diff options
| author | Juan Marin Noguera <juan@mnpi.eu> | 2022-11-30 10:34:03 +0100 |
|---|---|---|
| committer | Juan Marin Noguera <juan@mnpi.eu> | 2023-01-10 19:29:14 +0100 |
| commit | 996106289d0ae061b1850f957e75b68e13205e10 (patch) | |
| tree | 393c5263ba80fa2b81252108021eaf903aa8cf83 /af | |
| parent | de3e935e35f0fdad86aaf142e657cd9c0fbf0ef8 (diff) | |
Análisis funcional tema 1: Espacios de Hilbert
Diffstat (limited to 'af')
| -rw-r--r-- | af/n.lyx | 43 | ||||
| -rw-r--r-- | af/n1.lyx | 8135 |
2 files changed, 7546 insertions, 632 deletions
@@ -7,9 +7,7 @@ \textclass book \begin_preamble \input{../defs} -\newenvironment{nproof}{}{} -% So far I haven't found a way inside LaTeX to change nproofs to -% comments, so I guess that'll have to be a Perl or Awk script. +\usepackage{commath} \end_preamble \use_default_options true \begin_modules @@ -156,13 +154,17 @@ Análisis Funcional \begin_layout Itemize \lang english -Eric Shapiro. +Todd Kemp. \emph on -Constructing the real numbers (2)—Cauchy Sequences +Cauchy's construction of +\begin_inset Formula $\mathbb{R}$ +\end_inset + + \emph default -. - Algebrology. + (2016), pp. + 9–10. \lang spanish Recuperado de @@ -171,7 +173,7 @@ status open \begin_layout Plain Layout -https://algebrology.eshapiro.net/constructing-the-real-numbers-2-cauchy-sequences/ +https://mathweb.ucsd.edu/~tkemp/140A/Construction.of.R.pdf \end_layout \end_inset @@ -179,6 +181,31 @@ https://algebrology.eshapiro.net/constructing-the-real-numbers-2-cauchy-sequence . \end_layout +\begin_layout Itemize + +\lang english +Wikipedia, the Free Encyclopedia. + +\emph on +Locally integrable function +\emph default +. + +\lang spanish + Recuperado de +\begin_inset Flex URL +status open + +\begin_layout Plain Layout + +https://en.wikipedia.org/wiki/Locally_integrable_function +\end_layout + +\end_inset + + el 18 de diciembre de 2022. +\end_layout + \begin_layout Chapter Espacios de Hilbert \end_layout @@ -7,6 +7,7 @@ \textclass book \begin_preamble \input{../defs} +\usepackage{commath} \end_preamble \use_default_options true \maintain_unincluded_children false @@ -81,6 +82,31 @@ \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 $\mathbb{R}$ @@ -91,6 +117,31 @@ s que lo son sobre \end_inset . + Dados dos +\begin_inset Formula $\mathbb{K}$ +\end_inset + +-espacios vectoriales +\begin_inset Formula $X$ +\end_inset + + e +\begin_inset Formula $Y$ +\end_inset + +, +\begin_inset Formula $f:X\to Y$ +\end_inset + + es +\series bold +lineal conjugada +\series default + si +\begin_inset Formula $\forall a,b\in\mathbb{K},\forall x,y\in X,f(ax+by)=\overline{a}f(x)+\overline{b}f(y)$ +\end_inset + +. \end_layout \begin_layout Section @@ -187,18 +238,10 @@ norma \begin_layout Standard Toda norma es definida positiva -\begin_inset ERT +\begin_inset Note Comment status open \begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - , pues si \begin_inset Formula $x\in X\setminus0$ \end_inset @@ -216,14 +259,6 @@ begin{nproof} \end_inset -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} \end_layout \end_inset @@ -258,7 +293,7 @@ espacio normado \end_inset , y llamamos -\begin_inset Formula $B_{X}\coloneqq B[0,1]=\overline{B(0,1)}=\{x\in X\mid \Vert x\Vert\leq1\}$ +\begin_inset Formula $B_{X}\coloneqq B[0,1]=\overline{B(0,1)}=\{x\in X\mid\Vert x\Vert\leq1\}$ \end_inset y conjunto de @@ -266,23 +301,15 @@ espacio normado vectores unitarios \series default a -\begin_inset Formula $S_{X}\coloneqq\partial B(0,1)=\{x\in X\mid \Vert x\Vert=1\}$ +\begin_inset Formula $S_{X}\coloneqq\partial B(0,1)=\{x\in X\mid\Vert x\Vert=1\}$ \end_inset . La norma es uniformemente continua en este espacio métrico -\begin_inset ERT +\begin_inset Note Comment status open \begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - , pues para \begin_inset Formula $\varepsilon>0$ \end_inset @@ -304,14 +331,6 @@ begin{nproof} \end_inset -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} \end_layout \end_inset @@ -330,6 +349,22 @@ espacio de Banach \end_layout \begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{samepage} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard Sea \begin_inset Formula $(X,\Vert\cdot\Vert)$ \end_inset @@ -366,23 +401,10 @@ Todo subespacio vectorial de \end_inset son continuas. -\begin_inset ERT +\begin_inset Note Comment status open \begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - - -\end_layout - -\begin_deeper -\begin_layout Standard Sea \begin_inset Formula $A\subseteq X$ \end_inset @@ -464,14 +486,6 @@ con lo que \end_inset . -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} \end_layout \end_inset @@ -479,7 +493,6 @@ end{nproof} \end_layout -\end_deeper \begin_layout Enumerate \begin_inset Formula $s_{y}:X\to X$ \end_inset @@ -505,23 +518,10 @@ end{nproof} \end_inset son homeomorfismos. -\begin_inset ERT +\begin_inset Note Comment status open \begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - - -\end_layout - -\begin_deeper -\begin_layout Standard \begin_inset Formula $s_{y}$ \end_inset @@ -550,14 +550,6 @@ begin{nproof} \end_inset , que también son continuas. -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} \end_layout \end_inset @@ -565,7 +557,6 @@ end{nproof} \end_layout -\end_deeper \begin_layout Enumerate La suma de un abierto y un subconjunto cualquiera de \begin_inset Formula $X$ @@ -576,23 +567,10 @@ La suma de un abierto y un subconjunto cualquiera de \end_inset . -\begin_inset ERT +\begin_inset Note Comment status open \begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - - -\end_layout - -\begin_deeper -\begin_layout Standard Sean \begin_inset Formula $G\subseteq X$ \end_inset @@ -631,14 +609,6 @@ Sean \end_inset . -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} \end_layout \end_inset @@ -646,7 +616,6 @@ end{nproof} \end_layout -\end_deeper \begin_layout Enumerate La suma de un cerrado y un compacto de \begin_inset Formula $X$ @@ -657,23 +626,10 @@ La suma de un cerrado y un compacto de \end_inset . -\begin_inset ERT +\begin_inset Note Comment status open \begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - - -\end_layout - -\begin_deeper -\begin_layout Standard Sean \begin_inset Formula $F$ \end_inset @@ -728,14 +684,6 @@ Sean \end_inset . -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} \end_layout \end_inset @@ -743,7 +691,6 @@ end{nproof} \end_layout -\end_deeper \begin_layout Enumerate Si \begin_inset Formula $Y\subseteq X$ @@ -754,23 +701,10 @@ Si \end_inset . -\begin_inset ERT +\begin_inset Note Comment status open \begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - - -\end_layout - -\begin_deeper -\begin_layout Standard Dados \begin_inset Formula $a\in\mathbb{K}$ \end_inset @@ -800,14 +734,6 @@ Dados \end_inset . -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} \end_layout \end_inset @@ -815,29 +741,15 @@ end{nproof} \end_layout -\end_deeper \begin_layout Enumerate Un subespacio vectorial de \begin_inset Formula $X$ \end_inset es propio si y sólo si su interior es vacío. -\begin_inset ERT +\begin_inset Note Comment status open -\begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - - -\end_layout - -\begin_deeper \begin_layout Enumerate \begin_inset Argument item:1 status open @@ -886,6 +798,7 @@ Sea . \end_layout +\begin_deeper \begin_layout Enumerate \begin_inset Argument item:1 status open @@ -902,23 +815,12 @@ status open El contrarrecíproco es trivial. \end_layout -\begin_layout Standard -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} -\end_layout - +\end_deeper \end_inset \end_layout -\end_deeper \begin_layout Enumerate \begin_inset Formula $X$ \end_inset @@ -944,22 +846,9 @@ end{nproof} \end_inset . -\begin_inset ERT +\begin_inset Note Comment status open -\begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - - -\end_layout - -\begin_deeper \begin_layout Enumerate \begin_inset Argument item:1 status open @@ -1052,23 +941,11 @@ Sea es convergente. \end_layout -\begin_layout Standard -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} -\end_layout - \end_inset \end_layout -\end_deeper \begin_layout Enumerate Todo subespacio vectorial de Banach de \begin_inset Formula $X$ @@ -1079,23 +956,10 @@ Todo subespacio vectorial de Banach de \end_inset . -\begin_inset ERT +\begin_inset Note Comment status open \begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - - -\end_layout - -\begin_deeper -\begin_layout Standard Toda sucesión de Cauchy en \begin_inset Formula $Y\leq X$ \end_inset @@ -1109,14 +973,6 @@ Toda sucesión de Cauchy en \end_inset . -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} \end_layout \end_inset @@ -1124,30 +980,16 @@ end{nproof} \end_layout -\end_deeper \begin_layout Enumerate Si \begin_inset Formula $X$ \end_inset es de Banach, todo subespacio vectorial cerrado es de Banach. -\begin_inset ERT +\begin_inset Note Comment status open \begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - - -\end_layout - -\begin_deeper -\begin_layout Standard Toda sucesión de Cauchy en \begin_inset Formula $Y\leq X$ \end_inset @@ -1171,6 +1013,11 @@ Toda sucesión de Cauchy en . \end_layout +\end_inset + + +\end_layout + \begin_layout Standard \begin_inset ERT status open @@ -1179,7 +1026,7 @@ status open \backslash -end{nproof} +end{samepage} \end_layout \end_inset @@ -1187,11 +1034,6 @@ end{nproof} \end_layout -\end_deeper -\begin_layout Subsection -Operadores continuos -\end_layout - \begin_layout Standard Dado un espacio normado \begin_inset Formula $X$ @@ -1311,22 +1153,9 @@ Si \end_inset es uniformemente continuo. -\begin_inset ERT +\begin_inset Note Comment status open -\begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - - -\end_layout - -\begin_deeper \begin_layout Description \begin_inset Formula $1\implies2]$ \end_inset @@ -1460,23 +1289,11 @@ begin{nproof} Obvio. \end_layout -\begin_layout Standard -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} -\end_layout - \end_inset \end_layout -\end_deeper \begin_layout Enumerate \begin_inset Formula ${\cal L}(X,Y)$ \end_inset @@ -1506,23 +1323,10 @@ tomando \end_inset también lo es. -\begin_inset ERT +\begin_inset Note Comment status open \begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - - -\end_layout - -\begin_deeper -\begin_layout Standard Si \begin_inset Formula $a\in\mathbb{K}$ \end_inset @@ -1629,7 +1433,7 @@ Núcleo 0: Si . \end_layout -\begin_layout Standard +\begin_layout Plain Layout Finalmente, si \begin_inset Formula $\{T_{n}\}_{n}\subseteq{\cal L}(X,Y)$ \end_inset @@ -1788,7 +1592,7 @@ Para ver que es continua, como es continua en 0, por lo que es continua. \end_layout -\begin_layout Standard +\begin_layout Plain Layout Finalmente, para \begin_inset Formula $\varepsilon>0$ \end_inset @@ -1821,19 +1625,11 @@ Finalmente, para \end_inset - por lo que finalmente +por lo que finalmente \begin_inset Formula $\Vert T_{n}-T\Vert=\sup_{x\in S_{X}}\Vert T_{n}(x)-T(x)\Vert<\varepsilon$ \end_inset . -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} \end_layout \end_inset @@ -1841,30 +1637,16 @@ end{nproof} \end_layout -\end_deeper \begin_layout Enumerate La composición de operadores acotados es un operador acotado. \end_layout \begin_layout Enumerate En espacios de dimensión infinita hay operadores no acotados. -\begin_inset ERT +\begin_inset Note Comment status open \begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - - -\end_layout - -\begin_deeper -\begin_layout Standard Sean \begin_inset Formula $X$ \end_inset @@ -1914,14 +1696,6 @@ Sean \end_inset no tiene cota superior. -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} \end_layout \end_inset @@ -1929,7 +1703,6 @@ end{nproof} \end_layout -\end_deeper \begin_layout Standard Una \series bold @@ -1971,10 +1744,6 @@ dual topológico . \end_layout -\begin_layout Subsection -Isomorfismos topológicos -\end_layout - \begin_layout Standard Dados dos espacios normados \begin_inset Formula $X$ @@ -1998,21 +1767,9 @@ isomorfismo topológico \end_inset . -\begin_inset ERT +\begin_inset Note Comment status open -\begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - - -\end_layout - \begin_layout Itemize \begin_inset Argument item:1 status open @@ -2075,17 +1832,6 @@ Para son continuas. \end_layout -\begin_layout Standard -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} -\end_layout - \end_inset @@ -2144,18 +1890,10 @@ Si son espacios normados topológicamente isomorfos, la completitud de uno equivale a la del otro. -\begin_inset ERT +\begin_inset Note Comment status open \begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - En efecto, si \begin_inset Formula $X$ \end_inset @@ -2229,14 +1967,6 @@ En efecto, si \end_inset . -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} \end_layout \end_inset @@ -2244,10 +1974,6 @@ end{nproof} \end_layout -\begin_layout Subsection -Compleción -\end_layout - \begin_layout Standard Para todo espacio normado \begin_inset Formula $X$ @@ -2283,19 +2009,11 @@ compleción . -\begin_inset ERT +\begin_inset Note Comment status open \begin_layout Plain Layout - -\backslash -begin{nproof} -\end_layout - -\end_inset - - \series bold Demostración: \series default @@ -2392,7 +2110,7 @@ Demostración: \end_inset - El operador isométrico +El operador isométrico \begin_inset Formula $J$ \end_inset @@ -2459,153 +2177,108 @@ Demostración: \end_inset es completo, sea -\begin_inset Formula $\{\overline{x}_{n}\}_{n}\subseteq\hat{X}$ +\begin_inset Formula $\{\overline{x_{n}}\}_{n}\subseteq\hat{X}$ \end_inset de Cauchy, para -\begin_inset Formula $k\in\mathbb{N}$ +\begin_inset Formula $n\in\mathbb{N}$ \end_inset existe -\begin_inset Formula $N_{k}\in\mathbb{N}$ +\begin_inset Formula $y_{n}\in X$ \end_inset -, que podemos tomar mayor que -\begin_inset Formula $N_{k-1}$ + tal que +\begin_inset Formula $\Vert\overline{x_{n}}-y_{n}\Vert<\frac{1}{n}$ \end_inset -, tal que para -\begin_inset Formula $n,m\geq N_{k}$ +. + En efecto, existe +\begin_inset Formula $m_{0}\in\mathbb{N}$ \end_inset - es -\begin_inset Formula $\Vert x_{kn}-x_{km}\Vert<\frac{1}{k}$ + tal que, para +\begin_inset Formula $k,m\geq m_{0}$ \end_inset -, y llamamos -\begin_inset Formula $y_{k}\coloneqq x_{kN_{k}}\in X$ +, +\begin_inset Formula $\Vert x_{nk}-x_{nm}\Vert<\frac{1}{2n}$ \end_inset -. - Queremos ver que -\begin_inset Formula $(y_{k})_{k}$ +, luego si +\begin_inset Formula $y_{n}\coloneqq x_{nm_{0}}$ \end_inset - es de Cauchy y que -\begin_inset Formula $\overline{x}_{n}\to\overline{y}$ +, +\begin_inset Formula $\Vert x_{nk}-y_{n}\Vert<\frac{1}{n}$ \end_inset -. - Para lo primero, para -\begin_inset Formula $\varepsilon>0$ + y +\begin_inset Formula $\Vert\overline{x_{n}}-y_{n}\Vert=\lim_{k}\Vert x_{nk}-y_{n}\Vert\leq\frac{1}{2n}<\frac{1}{n}$ \end_inset -, existe -\begin_inset Formula $M_{1}$ +. + Entonces, para +\begin_inset Formula $\varepsilon>0$ \end_inset - tal que para -\begin_inset Formula $n,m\geq M_{1}$ +, sean +\begin_inset Formula $N\in\mathbb{N}$ \end_inset - es -\begin_inset Formula $\Vert x_{n}-x_{m}\Vert=\lim_{k}\Vert x_{nk}-x_{mk}\Vert<\frac{\varepsilon}{2}$ + con +\begin_inset Formula $\frac{1}{N}<\frac{\varepsilon}{4}$ \end_inset -, luego existe -\begin_inset Formula $M_{2}$ + y +\begin_inset Formula $M\in\mathbb{N}$ \end_inset tal que para -\begin_inset Formula $i\geq M_{2}$ +\begin_inset Formula $n,m>M$ \end_inset es -\begin_inset Formula $\Vert x_{ni}-x_{mi}\Vert<\frac{\varepsilon}{2}$ -\end_inset - -. - Sea entonces -\begin_inset Formula $M\coloneqq\max\{M_{1},M_{2}\}$ +\begin_inset Formula $\Vert\overline{x_{n}}-\overline{x_{m}}\Vert<\frac{\varepsilon}{3}$ \end_inset , para -\begin_inset Formula $n,m\geq M$ -\end_inset - -, como -\begin_inset Formula $n,m\geq M_{1}$ -\end_inset - - y -\begin_inset Formula $N_{n}\geq n\geq M_{2}$ +\begin_inset Formula $n,m>\max\{N,M\}$ \end_inset , -\begin_inset Formula $\Vert x_{nN_{n}}-x_{mN_{n}}\Vert<\frac{\varepsilon}{2}$ -\end_inset - -, con lo que \begin_inset Formula \[ -\Vert y_{n}-y_{m}\Vert=\Vert x_{nN_{n}}-x_{mN_{m}}\Vert\leq\Vert x_{nN_{n}}-x_{mN_{n}}\Vert+\Vert x_{mN_{n}}-x_{mN_{m}}\Vert<\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon. +\Vert y_{n}-y_{m}\Vert\leq\Vert y_{n}-\overline{u_{n}}\Vert+\Vert\overline{u_{n}}-\overline{u_{m}}\Vert+\Vert\overline{u_{m}}-y_{m}\Vert<\frac{\varepsilon}{3}+\frac{\varepsilon}{3}+\frac{\varepsilon}{3}=\varepsilon, \] \end_inset -Para lo segundo, para -\begin_inset Formula $\varepsilon>0$ -\end_inset - -, sean -\begin_inset Formula $n_{0}$ -\end_inset - - tal que para -\begin_inset Formula $n,m\geq0$ +luego +\begin_inset Formula $y_{n}$ \end_inset - es -\begin_inset Formula $\Vert y_{n}-y_{m}\Vert<\frac{\varepsilon}{3}$ + es de Cauchy y solo queda que ver que +\begin_inset Formula $(x_{n})_{n}$ \end_inset - y -\begin_inset Formula $k\in\mathbb{N}$ + converge a +\begin_inset Formula $L\coloneqq\overline{(y_{n})_{n}}$ \end_inset - con -\begin_inset Formula $k\geq n_{0}$ +, pero por construcción es +\begin_inset Formula $\lim_{n}\Vert y_{n}-L\Vert=0$ \end_inset y -\begin_inset Formula $\frac{1}{k}<\frac{\varepsilon}{3}$ -\end_inset - -, para -\begin_inset Formula $i\geq N_{k}$ -\end_inset - - es -\begin_inset Formula $\Vert x_{ki}-y_{k}\Vert=\Vert x_{ki}-x_{kN_{k}}\Vert<\frac{1}{k}$ +\begin_inset Formula $\lim_{n}\Vert\overline{x_{n}}-y_{n}\Vert\leq\lim_{n}\frac{1}{n}=0$ \end_inset , luego -\begin_inset Formula -\[ -\Vert\overline{x}_{k}-\overline{y}\Vert=\lim_{i}\Vert x_{ki}-y_{i}\Vert\leq\lim_{i}(\Vert x_{ki}-y_{k}\Vert+\Vert y_{k}-y_{i}\Vert)\leq\frac{\varepsilon}{3}+\frac{\varepsilon}{3}<\varepsilon -\] - +\begin_inset Formula $\lim_{n}\Vert\overline{x_{n}}-L\Vert\leq\lim_{n}(\Vert\overline{x_{n}}-y_{n}\Vert+\Vert y_{n}-L\Vert)=0+0=0$ \end_inset -y la completitud queda probada. -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} +. \end_layout \end_inset @@ -2613,10 +2286,6 @@ end{nproof} \end_layout -\begin_layout Subsection -Espacios cociente -\end_layout - \begin_layout Standard \begin_inset ERT status open @@ -2714,6 +2383,22 @@ espacio vectorial cociente \begin_inset Formula $X$ \end_inset +, y llamamos +\series bold +codimensión +\series default + de +\begin_inset Formula $Y$ +\end_inset + + en +\begin_inset Formula $X$ +\end_inset + + a la dimensión de +\begin_inset Formula $X/Y$ +\end_inset + . Si \begin_inset Formula $X$ @@ -2743,23 +2428,10 @@ norma cociente \end_inset . -\begin_inset ERT +\begin_inset Note Comment status open \begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - - -\end_layout - -\begin_deeper -\begin_layout Standard Si \begin_inset Formula $x,x'\in X$ \end_inset @@ -2842,14 +2514,6 @@ Finalmente, si \end_inset . -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} \end_layout \end_inset @@ -2857,30 +2521,16 @@ end{nproof} \end_layout -\end_deeper \begin_layout Enumerate La aplicación cociente \begin_inset Formula $X\to X/Y$ \end_inset es lineal, continua y abierta. -\begin_inset ERT +\begin_inset Note Comment status open \begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - - -\end_layout - -\begin_deeper -\begin_layout Standard Sea \begin_inset Formula $Q$ \end_inset @@ -2932,14 +2582,6 @@ Sea \end_inset . -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} \end_layout \end_inset @@ -2947,26 +2589,12 @@ end{nproof} \end_layout -\end_deeper \begin_layout Enumerate La norma cociente genera la topología cociente. -\begin_inset ERT +\begin_inset Note Comment status open \begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - - -\end_layout - -\begin_deeper -\begin_layout Standard Dado \begin_inset Formula $V\subseteq X/Y$ \end_inset @@ -2992,14 +2620,6 @@ Dado \end_inset abierta. -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} \end_layout \end_inset @@ -3007,7 +2627,6 @@ end{nproof} \end_layout -\end_deeper \begin_layout Enumerate Si \begin_inset Formula $X$ @@ -3018,23 +2637,10 @@ Si \end_inset . -\begin_inset ERT +\begin_inset Note Comment status open \begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - - -\end_layout - -\begin_deeper -\begin_layout Standard Sea \begin_inset Formula $\{\overline{x_{n}}\}_{n}\subseteq X/Y$ \end_inset @@ -3072,14 +2678,6 @@ Sea \end_inset converge. -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} \end_layout \end_inset @@ -3087,11 +2685,6 @@ end{nproof} \end_layout -\end_deeper -\begin_layout Subsection -Suma directa topológica -\end_layout - \begin_layout Standard Sean \begin_inset Formula $X$ @@ -3162,21 +2755,9 @@ complementario topológico \end_inset . -\begin_inset ERT +\begin_inset Note Comment status open -\begin_layout Plain Layout - - -\backslash -begin{nproof} -\end_layout - -\end_inset - - -\end_layout - \begin_layout Itemize \begin_inset Argument item:1 status open @@ -3226,17 +2807,6 @@ Que es continua, y análogamente lo es la otra proyección. \end_layout -\begin_layout Standard -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{nproof} -\end_layout - \end_inset @@ -3270,12 +2840,17 @@ Desigualdad de Hölder: , \begin_inset Formula \[ -\sum_{k=1}^{n}a_{k}b_{k}\leq\left(\sum_{k=1}^{n}a_{k}^{p}\right)^{\frac{1}{p}}\left(\sum_{k=1}^{n}b_{k}^{q}\right)^{\frac{1}{q}}. +\sum_{k}a_{k}b_{k}\leq\left(\sum_{k}a_{k}^{p}\right)^{\frac{1}{p}}\left(\sum_{k}b_{k}^{q}\right)^{\frac{1}{q}}. \] \end_inset +\begin_inset Note Comment +status open + +\begin_layout Plain Layout + \series bold Demostración: \series default @@ -3315,7 +2890,7 @@ y si y sumando, \begin_inset Formula \[ -\frac{\sum_{k=1}^{n}a_{k}b_{k}}{AB}=\sum_{k=1}^{n}\frac{a_{k}}{A}\frac{b_{k}}{B}\leq\frac{1}{p}\left(\sum_{k=1}^{n}\frac{a_{k}^{p}}{A^{p}}\right)+\frac{1}{q}\left(\sum_{k=1}^{n}\frac{b_{k}^{q}}{B^{q}}\right)=1. +\frac{\sum_{k}a_{k}b_{k}}{AB}=\sum_{k}\frac{a_{k}}{A}\frac{b_{k}}{B}\leq\frac{1}{p}\left(\sum_{k}\frac{a_{k}^{p}}{A^{p}}\right)+\frac{1}{q}\left(\sum_{k}\frac{b_{k}^{q}}{B^{q}}\right)=1. \] \end_inset @@ -3323,6 +2898,11 @@ y si \end_layout +\end_inset + + +\end_layout + \begin_layout Standard La \series bold @@ -3357,6 +2937,11 @@ Desigualdad de Minkowski: \end_inset +\begin_inset Note Comment +status open + +\begin_layout Plain Layout + \series bold Demostración: \series default @@ -3384,9 +2969,9 @@ Demostración: , y por la desigualdad de Hölder, \begin_inset Formula \begin{align*} -\alpha & \coloneqq\sum_{k=1}^{n}(a_{k}+b_{k})^{p}=\sum_{k=1}^{n}a_{k}(a_{k}+b_{k})^{p-1}+\sum_{k=1}^{n}b_{k}(a_{k}+b_{k})^{p-1}\leq\\ - & \leq\left(\sum_{k=1}^{n}a_{k}^{p}\right)^{\frac{1}{p}}\left(\sum_{k=1}^{n}(a_{k}+b_{k})^{q(p-1)}\right)^{\frac{1}{q}}+\left(\sum_{k=1}^{n}b_{k}^{p}\right)^{\frac{1}{p}}\left(\sum_{k=1}^{n}(a_{k}+b_{k})^{q(p-1)}\right)^{\frac{1}{q}}=\\ - & =\left(\left(\sum_{k=1}^{n}a_{k}^{p}\right)^{\frac{1}{p}}+\left(\sum_{k=1}^{n}b_{k}^{p}\right)^{\frac{1}{p}}\right)\left(\sum_{k=1}^{n}(a_{k}+b_{k})^{p}\right)^{\frac{1}{q}}=\alpha^{\frac{1}{q}}\left(\left(\sum_{k=1}^{n}a_{k}^{p}\right)^{\frac{1}{p}}+\left(\sum_{k=1}^{n}b_{k}^{p}\right)^{\frac{1}{p}}\right), +\alpha & \coloneqq\sum_{k}(a_{k}+b_{k})^{p}=\sum_{k}a_{k}(a_{k}+b_{k})^{p-1}+\sum_{k}b_{k}(a_{k}+b_{k})^{p-1}\leq\\ + & \leq\left(\sum_{k}a_{k}^{p}\right)^{\frac{1}{p}}\left(\sum_{k}(a_{k}+b_{k})^{q(p-1)}\right)^{\frac{1}{q}}+\left(\sum_{k}b_{k}^{p}\right)^{\frac{1}{p}}\left(\sum_{k}(a_{k}+b_{k})^{q(p-1)}\right)^{\frac{1}{q}}=\\ + & =\left(\left(\sum_{k}a_{k}^{p}\right)^{\frac{1}{p}}+\left(\sum_{k}b_{k}^{p}\right)^{\frac{1}{p}}\right)\left(\sum_{k}(a_{k}+b_{k})^{p}\right)^{\frac{1}{q}}=\alpha^{\frac{1}{q}}\left(\left(\sum_{k}a_{k}^{p}\right)^{\frac{1}{p}}+\left(\sum_{k}b_{k}^{p}\right)^{\frac{1}{p}}\right), \end{align*} \end_inset @@ -3398,7 +2983,24 @@ y dividiendo entre se obtiene el resultado. \end_layout +\end_inset + + +\end_layout + \begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{samepage} +\end_layout + +\end_inset + Para \begin_inset Formula $n\in\mathbb{N}$ \end_inset @@ -3417,16 +3019,16 @@ Para , donde \begin_inset Formula \[ -\Vert x\Vert_{p}\coloneqq\left(\sum_{k=1}^{n}|x_{k}|^{p}\right)^{\frac{1}{p}}. +\Vert x\Vert_{p}\coloneqq\left(\sum_{k\in\mathbb{N}_{n}}|x_{k}|^{p}\right)^{\frac{1}{p}}. \] \end_inset -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Formula $\Vert\cdot\Vert_{p}$ \end_inset @@ -3438,7 +3040,11 @@ Para . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula $\ell_{n}^{\infty}\coloneqq(\mathbb{K}^{n},\Vert\cdot\Vert_{p})$ \end_inset @@ -3448,10 +3054,10 @@ Para \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Formula $\Vert\cdot\Vert_{\infty}$ \end_inset @@ -3462,7 +3068,27 @@ Para . \end_layout -\end_deeper +\end_inset + + +\end_layout + +\begin_layout Standard +\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 $X$ @@ -3486,6 +3112,11 @@ Si es un isomorfismo topológico. +\begin_inset Note Comment +status open + +\begin_layout Plain Layout + \series bold Demostración: \series default @@ -3501,7 +3132,7 @@ Demostración: , \begin_inset Formula \[ -\Vert T(a)\Vert=\left\Vert \sum_{k=1}^{n}a_{k}e_{k}\right\Vert \leq\sum_{k=1}^{n}|a_{k}|\sup_{k=1}^{n}\{\Vert e_{k}\Vert\}=\sup_{k=1}^{n}\{\Vert e_{k}\Vert\}\Vert(a_{1},\dots,a_{n})\Vert_{1}. +\Vert T(a)\Vert=\left\Vert \sum_{k}a_{k}e_{k}\right\Vert \leq\sum_{k}|a_{k}|\sup_{k}\{\Vert e_{k}\Vert\}=\sup_{k}\{\Vert e_{k}\Vert\}\Vert(a_{1},\dots,a_{n})\Vert_{1}. \] \end_inset @@ -3541,6 +3172,11 @@ Para la otra cota, \end_layout +\end_inset + + +\end_layout + \begin_layout Standard Así: \end_layout @@ -3560,10 +3196,10 @@ Toda norma en \end_inset genera la topología producto. -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Las bolas de \begin_inset Formula $\ell_{n}^{\infty}$ \end_inset @@ -3575,13 +3211,17 @@ Las bolas de genera la misma topología. \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Todo espacio de dimensión finita es de Banach. -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Lo es \begin_inset Formula $\ell_{n}^{\infty}$ \end_inset @@ -3589,7 +3229,11 @@ Lo es por tener topología producto. \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Todo subespacio de dimensión finita de un espacio normado es cerrado. \end_layout @@ -3597,10 +3241,10 @@ Todo subespacio de dimensión finita de un espacio normado es cerrado. \begin_layout Enumerate Todo operador entre espacios normados con dominio de dimensión finita es continuo. -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Por isomorfismo podemos suponer que el dominio es \begin_inset Formula $\ell_{n}^{1}$ \end_inset @@ -3615,13 +3259,17 @@ Por isomorfismo podemos suponer que el dominio es \end_inset , -\begin_inset Formula $\sup_{x\in S_{\ell_{n}^{1}}}\Vert T(x)\Vert=\sup_{\{x\in\mathbb{K}^{n}\mid \sum_{i}x_{i}=1\}}\left\Vert \sum_{i}x_{i}a_{i}\right\Vert =\sup_{i=1}^{n}a_{i}<\infty$ +\begin_inset Formula $\sup_{x\in S_{\ell_{n}^{1}}}\Vert T(x)\Vert=\sup_{\{x\in\mathbb{K}^{n}\mid\sum_{i}x_{i}=1\}}\left\Vert \sum_{i}x_{i}a_{i}\right\Vert =\sup_{i=1}^{n}a_{i}<\infty$ \end_inset . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Standard \series bold @@ -3756,7 +3404,6 @@ Si y \begin_inset Formula $d(M_{n},y_{n+1})\geq\frac{1}{2}$ - \end_inset con cada @@ -3764,8 +3411,7 @@ Si \end_inset y -\begin_inset Formula $d(M_{n},x_{n+1})\geq\frac{1}{2}$ >>>>>>> af - +\begin_inset Formula $\ensuremath{d(M_{n},x_{n+1})\geq\frac{1}{2}}$ \end_inset . @@ -3801,6 +3447,18 @@ Demostración: \end_layout \begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{samepage} +\end_layout + +\end_inset + \series bold Teorema de Riesz: @@ -3862,6 +3520,22 @@ Teorema de Riesz: \end_layout \begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{samepage} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard Dado un espacio normado \begin_inset Formula $X$ \end_inset @@ -3874,12 +3548,7225 @@ Si \begin_inset Formula $Y\leq X$ \end_inset - tiene dimensión finita + es cerrado y +\begin_inset Formula $Z\leq X$ +\end_inset + + tiene dimensión finita, +\begin_inset Formula $Y+Z\leq X$ +\end_inset + + es cerrado. +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +Si +\begin_inset Formula $Q:X\to\frac{X}{Y}$ +\end_inset + + es la función cociente, +\begin_inset Formula $Q(Z)$ +\end_inset + + es de dimensión finita y por tanto cerrado, y por continuidad +\begin_inset Formula $Q^{-1}(Q(Z))=Y+Z$ +\end_inset + + es cerrado. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +Sean +\begin_inset Formula $Y$ +\end_inset + + un espacio normado y +\begin_inset Formula $T:X\to Y$ +\end_inset + + una aplicación lineal con imagen de dimensión finita, +\begin_inset Formula $T$ +\end_inset + + es continua si y sólo si +\begin_inset Formula $\ker T$ +\end_inset + + es cerrado. +\begin_inset Note Comment +status open + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +\begin_inset Formula $\implies]$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula $0\leq T(X)$ +\end_inset + + es cerrado y por tanto +\begin_inset Formula $T^{-1}(0)$ +\end_inset + + también. +\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 + +Sean +\begin_inset Formula $Q:X\to\frac{X}{\ker T}$ +\end_inset + + la función cociente e +\begin_inset Formula $\iota:T(X)\hookrightarrow Y$ +\end_inset + + la inclusión, ambas continuas, existe un isomorfismo +\begin_inset Formula $J:\frac{X}{\ker T}\to T(X)$ +\end_inset + + con +\begin_inset Formula $T=\iota\circ J\circ Q$ +\end_inset + +, y como +\begin_inset Formula $T(X)$ +\end_inset + + es de dimensión finita, +\begin_inset Formula $J$ +\end_inset + + es un isomorfismo topológico, luego +\begin_inset Formula $J$ +\end_inset + + es continua y también lo es +\begin_inset Formula $T$ +\end_inset + +. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +Si +\begin_inset Formula $X$ +\end_inset + + es de Banach e +\begin_inset Formula $Y\leq X$ +\end_inset + + tiene codimensión finita, todo complementario algebraico +\begin_inset Formula $Z$ +\end_inset + + de +\begin_inset Formula $Y$ +\end_inset + + es un complementario topológico. +\end_layout + +\begin_layout Section +Ejemplos de espacios de Banach +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{reminder}{TEM} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $D\subseteq X$ +\end_inset + + es +\series bold +denso +\series default + en +\begin_inset Formula $(X,{\cal T})$ +\end_inset + + si +\begin_inset Formula $\overline{D}=X$ +\end_inset + +, si y sólo si cualquier abierto no vacío corta a +\begin_inset Formula $D$ +\end_inset + +. + +\begin_inset Formula $(X,{\cal T})$ +\end_inset + + es +\series bold +separable +\series default + si admite un subconjunto denso y numerable. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{reminder} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{reminder}{FVV2} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Llamamos +\series bold +soporte +\series default + de una función +\begin_inset Formula $g:\Omega\rightarrow\mathbb{C}$ +\end_inset + + a +\begin_inset Formula $\text{sop}(g)\coloneqq\overline{\{g\neq0\}}$ +\end_inset + +[...]. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{reminder} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Para +\begin_inset Formula $S\neq\emptyset$ +\end_inset + +, definimos +\begin_inset Formula $\Vert\cdot\Vert_{\infty}:\mathbb{K}^{S}\to[0,+\infty]$ +\end_inset + + como +\begin_inset Formula $\Vert f\Vert_{\infty}\coloneqq\sup_{s\in S}|f(s)|$ +\end_inset + + y llamamos +\series bold +espacio de las funciones acotadas +\series default + en +\begin_inset Formula $S$ +\end_inset + + al espacio de Banach +\begin_inset Formula $\ell^{\infty}(S)\coloneqq(\{f\in\mathbb{K}^{S}\mid\Vert f\Vert_{\infty}<\infty\},\Vert\cdot\Vert_{\infty})$ +\end_inset + +. + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\paragraph_spacing single +Si además +\begin_inset Formula $S$ +\end_inset + + es un espacio topológico: +\end_layout + +\begin_layout Enumerate +El espacio +\begin_inset Formula $C_{b}(S)$ +\end_inset + + de funciones +\begin_inset Formula $S\to\mathbb{K}$ +\end_inset + + continuas y acotadas es un subespacio cerrado de +\begin_inset Formula $\ell^{\infty}(S)$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +Una función +\begin_inset Formula $S\to\mathbb{K}$ +\end_inset + + +\series bold +se anula en el infinito +\series default + si +\begin_inset Formula $\forall\varepsilon>0,\exists K\subseteq S\text{ compacto}:\Vert f|_{S\setminus K}\Vert_{\infty}<\varepsilon$ +\end_inset + +. + El espacio +\begin_inset Formula $C_{0}(S)$ +\end_inset + + de funciones +\begin_inset Formula $S\to\mathbb{K}$ +\end_inset + + continuas que se anulan en el infinito es un subespacio cerrado de +\begin_inset Formula $C_{b}(S)$ +\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 $S$ +\end_inset + + es localmente compacto y Hausdorff, el espacio +\begin_inset Formula $C_{c}(S)$ +\end_inset + + de funciones +\begin_inset Formula $S\to\mathbb{K}$ +\end_inset + + continuas con soporte compacto es un subespacio denso de +\begin_inset Formula $C_{0}(S)$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Llamando +\begin_inset Formula $\ell^{\infty}\coloneqq\ell^{\infty}(\mathbb{N})$ +\end_inset + +: +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $\ell^{\infty}$ +\end_inset + + es no separable. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $c_{0}\coloneqq\{x\in\mathbb{K}^{\mathbb{N}}\mid\lim_{n}|x_{n}|=0\}$ +\end_inset + + y +\begin_inset Formula $c\coloneqq\{x\in\mathbb{K}^{\mathbb{N}}\mid\exists\lim_{n}x_{n}\}$ +\end_inset + + son subespacios cerrados y separables de +\begin_inset Formula $\ell^{\infty}$ +\end_inset + + con +\begin_inset Formula $c_{0}\subsetneq c\subsetneq\ell^{\infty}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Dados un espacio métrico +\begin_inset Formula $(X,d)$ +\end_inset + + y +\begin_inset Formula $x_{0}\in X$ +\end_inset + +, +\begin_inset Formula $\phi:(X,d)\to\ell^{\infty}(X)$ +\end_inset + + dado por +\begin_inset Formula $\phi(x)(y)\coloneqq d(x,y)-d(x_{0},y)$ +\end_inset + + es una isometría. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{reminder}{FVV2} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Un +\series bold +álgebra de partes +\series default + de +\begin_inset Formula $\Omega\neq\emptyset$ +\end_inset + + es un conjunto no vacío +\begin_inset Formula ${\cal A}\subseteq{\cal P}(\Omega)$ +\end_inset + + tal que +\begin_inset Formula $A\in{\cal A}\implies A^{\complement}\in{\cal A}$ +\end_inset + + y +\begin_inset Formula $A,B\in{\cal A}\implies A\cup B\in{\cal A}$ +\end_inset + +. + [...] Una +\series bold + +\begin_inset Formula $\sigma$ +\end_inset + +-álgebra de partes +\series default + de +\begin_inset Formula $\Omega$ +\end_inset + + es un álgebra +\begin_inset Formula $\Sigma\subseteq{\cal P}(\Omega)$ +\end_inset + + tal que si +\begin_inset Formula $\{A_{n}\}_{n}\subseteq\Sigma$ +\end_inset + + entonces +\begin_inset Formula $\bigcup_{n\in\mathbb{N}}A_{n}\in\Sigma$ +\end_inset + + [...]. + [...] +\end_layout + +\begin_layout Standard +Dada una +\begin_inset Formula $\sigma$ +\end_inset + +-álgebra +\begin_inset Formula $\Sigma$ +\end_inset + + de partes de +\begin_inset Formula $\Omega$ +\end_inset + +, una función +\begin_inset Formula $\mu:\Sigma\rightarrow[0,+\infty]$ +\end_inset + + es una +\series bold +medida +\series default + [...] sobre +\begin_inset Formula $\Sigma$ +\end_inset + + si +\begin_inset Formula $\mu(\emptyset)=0$ +\end_inset + + y para toda familia +\begin_inset Formula $\{A_{n}\}_{n\in\mathbb{N}}\subseteq\Sigma$ +\end_inset + + de conjuntos disjuntos [...] +\begin_inset Formula $\sigma(\bigcup_{n\in\mathbb{N}}A_{n})=\sum_{n\in\mathbb{N}}\mu(A_{n})$ +\end_inset + +. + Si existe [...] +\begin_inset Formula $(\Omega,\Sigma)$ +\end_inset + + es un +\series bold +espacio medible +\series default +. +\end_layout + +\begin_layout Standard +Un +\series bold +espacio de medida +\series default + es una terna +\begin_inset Formula $(\Omega,\Sigma,\mu)$ +\end_inset + + donde +\begin_inset Formula $(\Omega,\Sigma)$ +\end_inset + + es un espacio medible y +\begin_inset Formula $\mu$ +\end_inset + + es una medida sobre este. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{reminder} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Dados un espacio de medida +\begin_inset Formula $(\Omega,\Sigma,\mu)$ +\end_inset + + y, para +\begin_inset Formula $p\geq1$ +\end_inset + +, +\begin_inset Formula $\Vert\cdot\Vert_{p}:\mathbb{K}^{\Omega}\to[0,+\infty]$ +\end_inset + + dada por +\begin_inset Formula +\[ +\Vert f\Vert_{p}\coloneqq\left(\int|f(x)|^{p}\dif\mu(x)\right)^{\frac{1}{p}}, +\] + +\end_inset + + +\begin_inset Formula $L^{p}(\Omega)\coloneqq L^{p}(\Omega,\Sigma,\mu)\coloneqq(\{f\in\mathbb{K}^{\Omega}\mid\Vert f\Vert_{p}<\infty\},\Vert\cdot\Vert_{p})$ +\end_inset + + es un espacio de Banach. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $\Omega$ +\end_inset + + es un abierto de +\begin_inset Formula $\mathbb{R}^{n}$ +\end_inset + + con la medida de Lebesgue inducida, +\begin_inset Formula $C_{c}(\Omega)$ +\end_inset + + es denso en +\begin_inset Formula $L^{p}(\Omega)$ +\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 $p\geq1$ +\end_inset + +, llamamos +\begin_inset Formula $\ell^{p}\coloneqq L^{p}(\mathbb{N})$ +\end_inset + +, y entonces: +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $\ell^{p}$ +\end_inset + + es separable. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +El +\series bold +espacio de las sucesiones finitamente no nulas +\series default +, +\begin_inset Formula +\[ +c_{00}\coloneqq\{x\in\mathbb{K}^{\mathbb{N}}\mid\exists n_{0}\in\mathbb{N}:\forall n\geq n_{0},x_{n}=0\}, +\] + +\end_inset + +es un subespacio denso de +\begin_inset Formula $\ell^{p}$ +\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 $1\leq p<q$ +\end_inset + +, +\begin_inset Formula $\ell^{p}\subsetneq\ell^{q}\subsetneq c_{0}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +La suma de subespacios cerrados puede no ser un subespacio cerrado. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Sean +\begin_inset Formula $\Omega\subseteq\mathbb{R}^{n}$ +\end_inset + + un abierto no vacío, +\begin_inset Formula $K\subseteq\Omega$ +\end_inset + + compacto y, para +\begin_inset Formula $m\in\mathbb{N}$ +\end_inset + +, +\begin_inset Formula $\Vert\cdot\Vert_{m}:{\cal C}^{m}(\Omega)\to[0,+\infty)$ +\end_inset + + dada por +\begin_inset Formula +\[ +\Vert f\Vert_{m}\coloneqq\sup_{x\in\Omega}\sup_{\begin{subarray}{c} +\alpha\in\mathbb{N}^{n}\\ +|\alpha|\coloneqq\alpha_{1}+\dots+\alpha_{n}\leq m +\end{subarray}}\left|\frac{\partial^{|\alpha|}f}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{n}}}\right|, +\] + +\end_inset + +entonces: +\end_layout + +\begin_layout Enumerate +Para +\begin_inset Formula $K\subseteq\Omega$ +\end_inset + + compacto, +\begin_inset Formula ${\cal D}_{K}^{m}(\Omega)\coloneqq(\{f\in{\cal C}^{m}(\Omega)\mid\text{sop}f\subseteq K\},\Vert\cdot\Vert_{m})$ +\end_inset + + es un espacio de Banach. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Formula ${\cal D}^{m}(\Omega)\coloneqq(\{f\in{\cal C}^{m}(\Omega)\mid\text{sop}f\text{ compacto}\},\Vert\cdot\Vert_{m})$ +\end_inset + + es un espacio normado, y en particular lo es +\begin_inset Formula $(C_{c}(\Omega),\Vert\cdot\Vert_{\infty})={\cal D}^{0}(\Omega)$ +\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 ${\cal D}(\Omega)\coloneqq\{f\in{\cal C}^{\infty}(\Omega)\mid\text{sop}f\text{ compacto}\}$ +\end_inset + + es un subespacio de +\begin_inset Formula ${\cal D}^{m}(\Omega)$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Sean +\begin_inset Formula $D\coloneqq\{x\in\mathbb{C}:|x|<1\}$ +\end_inset + + y +\begin_inset Formula ${\cal H}(D)$ +\end_inset + + el espacio de funciones holomorfas +\begin_inset Formula $D\to\mathbb{C}$ +\end_inset + + y, para +\begin_inset Formula $p\geq1$ +\end_inset + +, +\begin_inset Formula $\Vert\cdot\Vert_{H_{p}},\Vert\cdot\Vert_{H_{\infty}}:{\cal H}(D)\to[0,\infty]$ +\end_inset + + dadas por +\begin_inset Formula +\begin{align*} +\Vert f\Vert_{H_{p}} & \coloneqq\sup_{r\in(0,1)}\left(\frac{1}{2\pi}\int_{0}^{2\pi}|f(r\text{e}^{\text{i}\theta})|^{p}\dif\theta\right)^{\frac{1}{p}}, & \Vert f\Vert_{H_{\infty}} & \coloneqq\Vert f\Vert_{\infty}, +\end{align*} + +\end_inset + +entonces para +\begin_inset Formula $p\in[1,\infty]$ +\end_inset + +, el +\series bold +espacio de Hardy +\series default + +\begin_inset Formula $H^{p}(D)\coloneqq(\{f\in{\cal H}(D):\Vert f\Vert_{H_{p}}<\infty)$ +\end_inset + + es un espacio de Banach. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\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 + + +\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 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 $\dim H=\mathbb{N}$ +\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 -TODO +nproof \end_layout \end_inset |