begin{align*} +B(u,v) & \coloneqq\int_{G}\sum_{j}(\partial_{j}u)(\partial_{j}v), & b_{0}(v) & \coloneqq\int_{G}fv, & b(v) & \coloneqq b_{0}(v)-B(v,g). +\end{align*} + +\end_inset + + +\begin_inset Formula $B$ +\end_inset + + es bilineal y simétrica, y es acotada porque +\begin_inset Formula +\[ +|B(u,v)|=\left|\sum_{j}\int_{G}(\partial_{j}u)(\partial_{j}v)\right|\leq\sum_{j}\left|\int_{G}(\partial_{j}u)(\partial_{j}v)\right|\leq\sum_{j}\Vert\partial_{j}u\Vert_{2}\Vert\partial_{j}v\Vert_{2}\leq n\Vert u\Vert_{1,2}\Vert v\Vert_{1,2}. +\] + +\end_inset + +Por la desigualdad de Poincaré-Friedrichs, existe +\begin_inset Formula $C>0$ +\end_inset + + tal que, para todo +\begin_inset Formula $v\in H$ +\end_inset + +, +\begin_inset Formula +\[ +C\int_{G}v^{2}\leq\int_{G}\sum_{j}(\partial_{j}v)^{2}, +\] + +\end_inset + +luego +\begin_inset Formula +\[ +C\Vert v\Vert_{1,2}^{2}=C\left(\int_{G}v^{2}+\sum_{j}(\partial_{j}v)^{2}\right)\leq(1+C)\int_{G}\sum_{j}(\partial_{j}v)^{2}=(1+C)B(v,v) +\] + +\end_inset + +y +\begin_inset Formula $B$ +\end_inset + + es fuertemente positiva. + Además, +\begin_inset Formula $b_{0}$ +\end_inset + + es lineal y es acotada por la desigualdad de Cauchy-Schwartz, y como además + +\begin_inset Formula $B$ +\end_inset + + es bilineal y acotada, +\begin_inset Formula $b_{0}$ +\end_inset + + es lineal acotada y se dan las condiciones del teorema principal de los + problemas variacionales cuadráticos. + Ahora bien, si +\begin_inset Formula $w\coloneqq u-g\in H_{0}^{1}(G)$ +\end_inset + +, +\begin_inset Formula +\begin{multline*} +\frac{1}{2}B(w,w)-b(w)=\frac{1}{2}\int_{G}\sum_{j}(\partial_{j}(u-g))^{2}-\int_{G}f(u-g)+\int_{G}\sum_{j}(\partial_{j}(u-g))(\partial_{j}(g))=\\ +=\frac{1}{2}\int_{G}\sum_{j}(\partial_{j}(u-g))(\partial_{j}(u+g))-\int_{G}f(u-g)=\\ +=\frac{1}{2}\int_{G}\sum_{j}(\partial_{j}u)^{2}-\int_{G}fu+\frac{1}{2}\int_{G}\sum_{j}(\partial_{j}g)^{2}+\int_{G}fg, +\end{multline*} + +\end_inset + +luego minimizar +\begin_inset Formula $F$ +\end_inset + + equivale a minimizar +\begin_inset Formula $\frac{1}{2}B(w,w)-b(w)$ +\end_inset + +, y además +\begin_inset Formula +\begin{multline*} +B(w,v)=b(v)\iff B(u,v)-B(g,v)=b_{0}(v)-B(v,g)\iff B(u,v)=b_{0}(v)\iff\\ +\iff\int_{G}\sum_{j}(\partial_{j}u)(\partial_{j}v)=\int_{G}fv. +\end{multline*} + +\end_inset + +Para la última parte, si +\begin_inset Formula $u_{0}$ +\end_inset + + cumple esta última fórmula para todo +\begin_inset Formula $v\in H_{0}^{1}(G)$ +\end_inset + +, por integración por partes, +\begin_inset Formula +\[ +0=\int_{G}\sum_{j}(\partial_{j}u_{0})(\partial_{j}v)-\int_{G}fv=-\int_{G}\sum_{j}(\partial_{j}\partial_{j}u_{0})v-\int_{G}fv=-\int_{G}(\nabla^{2}u_{0}+f)v, +\] + +\end_inset + +con lo que +\begin_inset Formula $(\nabla^{2}u_{0}+f)\bot H_{0}^{1}(G)$ +\end_inset + + y, como +\begin_inset Formula ${\cal D}(G)\subseteq H_{0}^{1}(G)$ +\end_inset + + es denso en +\begin_inset Formula $L^{2}(G)$ +\end_inset + +, +\begin_inset Formula $\nabla^{2}u_{0}+f=0$ +\end_inset + +. +\end_layout + +\begin_layout Section +Soluciones débiles +\end_layout + +\begin_layout Standard +Dados +\begin_inset Formula $k,n\in\mathbb{N}$ +\end_inset + + y +\begin_inset Formula $a_{\alpha}\in\mathbb{K}^{n}$ +\end_inset + + para cada +\begin_inset Formula $\alpha\in\mathbb{N}^{n}$ +\end_inset + + con +\begin_inset Formula $|\alpha|<k$ +\end_inset + +, un +\series bold +operador diferencial lineal de coeficientes constantes +\series default + es uno de la forma +\begin_inset Formula +\[ +L\coloneqq\sum_{|\alpha|\leq k}a_{\alpha}\left(\frac{\partial}{\partial x}\right)^{\alpha}\coloneqq\sum_{|\alpha|\leq k}a_{\alpha}\frac{\partial^{|\alpha|}}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{n}}}, +\] + +\end_inset + +y su +\series bold +operador adjunto +\series default + es +\begin_inset Formula +\[ +L^{*}\coloneqq\sum_{|\alpha|\leq k}(-1)^{|\alpha|}\overline{a_{\alpha}}\left(\frac{\partial}{\partial x}\right)^{\alpha}. +\] + +\end_inset + +Si +\begin_inset Formula $G\subseteq\mathbb{R}^{n}$ +\end_inset + + es abierto, +\begin_inset Formula $\varphi,\psi\in L^{2}(G)$ +\end_inset + + son de clase +\begin_inset Formula ${\cal C}^{k}$ +\end_inset + + y una de las dos tiene soporte compacto, entonces +\begin_inset Formula $\langle L\psi,\varphi\rangle=\langle\psi,L^{*}\varphi\rangle$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Así, si +\begin_inset Formula $G$ +\end_inset + + es un abierto en +\begin_inset Formula $\mathbb{R}^{n}$ +\end_inset + +, +\begin_inset Formula $f,u\in L^{2}(G)$ +\end_inset + + son de clase +\begin_inset Formula ${\cal C}^{k}$ +\end_inset + + y +\begin_inset Formula $Lu=f$ +\end_inset + +, entonces +\begin_inset Formula $\langle f,\psi\rangle=\langle u,L^{*}\psi\rangle$ +\end_inset + + para todo +\begin_inset Formula $\psi\in{\cal D}(G)$ +\end_inset + +. + Para +\begin_inset Formula $f\in L^{2}(G)$ +\end_inset + +, +\begin_inset Formula $u\in L^{2}(G)$ +\end_inset + + es +\series bold +solución débil +\series default + de la ecuación en derivadas parciales +\begin_inset Formula $Lu=f$ +\end_inset + + si para todo +\begin_inset Formula $\psi\in{\cal D}(G)$ +\end_inset + + es +\begin_inset Formula $\langle f,\psi\rangle=\langle u,L^{*}\psi\rangle$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $L=\od{}{x}$ +\end_inset + + y +\begin_inset Formula $u,f\in L^{2}((0,1))$ +\end_inset + +, +\begin_inset Formula $Lu=f$ +\end_inset + + en sentido débil si y sólo si existe +\begin_inset Formula $F:(0,1)\to\mathbb{R}$ +\end_inset + + absolutamente continua con +\begin_inset Formula $F=u$ +\end_inset + + y +\begin_inset Formula $F'=f$ +\end_inset + + para casi todo +\begin_inset Formula $x\in(0,1)$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +La ecuación de ondas en una dimensión, +\begin_inset Formula +\[ +\left\{ \begin{array}{rlrl} +\frac{\partial^{2}u}{\partial x^{2}}-\frac{\partial^{2}u}{\partial t^{2}} & =0, & t & \in[0,+\infty),\\ +u(x,0) & \equiv f(x), & x & \in[0,\pi],\\ +\frac{\partial u}{\partial t}(x,0) & \equiv0, +\end{array}\right. +\] + +\end_inset + +siendo +\begin_inset Formula $f:[0,\pi]\to\mathbb{R}$ +\end_inset + + una función lineal a trozos, admite soluciones débiles que no son soluciones + ordinarias. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\series bold +Teorema de Malgrange-Ehrenpreis: +\series default + Sean +\begin_inset Formula $G$ +\end_inset + + un abierto acotado de +\begin_inset Formula $\mathbb{R}^{n}$ +\end_inset + + y +\begin_inset Formula $L$ +\end_inset + + un operador en derivadas parciales lineal con coeficientes constantes, + existe un operador lineal continuo +\begin_inset Formula $K:L^{2}(G)\to L^{2}(G)$ +\end_inset + + tal que para todo +\begin_inset Formula $f\in L^{2}(G)$ +\end_inset + +, +\begin_inset Formula $u\coloneqq K(f)$ +\end_inset + + es solución débil de +\begin_inset Formula $Lu=f$ +\end_inset + +. +\end_layout + +\begin_layout Standard + +\series bold +Demostración: +\series default + Definimos +\begin_inset Formula $\langle\varphi,\psi\rangle_{L}\coloneqq\langle L^{*}\varphi,L^{*}\psi\rangle_{2}$ +\end_inset + +, y para ver que es un producto escalar sobre +\begin_inset Formula ${\cal D}(G)$ +\end_inset + + vemos que existe +\begin_inset Formula $C>0$ +\end_inset + + tal que, para +\begin_inset Formula $\psi\in{\cal D}(G)$ +\end_inset + +, +\begin_inset Formula $\Vert\psi\Vert_{2}\leq C\Vert L^{*}\psi\Vert_{2}$ +\end_inset + +. + Si +\begin_inset Formula $L^{*}=\frac{\partial}{\partial x_{1}}$ +\end_inset + +, llamando +\begin_inset Formula $\psi(x)\coloneqq0$ +\end_inset + + para +\begin_inset Formula $x\notin G$ +\end_inset + +, para +\begin_inset Formula $x\in G$ +\end_inset + +, como +\begin_inset Formula $\text{sop}\psi\subseteq G$ +\end_inset + + es compacto, sea +\begin_inset Formula $m\coloneqq\inf_{x\in G}x_{1}$ +\end_inset + +, +\begin_inset Formula +\begin{align*} +\psi(x)^{2} & =\left(\int_{m}^{x_{1}}\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\dif t\right)^{2}\leq\left(\int_{m}^{x_{1}}\left|\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\right|\cdot1\dif t\right)\leq\\ + & \leq\int_{m}^{x_{1}}\dif t\int_{m}^{x_{1}}\left|\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\right|^{2}\dif t\leq d\int_{m}^{x_{1}}\left|\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\right|^{2}, +\end{align*} + +\end_inset + +donde +\begin_inset Formula $d$ +\end_inset + + es el diámetro de +\begin_inset Formula $G$ +\end_inset + +, e integrando de nuevo, +\begin_inset Formula +\begin{align*} +\Vert\psi\Vert_{2}^{2} & =\int_{G}\psi(x)^{2}\dif x\leq d\int_{m}^{x_{1}}\int_{-\infty}^{x_{2}}\cdots\int_{-\infty}^{x_{n}}\int_{m}^{x_{1}}\left|\frac{\partial\psi}{\partial x_{1}}(t,x_{2},\dots,x_{n})\right|^{2}\dif t\dif x_{n}\cdots\dif x_{1}\leq\\ + & \leq d^{2}\int_{G}\left|\frac{\partial\psi}{\partial x_{1}}(x)\right|^{2}\dif x=d^{2}\Vert L^{*}\psi\Vert_{2}^{2}. +\end{align*} + +\end_inset + +Si +\begin_inset Formula $L^{*}=\frac{\partial}{\partial x_{i}}$ +\end_inset + + para otro +\begin_inset Formula $i$ +\end_inset + +, es análogo, y si +\begin_inset Formula $L^{*}=\left(\frac{\partial}{\partial x}\right)^{|\alpha|}$ +\end_inset + +, por inducción, +\begin_inset Formula $\Vert\psi\Vert_{2}\leq d^{|\alpha|}\Vert L^{*}\psi\Vert_{2}$ +\end_inset + +. + Para +\begin_inset Formula $L$ +\end_inset + + arbitrario basta hacer combinaciones lineales. + Visto esto, sean +\begin_inset Formula $H_{0}\coloneqq({\cal D}(G),\langle\cdot,\cdot\rangle_{L})$ +\end_inset + + y +\begin_inset Formula $H$ +\end_inset + + su compleción, +\begin_inset Formula $L^{*}:H_{0}\to L^{2}(G)$ +\end_inset + + es lineal y continuo y por tanto admite una extensión lineal y continua + +\begin_inset Formula $\hat{L}^{*}:H\to L^{2}(G)$ +\end_inset + +. + Sea ahora +\begin_inset Formula $f\in L^{2}(G)$ +\end_inset + + y +\begin_inset Formula $l_{0}:H_{0}\to\mathbb{K}$ +\end_inset + + dada por +\begin_inset Formula $l_{0}(\psi)\coloneqq\langle\psi,f\rangle_{2}$ +\end_inset + +, +\begin_inset Formula +\[ +|l_{0}(\psi)|=|\langle\psi,f\rangle_{2}|\leq\Vert\psi\Vert_{2}\Vert f\Vert_{2}\leq C\Vert f\Vert_{2}\Vert L^{*}\psi\Vert_{2}, +\] + +\end_inset + +donde +\begin_inset Formula $C$ +\end_inset + + es tal que +\begin_inset Formula $\Vert\psi\Vert_{2}\leq C\Vert L^{*}\psi\Vert_{2}$ +\end_inset + + para todo +\begin_inset Formula $C$ +\end_inset + +, de modo que +\begin_inset Formula $l_{0}$ +\end_inset + + es lineal continua por la cota +\begin_inset Formula $C\Vert f\Vert_{2}$ +\end_inset + + y se puede extender a una forma lineal y continua +\begin_inset Formula $l:H\to\mathbb{K}$ +\end_inset + + con +\begin_inset Formula $\Vert l\Vert\leq C\Vert f\Vert_{2}$ +\end_inset + +. + Por el teorema de Riesz, existe un único +\begin_inset Formula $\hat{u}\in H$ +\end_inset + + con +\begin_inset Formula $l(h)\equiv\langle h,\hat{u}\rangle_{L}$ +\end_inset + + para +\begin_inset Formula $h\in H$ +\end_inset + + y además +\begin_inset Formula $\Vert\hat{u}\Vert_{H}=\Vert l\Vert_{H}$ +\end_inset + +, y tomando +\begin_inset Formula $u\coloneqq\hat{L}^{*}\hat{u}$ +\end_inset + +, +\begin_inset Formula $l(h)=\langle\hat{L}^{*}h,\hat{L}^{*}\hat{u}\rangle=\langle\hat{L}^{*}h,u\rangle_{2}$ +\end_inset + +, pero para +\begin_inset Formula $\psi\in{\cal D}(G)$ +\end_inset + +, +\begin_inset Formula $l(\psi)=\langle\psi,f\rangle_{2}$ +\end_inset + + y +\begin_inset Formula $\hat{L}^{*}(\psi)=L^{*}\psi$ +\end_inset + +, con lo que +\begin_inset Formula $\langle L^{*}\psi,u\rangle_{2}=l(\psi)=\langle\psi,f\rangle_{2}$ +\end_inset + +, y basta llamar +\begin_inset Formula $K(f)\coloneqq u$ +\end_inset + +. + Para la continuidad de +\begin_inset Formula $K$ +\end_inset + +, +\begin_inset Formula +\[ +\Vert K(f)\Vert_{2}=\Vert u\Vert_{2}=\Vert\hat{L}^{*}\hat{u}\Vert_{2}=\Vert\hat{u}\Vert_{H}=\Vert l\Vert_{H}=\sup_{\Vert\psi\Vert_{H}=\Vert L^{*}\psi\Vert_{2}=1}|l(\psi)|\leq C\Vert f\Vert_{2}. +\] + +\end_inset + + +\end_layout + +\begin_layout Section +Método de Galerkin +\end_layout + +\begin_layout Standard +Sean +\begin_inset Formula $M_{1}\subseteq M_{2}\subseteq\dots\subseteq M_{n}\subseteq\dots$ +\end_inset + + una sucesión de subespacios cerrados de un espacio de Hilbert +\begin_inset Formula $H$ +\end_inset + + con unión densa en +\begin_inset Formula $H$ +\end_inset + +, +\begin_inset Formula $a:H\times H\to\mathbb{R}$ +\end_inset + + bilineal, simétrica, continua y fuertemente positiva, +\begin_inset Formula $b:H\to\mathbb{R}$ +\end_inset + + lineal continua, +\begin_inset Formula +\[ +J(x)\coloneqq\frac{1}{2}a(x,x)-b(x) +\] + +\end_inset + +para +\begin_inset Formula $x\in H$ +\end_inset + +, +\begin_inset Formula $u\in H$ +\end_inset + + con +\begin_inset Formula $J(u)$ +\end_inset + + mínimo y, para +\begin_inset Formula $n\in\mathbb{N}$ +\end_inset + +, +\begin_inset Formula $u_{n}\in M_{n}$ +\end_inset + + con +\begin_inset Formula $J(u_{n})$ +\end_inset + + mínimo, de modo que +\begin_inset Formula $a(x,u_{n})=b(x)$ +\end_inset + + para todo +\begin_inset Formula $x\in M_{n}$ +\end_inset + + y +\begin_inset Formula $a(x,u)=b(x)$ +\end_inset + + para todo +\begin_inset Formula $x\in H$ +\end_inset + +: +\end_layout + +\begin_layout Enumerate + +\series bold +Teorema de Galerkin-Ritz: +\series default + +\begin_inset Formula $\lim_{n}u_{n}=u$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +Para +\begin_inset Formula $x\in M_{n}$ +\end_inset + +, +\begin_inset Formula $a(x,u_{n})=b(x)$ +\end_inset + +, y para +\begin_inset Formula $x\in H$ +\end_inset + +, +\begin_inset Formula $a(x,u)=f(x)$ +\end_inset + +, luego +\begin_inset Formula $a(x,u-u_{n})=b(x)-b(x)=0$ +\end_inset + + para +\begin_inset Formula $x\in M_{n}$ +\end_inset + +. + Pero +\begin_inset Formula $a$ +\end_inset + + es un producto escalar equivalente al de +\begin_inset Formula $H$ +\end_inset + +, luego +\begin_inset Formula $u-u_{n}\bot M_{n}$ +\end_inset + + y, si +\begin_inset Formula $P_{n}:H\to M_{n}$ +\end_inset + + es la proyección ortogonal, +\begin_inset Formula $P_{n}(u)=u_{n}$ +\end_inset + +. + Por el teorema de la proyección, +\begin_inset Formula $\Vert u-u_{n}\Vert=\Vert u-P_{n}(u)\Vert=d(u,M_{n})$ +\end_inset + +, pero por la densidad es +\begin_inset Formula $d(u,\bigcup_{n}M_{n})=0$ +\end_inset + +, y para +\begin_inset Formula $\varepsilon>0$ +\end_inset + + existen +\begin_inset Formula $n_{0}\in\mathbb{N}$ +\end_inset + + e +\begin_inset Formula $y\in M_{n_{0}}$ +\end_inset + + con +\begin_inset Formula $\Vert u-y\Vert<\varepsilon$ +\end_inset + +, y como la sucesión es creciente, para +\begin_inset Formula $n\geq n_{0}$ +\end_inset + +, +\begin_inset Formula $\Vert u-u_{n}\Vert=d(u,M_{n})\leq d(u,M_{n_{0}})\leq\Vert u-y\Vert<\varepsilon$ +\end_inset + +, con lo que +\begin_inset Formula $\lim_{n}u_{n}=u$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +Dados +\begin_inset Formula $c,d>0$ +\end_inset + + con +\begin_inset Formula $a(x,y)\leq d\Vert x\Vert\Vert y\Vert$ +\end_inset + + y +\begin_inset Formula $c\Vert x\Vert^{2}\leq a(x,x)$ +\end_inset + + para todo +\begin_inset Formula $x,y\in H$ +\end_inset + +, +\begin_inset Formula $c\Vert u\Vert\leq\Vert b\Vert$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate + +\series bold +Razón de convergencia: +\series default + +\begin_inset Formula $\Vert u-u_{n}\Vert\leq\frac{d}{c}d(u,M_{n})$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate + +\series bold +Estimación del error: +\series default + Si +\begin_inset Formula $\beta\leq J(x)$ +\end_inset + + para todo +\begin_inset Formula $x\in H$ +\end_inset + +, para +\begin_inset Formula $n\in\mathbb{N}$ +\end_inset + + es +\begin_inset Formula $\frac{c}{2}\Vert u-u_{n}\Vert^{2}\leq J(u_{n})-\beta$ +\end_inset + +. +\end_layout + +\begin_layout Standard +El +\series bold +método de Galerkin +\series default + para resolver un problema de esta forma consiste en tomar en el teorema + anterior los +\begin_inset Formula $M_{n}$ +\end_inset + + de dimensión finita y resolver los sistemas de ecuaciones lineales resultantes, + con matriz de coeficientes simétrica y definida positiva de tamaño +\begin_inset Formula $\dim M_{n}$ +\end_inset + +. + Tomando adecuadamente las bases de los +\begin_inset Formula $M_{n}$ +\end_inset + + se puede conseguir que las matrices tengan muchas entradas nulas. +\end_layout + +\begin_layout Section +Bases hilbertianas +\end_layout + +\begin_layout Standard +Sean +\begin_inset Formula $(H_{i})_{i\in I}$ +\end_inset + + una familia de +\begin_inset Formula $\mathbb{K}$ +\end_inset + +-espacios de Hilbert, +\begin_inset Formula $H_{0}\coloneqq\prod_{i\in I}H_{i}$ +\end_inset + + y +\begin_inset Formula $\langle\cdot,\cdot\rangle:H_{0}\times H_{0}\to[0,+\infty]$ +\end_inset + + dada por +\begin_inset Formula +\[ +\langle x,y\rangle\coloneqq\sum_{i\in I}\langle x_{i},y_{i}\rangle_{H_{i}}, +\] + +\end_inset + +llamamos +\series bold +suma directa hilbertiana +\series default + o +\series bold +suma +\begin_inset Formula $\ell^{2}$ +\end_inset + + +\series default + de +\begin_inset Formula $\{H_{i}\}_{i\in I}$ +\end_inset + + al espacio de Hilbert +\begin_inset Formula +\[ +\bigoplus_{i\in I}H_{i}\coloneqq\ell^{2}((H_{i})_{i\in I})\coloneqq(\{x\in H_{0}\mid\langle x,x\rangle<\infty\},\langle\cdot,\cdot\rangle). +\] + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Cada +\begin_inset Formula $H_{i}$ +\end_inset + + es isométricamente isomorfo al subespacio de +\begin_inset Formula $H$ +\end_inset + + de los vectores con todas las coordenadas nulas salvo la +\begin_inset Formula $i$ +\end_inset + +, los +\begin_inset Formula $H_{i}$ +\end_inset + + son mutuamente ortogonales en +\begin_inset Formula $H$ +\end_inset + +, +\begin_inset Formula $H$ +\end_inset + + es la clausura lineal cerrada de los +\begin_inset Formula $H_{i}$ +\end_inset + + y cada +\begin_inset Formula $x\in H$ +\end_inset + + se puede expresar de forma única como +\begin_inset Formula $\sum_{i\in I}x_{i}$ +\end_inset + + con cada +\begin_inset Formula $x_{i}\in H_{i}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $H$ +\end_inset + + es un +\begin_inset Formula $\mathbb{K}$ +\end_inset + +-espacio de Hilbert y +\begin_inset Formula $(H_{i})_{i\in I}$ +\end_inset + + es una familia de subespacios cerrados de +\begin_inset Formula $H$ +\end_inset + + mutuamente ortogonales con +\begin_inset Formula $H=\overline{\text{span}\{H_{i}\}_{i\in I}}$ +\end_inset + +, entonces +\begin_inset Formula $H$ +\end_inset + + es isométricamente isomorfo a +\begin_inset Formula $\bigoplus_{i\in I}H_{i}$ +\end_inset + +, e identificamos +\begin_inset Formula $H$ +\end_inset + + con +\begin_inset Formula $\bigoplus_{i\in I}H_{i}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\series bold +Desigualdad de Bessel: +\series default + Sean +\begin_inset Formula $H$ +\end_inset + + un espacio prehilbertiano y +\begin_inset Formula $\{e_{i}\}_{i\in I}\subseteq H$ +\end_inset + + una familia ortonormal, para +\begin_inset Formula $x\in H$ +\end_inset + +, +\begin_inset Formula +\[ +\sum_{i\in I}|\langle x,e_{i}\rangle|^{2}\leq\Vert x\Vert^{2}. +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Para un conjunto +\begin_inset Formula $I$ +\end_inset + + arbitrario, llamamos +\begin_inset Formula $\ell^{2}(I)\coloneqq\bigoplus_{i\in I}\mathbb{K}$ +\end_inset + +. +\end_layout + +\begin_layout Standard + +\series bold +Teorema de la base hilbertiana: +\series default + Sean +\begin_inset Formula $H$ +\end_inset + + un espacio de Hilbert y +\begin_inset Formula $\{e_{i}\}_{i\in I}\subseteq H$ +\end_inset + + una familia ortonormal, +\begin_inset Formula $\{e_{i}\}_{i\in I}$ +\end_inset + + es ortonormal maximal (por inclusión) si y sólo si +\begin_inset Formula $\forall x\in H,(\forall i\in I,\langle x,e_{i}\rangle=0\implies x=0)$ +\end_inset + +, si y sólo si es un conjunto total, si y sólo si +\begin_inset Formula $\hat{}:H\to\ell^{2}(I)$ +\end_inset + + dada por +\begin_inset Formula $\hat{x}\coloneqq(\langle x,e_{i}\rangle)_{i\in I}$ +\end_inset + + es inyectiva, si y sólo si todo +\begin_inset Formula $x\in H$ +\end_inset + + admite un +\series bold +desarrollo de Fourier +\series default + +\begin_inset Formula $x=\sum_{i\in I}\langle x,e_{i}\rangle e_{i}$ +\end_inset + +, si y sólo si +\begin_inset Formula $\forall x,y\in H,\langle x,y\rangle=\sum_{i\in I}\langle x,e_{i}\rangle\overline{\langle y,e_{i}\rangle}$ +\end_inset + +, si y sólo si todo +\begin_inset Formula $x\in H$ +\end_inset + + cumple la +\series bold +identidad de Parseval +\series default +, +\begin_inset Formula $\Vert x\Vert^{2}=\sum_{i\in I}|\langle x,e_{i}\rangle|^{2}$ +\end_inset + +, y entonces decimos que +\begin_inset Formula $(e_{i})_{i\in I}$ +\end_inset + + es una +\series bold +base hilbertiana +\series default + de +\begin_inset Formula $H$ +\end_inset + + o un +\series bold +sistema ortonormal completo +\series default +. +\end_layout + +\begin_layout Description +\begin_inset Formula $1\implies2]$ +\end_inset + + Entonces +\begin_inset Formula $x\bot\{e_{i}\}_{i\in I}$ +\end_inset + +, por lo que si +\begin_inset Formula $x\neq0$ +\end_inset + +, +\begin_inset Formula $\{e_{i}\}_{i\in I}\cup\{x\}$ +\end_inset + + sería ortogonal. +\begin_inset Formula $\#$ +\end_inset + + +\end_layout + +\begin_layout Description +\begin_inset Formula $2\iff3]$ +\end_inset + + Sabemos que un +\begin_inset Formula $S\subseteq H$ +\end_inset + + es total si y sólo si +\begin_inset Formula $S^{\bot}=0$ +\end_inset + +. +\end_layout + +\begin_layout Description +\begin_inset Formula $2\iff4]$ +\end_inset + + Por ser +\begin_inset Formula $\hat{}$ +\end_inset + + lineal. +\end_layout + +\begin_layout Description +\begin_inset Formula $4\implies5]$ +\end_inset + + +\begin_inset Formula $\widehat{\sum_{i}\langle x,e_{i}\rangle e_{i}}=\sum_{i}\langle x,e_{i}\rangle\hat{e}_{i}=\sum_{i}\langle x,e_{i}\rangle e_{i}=\hat{x}$ +\end_inset + +, y por inyectividad +\begin_inset Formula $x=\sum_{i\in I}\langle x,e_{i}\rangle e_{i}$ +\end_inset + +. +\end_layout + +\begin_layout Description +\begin_inset Formula $5\implies6]$ +\end_inset + + +\begin_inset Formula $\langle x,y\rangle=\sum_{i,j\in I}\langle\langle x,e_{i}\rangle e_{i},\langle y,e_{j}\rangle e_{j}\rangle=\sum_{i,j\in I}\langle x,e_{i}\rangle\overline{\langle y,e_{j}\rangle}\langle e_{i},e_{j}\rangle=\sum_{i\in I}\langle x,e_{i}\rangle\overline{\langle y,e_{j}\rangle}$ +\end_inset + +. +\end_layout + +\begin_layout Description +\begin_inset Formula $6\implies7]$ +\end_inset + + Basta tomar +\begin_inset Formula $x=y$ +\end_inset + +. +\end_layout + +\begin_layout Description +\begin_inset Formula $7\implies1]$ +\end_inset + + Si fuera +\begin_inset Formula $\{e_{i}\}_{i}\subsetneq M\subseteq H$ +\end_inset + + con +\begin_inset Formula $M$ +\end_inset + + ortonormal, para +\begin_inset Formula $x\in M\setminus\{e_{i}\}_{i}$ +\end_inset + +, +\begin_inset Formula $1=\Vert x\Vert^{2}=\sum_{i\in I}|\langle x,e_{i}\rangle|^{2}=0\#$ +\end_inset + +. +\end_layout + +\begin_layout Standard + +\series bold +Primer teorema de Riesz-Fischer: +\series default + Si +\begin_inset Formula $H$ +\end_inset + + es un espacio prehilbertiano con una familia ortonormal +\begin_inset Formula $\{e_{i}\}_{i\in I}$ +\end_inset + + y +\begin_inset Formula $\hat{}:H\to\mathbb{K}^{I}$ +\end_inset + + viene dada por +\begin_inset Formula $\hat{x}\coloneqq(\langle x,e_{i}\rangle)_{i\in I}$ +\end_inset + +, +\begin_inset Formula $\hat{}$ +\end_inset + + es lineal y continua con imagen contenida en +\begin_inset Formula $\ell^{2}(I)$ +\end_inset + + e igual a +\begin_inset Formula $\ell^{2}(I)$ +\end_inset + + si +\begin_inset Formula $H$ +\end_inset + + es de Hilbert. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $H$ +\end_inset + + es un espacio de Hilbert, todo espacio ortonormal de vectores en +\begin_inset Formula $H$ +\end_inset + + se puede completar a una base hilbertiana de +\begin_inset Formula $H$ +\end_inset + +, y en particular todo espacio de Hilbert posee una base hilbertiana y es + isométricamente isomorfo a un +\begin_inset Formula $\ell^{2}(I)$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Los espacios de Hilbert +\begin_inset Formula $\ell^{2}(I)$ +\end_inset + + y +\begin_inset Formula $\ell^{2}(J)$ +\end_inset + + son topológicamente isomorfos si y sólo si +\begin_inset Formula $|I|=|J|$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Llamamos +\series bold +dimensión hilbertiana +\series default + de un espacio de Hilbert al cardinal de cualquier base hilbertiana. + +\series bold +Segundo teorema de Riesz-Fischer: +\series default + Si +\begin_inset Formula $H$ +\end_inset + + es de dimensión infinita, +\begin_inset Formula $\dim H=\aleph_{0}\coloneqq|\mathbb{N}|$ +\end_inset + + si y sólo si +\begin_inset Formula $H\cong\ell^{2}$ +\end_inset + +, si y sólo si +\begin_inset Formula $H$ +\end_inset + + es separable. +\end_layout + +\begin_layout Description +\begin_inset Formula $1\iff2]$ +\end_inset + + Por lo anterior. +\end_layout + +\begin_layout Description +\begin_inset Formula $2\implies3]$ +\end_inset + + Visto. +\end_layout + +\begin_layout Description +\begin_inset Formula $3\implies2]$ +\end_inset + + Dado +\begin_inset Formula $\{x_{n}\}_{n\in\mathbb{N}}\subseteq H$ +\end_inset + + denso, como +\begin_inset Formula $H$ +\end_inset + + es de dimensión infinita, existe una subsucesión +\begin_inset Formula $(x_{n_{k}})_{k}$ +\end_inset + + linealmente independiente de +\begin_inset Formula $(x_{n})_{n}$ +\end_inset + + con +\begin_inset Formula $\text{span}\{x_{n}\}_{n}=\text{span}\{x_{n_{k}}\}_{k}$ +\end_inset + +, luego +\begin_inset Formula $\overline{\text{span}\{x_{n_{k}}\}_{k}}=H$ +\end_inset + + y el proceso de ortonormalización de Gram-Schmidt nos da una base hilbertiana + numerable de +\begin_inset Formula $H$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Así, si +\begin_inset Formula $Z\leq_{\mathbb{K}}\ell^{2}$ +\end_inset + + es cerrado de dimensión infinita, +\begin_inset Formula $Z\cong\ell^{2}$ +\end_inset + +. +\end_layout + +\begin_layout Section +Aproximaciones por polinomios +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $I\subseteq\mathbb{R}$ +\end_inset + + es un intervalo cerrado, llamamos +\begin_inset Formula ${\cal C}(I)$ +\end_inset + + al conjunto de funciones +\begin_inset Formula $I\to\mathbb{R}$ +\end_inset + + continuas en el interior de +\begin_inset Formula $I$ +\end_inset + +. +\end_layout + +\begin_layout Standard + +\series bold +Teorema de Korovkin: +\series default + Sean +\begin_inset Formula $p_{0},p_{1},p_{2}:[a,b]\subseteq\mathbb{R}\to\mathbb{R}$ +\end_inset + + dadas por +\begin_inset Formula $p_{k}(t)\coloneqq t^{k}$ +\end_inset + + y +\begin_inset Formula $(P_{n}:{\cal C}([a,b])\to{\cal C}([a,b]))_{n}$ +\end_inset + + una sucesión de funciones lineales positivas ( +\begin_inset Formula $\forall f\in{\cal C}([a,b]),(f\geq0\implies P_{n}(f)\geq0)$ +\end_inset + +) con +\begin_inset Formula $\lim_{n}\Vert P_{n}(p_{k})-p_{k}\Vert_{\infty}=0$ +\end_inset + + para +\begin_inset Formula $k\in\{0,1,2\}$ +\end_inset + +, entonces, para +\begin_inset Formula $f\in{\cal C}([a,b])$ +\end_inset + +, +\begin_inset Formula $\lim_{n}\Vert P_{n}(f)-f\Vert_{\infty}=0$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\series bold +Teorema de Weierstrass: +\series default + El conjunto de polinomios en una variable es denso +\begin_inset Formula $({\cal C}([a,b]),\Vert\cdot\Vert_{\infty})$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Así, para +\begin_inset Formula $f\in{\cal C}([a,b])$ +\end_inset + +, se puede encontrar una sucesión de polinomios que converja uniformemente + a +\begin_inset Formula $f$ +\end_inset + +. + Hacerlo con polinomios de interpolación por nodos prefijados no es una + buena estrategia ya que para toda secuencia de nodos de interpolación en + +\begin_inset Formula $[a,b]$ +\end_inset + +, existe +\begin_inset Formula $f\in{\cal C}([a,b])$ +\end_inset + + para la que los polinomios de interpolación en dichos nodos no converge + uniformemente a +\begin_inset Formula $f$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + Si se hace con nodos equidistantes se da el fenómeno de Runge. +\end_layout + +\begin_layout Standard + +\series bold +Teorema de Čebyšev: +\series default + Para +\begin_inset Formula $f\in{\cal C}([a,b])$ +\end_inset + + y +\begin_inset Formula $n\in\mathbb{N}$ +\end_inset + +, si +\begin_inset Formula $K_{n}\subseteq\mathbb{K}[X]$ +\end_inset + + es el conjunto de polinomio de grado máximo +\begin_inset Formula $n$ +\end_inset + +, +\begin_inset Formula $p:K_{n}\mapsto\Vert f-p\Vert_{\infty}$ +\end_inset + + tiene un único mínimo +\begin_inset Formula $p_{n}$ +\end_inset + +, y +\begin_inset Formula $(p_{n})_{n}$ +\end_inset + + converge uniformemente a +\begin_inset Formula $f$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Un +\series bold +polinomio trigonométrico real +\series default + es una función +\begin_inset Formula $p:\mathbb{R}\to\mathbb{R}$ +\end_inset + + de la forma +\begin_inset Formula +\[ +p(x)\coloneqq\sum_{n=0}^{m}(a_{n}\cos(nx)+b_{n}\sin(nx)) +\] + +\end_inset + +para ciertos +\begin_inset Formula $a_{n},b_{n}\in\mathbb{R}$ +\end_inset + +. + +\series bold +Teorema de Weierstrass: +\series default + Si +\begin_inset Formula $f:[-\pi,\pi]\to\mathbb{R}$ +\end_inset + + es continua con +\begin_inset Formula $f(-\pi)=f(\pi)$ +\end_inset + +, para cada +\begin_inset Formula $\varepsilon>0$ +\end_inset + + existe un polinomio trigonométrico real +\begin_inset Formula $p$ +\end_inset + + con +\begin_inset Formula $\Vert f-p\Vert_{\infty}<\varepsilon$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Para +\begin_inset Formula $f:[-\pi,\pi]\to\mathbb{C}$ +\end_inset + + integrable y +\begin_inset Formula $r\in\mathbb{Z}$ +\end_inset + +, llamamos +\series bold + +\begin_inset Formula $r$ +\end_inset + +-ésimo coeficiente de Fourier +\series default + de +\begin_inset Formula $f$ +\end_inset + + a +\begin_inset Formula +\[ +\hat{f}(r)\coloneqq\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)\text{e}^{-\text{i}rt}\dif t, +\] + +\end_inset + +y +\series bold +serie de Fourier +\series default + de +\begin_inset Formula $f$ +\end_inset + + a la serie formal +\begin_inset Formula +\[ +\sum_{r\in\mathbb{Z}}\hat{f}(r)\text{e}^{-\text{i}rt}. +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Para +\begin_inset Formula $f:[-\pi,\pi]\to\mathbb{R}$ +\end_inset + + integrable y +\begin_inset Formula $n\in\mathbb{N}^{*}$ +\end_inset + +, llamando +\begin_inset Formula +\begin{align*} +a_{0} & \coloneqq\frac{1}{2\pi}\int_{-\pi}^{\pi}f, & a_{n} & \coloneqq\frac{1}{\pi}\int_{-\pi}^{\pi}f(t)\cos(nt)\dif t, & b_{n} & \coloneqq\frac{1}{\pi}\int_{-\pi}^{\pi}f(t)\sin(nt)\dif t, +\end{align*} + +\end_inset + +la +\series bold +serie de Fourier real +\series default + de +\begin_inset Formula $f$ +\end_inset + + es +\begin_inset Formula +\[ +\sum_{n=0}^{\infty}a_{n}\cos(nt)+\sum_{n=1}^{\infty}b_{n}\sin(nt). +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Como +\series bold +teorema +\series default +, sean +\begin_inset Formula $([-\pi,\pi],\Sigma,\mu)$ +\end_inset + + es el espacio de medida usual en +\begin_inset Formula $[-\pi,\pi]$ +\end_inset + +, +\begin_inset Formula $M_{\mathbb{R}}\coloneqq L_{\mathbb{R}}^{2}([-\pi,\pi],\Sigma,\frac{\mu}{\pi})$ +\end_inset + + y +\begin_inset Formula $M_{\mathbb{C}}\coloneqq L_{\mathbb{C}}^{2}([-\pi,\pi],\Sigma,\frac{\mu}{2\pi})$ +\end_inset + +: +\end_layout + +\begin_layout Enumerate +El +\series bold +sistema trigonométrico +\series default + +\begin_inset Formula $(\text{e}^{\text{i}rt})_{r\in\mathbb{Z}}$ +\end_inset + + es una base hilbertiana de +\begin_inset Formula $M_{\mathbb{C}}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $(\cos(nt))_{n\in\mathbb{N}}\star(\sin(nt))_{n\in\mathbb{N}^{*}}$ +\end_inset + + es una base hilbertiana de +\begin_inset Formula $M_{\mathbb{R}}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +Para +\begin_inset Formula $f\in M_{\mathbb{C}}$ +\end_inset + +, +\begin_inset Formula $f$ +\end_inset + + coincide con su serie de Fourier en +\begin_inset Formula $\Vert\cdot\Vert_{2}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +Para +\begin_inset Formula $f\in M_{\mathbb{R}}$ +\end_inset + +, +\begin_inset Formula $f$ +\end_inset + + coincide con su serie de Fourier real en +\begin_inset Formula $\Vert\cdot\Vert_{2}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Formula ${\cal F}:M_{\mathbb{C}}\to\ell^{2}(\mathbb{Z})$ +\end_inset + + que asigna a cada función su familia de coeficientes de Fourier +\begin_inset Formula $(\hat{f}(n))_{n\in\mathbb{Z}}$ +\end_inset + + es un isomorfismo de espacios de Hilbert. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Un +\series bold +peso +\series default + en un intervalo cerrado +\begin_inset Formula $I\subseteq\mathbb{R}$ +\end_inset + + es una +\begin_inset Formula $p\in{\cal C}(I)$ +\end_inset + + estrictamente positiva tal que +\begin_inset Formula +\[ +\forall n\in\mathbb{N},\int_{I}|t|^{n}p(t)\dif t<\infty. +\] + +\end_inset + +Entonces +\begin_inset Formula $\langle\cdot,\cdot\rangle:{\cal C}(I)\times{\cal C}(I)\to[-\infty,+\infty]$ +\end_inset + + dada por +\begin_inset Formula +\[ +\langle f,g\rangle\coloneqq\int_{I}f\overline{g}p +\] + +\end_inset + +es un producto escalar en +\begin_inset Formula $H_{p}\coloneqq\{f\in{\cal C}(I)\mid\langle f,f\rangle<\infty\}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Llamamos +\series bold +sucesión de polinomios ortonormales +\series default + asociada a +\begin_inset Formula $\langle\cdot,\cdot\rangle$ +\end_inset + + o al peso +\begin_inset Formula $p$ +\end_inset + + en +\begin_inset Formula $I$ +\end_inset + + a una sucesión +\begin_inset Formula $\{P_{n}\}_{n\in\mathbb{N}}\subseteq H_{p}$ +\end_inset + + de polinomios con +\begin_inset Formula $\text{span}\{1,t,\dots,t^{n}\}=\text{span}\{P_{0},P_{1},\dots,P_{n}\}$ +\end_inset + + para cada +\begin_inset Formula $n\in\mathbb{N}$ +\end_inset + +, y entonces, para +\begin_inset Formula $n\in\mathbb{N}$ +\end_inset + +: +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $P_{n}$ +\end_inset + + es un polinomio de grado +\begin_inset Formula $n$ +\end_inset + + con coeficientes reales. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $P_{n}$ +\end_inset + + es ortogonal en +\begin_inset Formula $H_{p}$ +\end_inset + + al subespacio de polinomios de grado menor que +\begin_inset Formula $n$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $P_{n}$ +\end_inset + + tiene +\begin_inset Formula $n$ +\end_inset + + raíces distintas en +\begin_inset Formula $(a,b)$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Ejemplos: +\end_layout + +\begin_layout Enumerate + +\series bold +Polinomios de Legendre. +\series default + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{align*} +I & =[-1,1], & p(t) & =1, & P_{n}(t) & =\frac{\sqrt{\frac{2n+1}{2}}}{2^{n}n!}\od[n]{(t^{2}-1)^{n}}{t}. +\end{align*} + +\end_inset + + +\end_layout + +\begin_layout Enumerate + +\series bold +Polinomios de Laguerre. +\series default + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\series bold + +\begin_inset Formula +\begin{align*} +I & =[0,\infty), & p(t) & =\text{e}^{-t}, & P_{n}(t) & =\frac{\text{e}^{t}}{n!}\od[n]{\text{e}^{-t}t^{n}}{t}. +\end{align*} + +\end_inset + + +\end_layout + +\begin_layout Enumerate + +\series bold +Polinomios de Hermite. +\series default + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{align*} +I & =(-\infty,\infty), & p(t) & =\text{e}^{-t^{2}}, & P_{n}(t) & =\frac{\text{e}^{t^{2}}}{\sqrt[4]{\pi}\sqrt{2^{n}n!}}\od[n]{\text{e}^{-t^{2}}}{t}. +\end{align*} + +\end_inset + + +\end_layout + +\begin_layout Enumerate + +\series bold +Polinomios de Čebyšev. +\series default + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{align*} +I & =[-1,1], & p(t) & =\frac{1}{\sqrt{1-t^{2}}}, & P_{n}(t) & =\cos(n\arccos t), +\end{align*} + +\end_inset + +siendo +\begin_inset Formula $\arccos:[-1,1]\to[0,\pi]$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Una sucesión de polinomios ortonormales asociada a un peso +\begin_inset Formula $p$ +\end_inset + + en un intervalo compacto es total en +\begin_inset Formula $H_{p}$ +\end_inset + +, y en particular los polinomios de Legendre forman una base hilbertiana + en +\begin_inset Formula $L^{2}([-1,1]).$ +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $p$ +\end_inset + + es un peso en +\begin_inset Formula $[a,b]$ +\end_inset + + y +\begin_inset Formula $a\leq t_{1}<\dots<t_{n}\leq b$ +\end_inset + +, se tiene una +\series bold +fórmula de cuadratura gaussiana +\series default +, +\begin_inset Formula +\[ +\int_{a}^{b}fp\approx\sum_{k=1}^{n}A_{k}f(t_{k}) +\] + +\end_inset + +para ciertos +\begin_inset Formula $A_{1},\dots,A_{n}\in\mathbb{R}$ +\end_inset + +, y se alcanza la igualdad si +\begin_inset Formula $f$ +\end_inset + + es un polinomio de grado menor que +\begin_inset Formula $n$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\series bold +Teorema de Gauss: +\series default + Dados un peso +\begin_inset Formula $p$ +\end_inset + + en +\begin_inset Formula $[a,b]$ +\end_inset + + con una sucesión de polinomios ortonormales +\begin_inset Formula $(P_{n})_{n}$ +\end_inset + +, +\begin_inset Formula $n\in\mathbb{N}^{*}$ +\end_inset + +, +\begin_inset Formula $a<t_{1}<\dots<t_{n}<b$ +\end_inset + + y +\begin_inset Formula $A_{1},\dots,A_{n}\in\mathbb{R}$ +\end_inset + +, si +\begin_inset Formula +\[ +\int_{a}^{b}fp=\sum_{k=1}^{n}A_{k}f(t_{k}) +\] + +\end_inset + +para todo polinomio +\begin_inset Formula $f$ +\end_inset + + de grado menor que +\begin_inset Formula $n$ +\end_inset + +, esta fórmula se para polinomios de grado menor que +\begin_inset Formula $2n$ +\end_inset + + si y sólo si +\begin_inset Formula $t_{1},\dots,t_{n}$ +\end_inset + + son los ceros de +\begin_inset Formula $P_{n}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\series bold +Teorema de Stieltjes: +\series default + Sean +\begin_inset Formula $p$ +\end_inset + + un peso en +\begin_inset Formula $[a,b]$ +\end_inset + + con una sucesión de polinomios ortonormales +\begin_inset Formula $(P_{n})_{n}$ +\end_inset + + y, para +\begin_inset Formula $n\in\mathbb{N}$ +\end_inset + +, +\begin_inset Formula $t_{n1}<\dots<t_{nn}$ +\end_inset + + los ceros de +\begin_inset Formula $P_{n}$ +\end_inset + + y +\begin_inset Formula $A_{n1},\dots,A_{nn}\in\mathbb{R}$ +\end_inset + + los correspondientes coeficientes en la fórmula de cuadratura gaussiana, + para +\begin_inset Formula $f\in{\cal C}([a,b])$ +\end_inset + +, +\begin_inset Formula +\[ +\int_{a}^{b}fp=\lim_{n}\sum_{k=1}^{n}A_{nk}f(t_{nk}). +\] + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Section +El espacio de Bergman +\end_layout + +\begin_layout Standard +Llamamos +\begin_inset Formula $D(a,r)\coloneqq B(a,r)\subseteq\mathbb{C}$ +\end_inset + +. + Si +\begin_inset Formula $\Omega\subseteq\mathbb{C}$ +\end_inset + + es abierto, +\begin_inset Formula ${\cal H}(\Omega)$ +\end_inset + + es el conjunto de las funciones holomorfas en +\begin_inset Formula $\Omega$ +\end_inset + +, y para +\begin_inset Formula $f\in{\cal H}(\Omega)$ +\end_inset + + y +\begin_inset Formula $\overline{D(a,r)}\subseteq\Omega$ +\end_inset + +, la serie +\begin_inset Formula $\sum_{n\in\mathbb{N}}a_{n}(z-a)^{n}$ +\end_inset + + con +\begin_inset Formula $z\in D(a,r)$ +\end_inset + + converge uniformemente a +\begin_inset Formula $f$ +\end_inset + + en compactos de +\begin_inset Formula $D(a,r)$ +\end_inset + + para ciertos +\begin_inset Formula $a_{n}\in\mathbb{C}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $\Omega\subseteq\mathbb{C}$ +\end_inset + + es abierto, llamamos +\begin_inset Formula ${\cal T}_{\text{K}}$ +\end_inset + + a la topología en +\begin_inset Formula ${\cal H}(\Omega)$ +\end_inset + + de convergencia uniforme sobre compactos, y +\series bold +espacio de Bergman +\series default + en el abierto +\begin_inset Formula $\Omega\subseteq\mathbb{C}$ +\end_inset + + a +\begin_inset Formula +\[ +A^{2}(\Omega)\coloneqq\left\{ f\in{\cal H}(\Omega)\;\middle|\;\int_{\Omega}|f|^{2}<\infty\right\} , +\] + +\end_inset + +un subespacio cerrado y separable de +\begin_inset Formula $L^{2}(\Omega)$ +\end_inset + + que es pues un espacio de Hilbert numerable con +\begin_inset Formula $\langle\cdot,\cdot\rangle_{2}$ +\end_inset + +, y en el que la topología inducida por +\begin_inset Formula $L^{2}(\Omega)$ +\end_inset + + es más fina que la inducida por +\begin_inset Formula ${\cal T}_{\text{K}}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $\Omega\subseteq\mathbb{C}$ +\end_inset + + es abierto, +\begin_inset Formula $(\omega_{n})_{n}$ +\end_inset + + es base hilbertiana de +\begin_inset Formula $A^{2}(\Omega)$ +\end_inset + + y +\begin_inset Formula $f\in A^{2}(\Omega)$ +\end_inset + +, el desarrollo en serie de Fourier de +\begin_inset Formula $f$ +\end_inset + +, +\begin_inset Formula $\sum_{n}\langle f,\omega_{n}\rangle\omega_{n}$ +\end_inset + +, converge uniformemente a +\begin_inset Formula $f$ +\end_inset + + en compactos de +\begin_inset Formula $\Omega$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $\psi_{n}(z)\coloneqq(z-a)^{n}$ +\end_inset + +, +\begin_inset Formula $(\frac{\psi_{n}}{\Vert\psi_{n}\Vert})_{n}$ +\end_inset + + es una base hilbertiana de +\begin_inset Formula $A^{2}(D(a,r))$ +\end_inset + +, y el desarrollo en serie de potencias es el desarrollo en serie de Fourier + sobre esta base. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Como +\series bold +teorema +\series default +, si +\begin_inset Formula $\Omega\subsetneq\mathbb{C}$ +\end_inset + + es un abierto simplemente conexo y +\begin_inset Formula $f:\Omega\to D(0,1)$ +\end_inset + + es un isomorfismo, +\begin_inset Formula +\[ +\left(z\mapsto\sqrt{\frac{n}{\pi}}(f(z))^{n-1}\dot{f}(z)\right)_{n} +\] + +\end_inset + +es base hilbertiana de +\begin_inset Formula $A^{2}(\Omega)$ +\end_inset + +, y en particular para +\begin_inset Formula $R>0$ +\end_inset + +, +\begin_inset Formula +\[ +\left(z\mapsto\sqrt{\frac{n}{\pi}}R^{-n}z^{n-1}\right)_{n} +\] + +\end_inset + + es base hilbertiana de +\begin_inset Formula $A^{2}(D(0,R))$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\end_body +\end_document |