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Diffstat (limited to 'af')
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| -rw-r--r-- | af/n1.lyx | 1133 |
2 files changed, 1132 insertions, 3 deletions
@@ -116,7 +116,7 @@ status open \backslash def \backslash -cryear{2021} +cryear{2023} \end_layout \end_inset @@ -9243,7 +9243,898 @@ método de Galerkin \end_layout \begin_layout Section -Bases hilbertianas +Redes +\end_layout + +\begin_layout Standard +Un +\series bold +conjunto dirigido +\series default + es un par +\begin_inset Formula $(D,\geq)$ +\end_inset + + formado por un conjunto +\begin_inset Formula $D$ +\end_inset + + y una relación +\begin_inset Formula $\geq$ +\end_inset + + transitiva reflexiva y tal que +\begin_inset Formula $\forall i,j\in D,\exists k\in D:k\geq i,j$ +\end_inset + +. + Una +\series bold +red +\series default + en un conjunto +\begin_inset Formula $Y$ +\end_inset + + es una función +\begin_inset Formula $\phi:D\to Y$ +\end_inset + + donde +\begin_inset Formula $(D,\geq)$ +\end_inset + + es un conjunto dirigido, que escribimos como +\begin_inset Formula $\phi\eqqcolon\{\omega_{i}\}_{i\in D}$ +\end_inset + + con +\begin_inset Formula $\omega_{i}\coloneqq\phi(i)$ +\end_inset + +. + Todo conjunto totalmente ordenado es dirigido, y en particular +\begin_inset Formula $(\mathbb{N},\geq)$ +\end_inset + + lo es y así las sucesiones son redes. +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $X$ +\end_inset + + es un espacio topológico, la red +\begin_inset Formula $\{x_{i}\}_{i\in D}\subseteq X$ +\end_inset + + +\series bold +converge +\series default + a +\begin_inset Formula $z\in T$ +\end_inset + + (con +\series bold +convergencia de Moore-Smith +\series default +) si +\begin_inset Formula $\forall V\in{\cal E}(z),\exists i_{0}\in D:\forall i\geq i_{0},x_{i}\in V$ +\end_inset + +, y +\begin_inset Formula $s\in X$ +\end_inset + + es +\series bold +de aglomeración +\series default + de +\begin_inset Formula $(x_{i})_{i\in D}$ +\end_inset + + si +\begin_inset Formula $\forall V\in{\cal E}(s),\forall j\in D,\exists i\geq j:x_{i}\in V$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $(X,\Vert\cdot\Vert)$ +\end_inset + + es un espacio normado, una red +\begin_inset Formula $\{x_{i}\}_{i\in D}\subseteq X$ +\end_inset + + es +\series bold +de Cauchy +\series default + o satisface la +\series bold +condición de Cauchy +\series default + si +\begin_inset Formula +\[ +\forall\varepsilon>0,\exists i_{0}\in D:\forall i,j\geq i_{0},\Vert x_{i}-x_{j}\Vert<\varepsilon. +\] + +\end_inset + +Si +\begin_inset Formula $X$ +\end_inset + + es de Banach, toda red de Cauchy es convergente. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Una +\series bold +subred +\series default + de la red +\begin_inset Formula $\phi:D\to Y$ +\end_inset + + es una red +\begin_inset Formula $\phi\circ\rho:J\to Y$ +\end_inset + + para cierta +\begin_inset Formula $\rho:J\to D$ +\end_inset + + que cumple que +\begin_inset Formula $\forall i_{0}\in D,\exists j_{0}\in J:\forall j\geq j_{0},\rho(j)\geq i_{0}$ +\end_inset + +. + Si +\begin_inset Formula $D$ +\end_inset + + es un conjunto dirigido, +\begin_inset Formula $J\subseteq D$ +\end_inset + + es +\series bold +cofinal +\series default + en +\begin_inset Formula $D$ +\end_inset + + si +\begin_inset Formula $\forall i\in D,\exists j\in J:j\geq i$ +\end_inset + +, y entonces, si +\begin_inset Formula $(\omega_{i})_{i\in D}$ +\end_inset + + es una red, +\begin_inset Formula $(\omega_{i})_{i\in J}$ +\end_inset + + es una subred suya. + Si una red converge en un espacio topológico, toda subred suya converge + al mismo punto. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Sea +\begin_inset Formula $X$ +\end_inset + + un espacio topológico: +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $X$ +\end_inset + + es de Hausdorff si y sólo si toda red +\begin_inset Formula $\{x_{i}\}_{i\in D}\subseteq X$ +\end_inset + + convergente converge a un único +\begin_inset Formula $z\in X$ +\end_inset + +, en cuyo caso +\begin_inset Formula $z$ +\end_inset + + es el +\series bold +límite +\series default + de la red, +\begin_inset Formula $\lim_{i\in D}x_{i}=z$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +Un +\begin_inset Formula $s\in X$ +\end_inset + + es de aglomeración de una red en +\begin_inset Formula $X$ +\end_inset + + si y sólo si esta admite una subred convergente a +\begin_inset Formula $s$ +\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 $A\subseteq X$ +\end_inset + + es cerrado si y sólo si toda red convergente en +\begin_inset Formula $A$ +\end_inset + + tiene límite en +\begin_inset Formula $A$ +\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 $s\in X$ +\end_inset + + y +\begin_inset Formula $A\subseteq X$ +\end_inset + +, +\begin_inset Formula $s\in\overline{A}$ +\end_inset + + si y sólo si es límite de una red en +\begin_inset Formula $A$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Si +\begin_inset Formula $Y$ +\end_inset + + es otro espacio topológico, +\begin_inset Formula $f:X\to Y$ +\end_inset + + es continua en +\begin_inset Formula $a\in Y$ +\end_inset + + si y sólo si lleva redes en +\begin_inset Formula $X$ +\end_inset + + convergentes a +\begin_inset Formula $a$ +\end_inset + + a redes en +\begin_inset Formula $Y$ +\end_inset + + convergentes a +\begin_inset Formula $f(a)$ +\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 $A\subseteq X$ +\end_inset + + es compacto si y sólo si toda red en +\begin_inset Formula $A$ +\end_inset + + posee una subred convergente a un punto de +\begin_inset Formula $A$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $X$ +\end_inset + + es un espacio métrico, +\begin_inset Formula $s\in X$ +\end_inset + + es de aglomeración de una sucesión si y sólo si esta posee una subsucesión + convergente a +\begin_inset Formula $s$ +\end_inset + +, pero esto no es cierto en espacios topológicos arbitrarios. + Sean +\begin_inset Formula $X\coloneqq\mathbb{N}\times\mathbb{N}$ +\end_inset + + y +\begin_inset Formula +\[ +{\cal T}\coloneqq\{\{x\}\}_{x\in X\setminus0}\cup\{V\subseteq X\mid(0,0)\in V\land\exists n_{0}\in\mathbb{N}:\forall n\geq n_{0},\{m\in\mathbb{N}\mid(n,m)\notin V\}\text{ es finito}\}, +\] + +\end_inset + + +\begin_inset Formula $(X,{\cal T})$ +\end_inset + + es un espacio de Hausdorff en el que ninguna sucesión converge a +\begin_inset Formula $(0,0)$ +\end_inset + + pero la sucesión resultante de enumerar +\begin_inset Formula $A$ +\end_inset + + según el proceso diagonal de Cantor tiene a +\begin_inset Formula $(0,0)$ +\end_inset + + como punto de aglomeración. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Sean +\begin_inset Formula $Y\coloneqq[0,1]^{\mathbb{R}}$ +\end_inset + + con la topología producto y, para +\begin_inset Formula $y\in Y$ +\end_inset + +, +\begin_inset Formula $\text{sop}(y)\coloneqq\{\gamma\in\mathbb{R}\mid y_{\gamma}\neq0\}$ +\end_inset + +, +\begin_inset Formula $D\coloneqq\{y\in Y\mid\text{sop}(x_{\gamma})_{\gamma\in\mathbb{R}}\text{ contable}\}$ +\end_inset + + es denso en +\begin_inset Formula $Y$ +\end_inset + + y cerrado por sucesiones, y de hecho, toda sucesión en +\begin_inset Formula $D$ +\end_inset + + tiene una subsucesión convergente a un punto de +\begin_inset Formula $D$ +\end_inset + +, pero +\begin_inset Formula $D$ +\end_inset + + no es cerrado ni compacto. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Section +Familias sumables +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $I\neq\emptyset$ +\end_inset + +, llamamos +\begin_inset Formula ${\cal P}_{0}(I)\coloneqq\{J\subseteq I\mid J\text{ finito}\}$ +\end_inset + +, que es un conjunto dirigido por +\begin_inset Formula $\supseteq$ +\end_inset + +. + Si +\begin_inset Formula $(X,\Vert\cdot\Vert)$ +\end_inset + + es un espacio normado e +\begin_inset Formula $I\neq\emptyset$ +\end_inset + +, +\begin_inset Formula $\{x_{i}\}_{i\in I}\subseteq X$ +\end_inset + + es +\series bold +sumable +\series default + con +\series bold +suma +\series default + +\begin_inset Formula $s\in X$ +\end_inset + + si +\begin_inset Formula $(\sum_{i\in J}x_{i})_{J\in{\cal P}_{0}(I)}$ +\end_inset + + converge a +\begin_inset Formula $s$ +\end_inset + +, y es +\series bold +absolutamente sumable +\series default + si +\begin_inset Formula $(\Vert x_{i}\Vert)_{i\in I}$ +\end_inset + + es sumable en +\begin_inset Formula $\mathbb{R}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $(X,\Vert\cdot\Vert)$ +\end_inset + + es un espacio normado, +\begin_inset Formula $I\neq\emptyset$ +\end_inset + + y +\begin_inset Formula $\{x_{i}\}_{i\in I}\subseteq X$ +\end_inset + +: +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $(\sum_{i\in J}x_{i})_{J\in{\cal P}_{0}(I)}$ +\end_inset + + es de Cauchy si y sólo si +\begin_inset Formula +\[ +\forall\varepsilon>0,\exists J_{0}\in{\cal P}_{0}(I):\forall J\in{\cal P}_{0}(I\setminus J_{0}),\left\Vert \sum_{i\in J}x_{i}\right\Vert <\varepsilon, +\] + +\end_inset + +y entonces +\begin_inset Formula $\{i\in I\mid x_{i}\neq0\}$ +\end_inset + + es contable y +\begin_inset Formula $\sup_{J\in{\cal P}_{0}(I)}\left\Vert \sum_{i\in J}x_{i}\right\Vert <\infty$ +\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 $(x_{i})_{i\in I}$ +\end_inset + + es absolutamente sumable si y sólo si +\begin_inset Formula $\sup_{J\in{\cal P}_{0}(I)}\sum_{i\in J}\Vert x_{i}\Vert<\infty$ +\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 $X$ +\end_inset + + es de Banach, toda familia absolutamente sumable es sumable. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $X$ +\end_inset + + es de Banach si y sólo si toda familia sumable es absolutamente sumable. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $(X,\Vert\cdot\Vert)$ +\end_inset + + es un espacio de Banach, llamamos +\begin_inset Formula +\[ +C(S)\coloneqq\sup\left\{ C\in[0,1)\;\middle|\;\forall n\in\mathbb{N},\forall z\in X^{n},\exists S\subseteq\mathbb{N}_{n}:C\sum_{j\in\mathbb{N}_{n}}\Vert z_{j}\Vert\leq\left\Vert \sum_{j\in S}z_{j}\right\Vert \right\} , +\] + +\end_inset + +y +\begin_inset Formula $\Vert\cdot\Vert$ +\end_inset + + en +\begin_inset Formula $X$ +\end_inset + + tiene la +\series bold +propiedad S +\series default + si +\begin_inset Formula $C(S)>0$ +\end_inset + +. + Por ejemplo +\begin_inset Formula $\Vert\cdot\Vert_{2}$ +\end_inset + + en +\begin_inset Formula $\mathbb{R}^{n}$ +\end_inset + + tiene la propiedad S y +\begin_inset Formula $C(S)\in[\frac{1}{2n\sqrt{n}},\frac{1}{\sqrt{n}}]$ +\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 $(X,\Vert\cdot\Vert)$ +\end_inset + + es de dimensión finita y +\begin_inset Formula $\{x_{i}\}_{i\in I}\subseteq X$ +\end_inset + + no es vacía: +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $(x_{i})_{i\in I}$ +\end_inset + + es absolutamente sumable si y sólo si es sumable. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +Si +\begin_inset Formula $I=(\mathbb{N},\geq)$ +\end_inset + +, +\begin_inset Formula $(x_{n})_{n\in\mathbb{N}}$ +\end_inset + + es sumable si y sólo si +\begin_inset Formula $\sum_{n}x_{n}$ +\end_inset + + es absolutamente convergente, si y sólo si +\begin_inset Formula $\sup_{n\in\mathbb{N}}\sum_{i=1}^{n}\Vert x_{i}\Vert<\infty$ +\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 reordenación de Riemann: +\series default + Si la serie real +\begin_inset Formula $\sum_{n}x_{n}$ +\end_inset + + es convergente pero no absolutamente convergente, para +\begin_inset Formula $x\in[-\infty,\infty]$ +\end_inset + +, existe +\begin_inset Formula $\pi:\mathbb{N}\to\mathbb{N}$ +\end_inset + + tal que +\begin_inset Formula $x=\sum_{n=1}^{\infty}x_{\pi(n)}$ +\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 $X$ +\end_inset + + es un espacio de Banach y +\begin_inset Formula $\{x_{n}\}_{n}\subseteq X$ +\end_inset + + es una sucesión, +\begin_inset Formula $\sum_{n}x_{n}$ +\end_inset + + es +\series bold +incondicionalmente convergente +\series default + si para +\begin_inset Formula $\pi:\mathbb{N}\to\mathbb{N}$ +\end_inset + +, +\begin_inset Formula $\sum_{n}x_{\pi(n)}$ +\end_inset + + converge. + Si +\begin_inset Formula $X$ +\end_inset + + es de Banach, esto ocurre si y sólo si +\begin_inset Formula $(x_{n})_{n}$ +\end_inset + + es sumable, en cuyo caso todas las +\begin_inset Formula $\sum_{n}x_{\pi(n)}$ +\end_inset + + convergen al mismo número. +\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 +,un espacio de Banach +\begin_inset Formula $(X,\Vert\cdot\Vert)$ +\end_inset + + es de dimensión finita si y sólo si tiene la propiedad +\begin_inset Formula $S$ +\end_inset + +, si y sólo si +\begin_inset Formula +\[ +\forall\varepsilon>0,\exists\delta>0:\forall\{z_{j}\}_{j\in\mathbb{N}_{n}}\subseteq X,\left(\sup_{S\subseteq\mathbb{N}_{n}}\left\Vert \sum_{j\in S}z_{j}\right\Vert <\delta\implies\sum_{j\in J}\Vert z_{j}\Vert<\varepsilon\right), +\] + +\end_inset + +si y sólo si toda serie sumable en +\begin_inset Formula $X$ +\end_inset + + es absolutamente convergente. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Section +Bases hilbertiana \end_layout \begin_layout Standard @@ -9749,7 +10640,7 @@ Segundo teorema de Riesz-Fischer: \end_inset es de dimensión infinita, -\begin_inset Formula $\dim H=\mathbb{N}$ +\begin_inset Formula $\dim H=\aleph_{0}\coloneqq|\mathbb{N}|$ \end_inset si y sólo si @@ -10774,5 +11665,243 @@ nproof \end_layout +\begin_layout Section +El espacio de Bergman +\end_layout + +\begin_layout Standard +Llamamos +\begin_inset Formula $D(a,r)\coloneqq B(a,r)\subseteq\mathbb{C}$ +\end_inset + +. + Si +\begin_inset Formula $\Omega\subseteq\mathbb{C}$ +\end_inset + + es abierto, +\begin_inset Formula ${\cal H}(\Omega)$ +\end_inset + + es el conjunto de las funciones holomorfas en +\begin_inset Formula $\Omega$ +\end_inset + +, y para +\begin_inset Formula $f\in{\cal H}(\Omega)$ +\end_inset + + y +\begin_inset Formula $\overline{D(a,r)}\subseteq\Omega$ +\end_inset + +, la serie +\begin_inset Formula $\sum_{n\in\mathbb{N}}a_{n}(z-a)^{n}$ +\end_inset + + con +\begin_inset Formula $z\in D(a,r)$ +\end_inset + + converge uniformemente a +\begin_inset Formula $f$ +\end_inset + + en compactos de +\begin_inset Formula $D(a,r)$ +\end_inset + + para ciertos +\begin_inset Formula $a_{n}\in\mathbb{C}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $\Omega\subseteq\mathbb{C}$ +\end_inset + + es abierto, llamamos +\begin_inset Formula ${\cal T}_{\text{K}}$ +\end_inset + + a la topología en +\begin_inset Formula ${\cal H}(\Omega)$ +\end_inset + + de convergencia uniforme sobre compactos, y +\series bold +espacio de Bergman +\series default + en el abierto +\begin_inset Formula $\Omega\subseteq\mathbb{C}$ +\end_inset + + a +\begin_inset Formula +\[ +A^{2}(\Omega)\coloneqq\left\{ f\in{\cal H}(\Omega)\;\middle|\;\int_{\Omega}|f|^{2}<\infty\right\} , +\] + +\end_inset + +un subespacio cerrado y separable de +\begin_inset Formula $L^{2}(\Omega)$ +\end_inset + + que es pues un espacio de Hilbert numerable con +\begin_inset Formula $\langle\cdot,\cdot\rangle_{2}$ +\end_inset + +, y en el que la topología inducida por +\begin_inset Formula $L^{2}(\Omega)$ +\end_inset + + es más fina que la inducida por +\begin_inset Formula ${\cal T}_{\text{K}}$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $\Omega\subseteq\mathbb{C}$ +\end_inset + + es abierto, +\begin_inset Formula $(\omega_{n})_{n}$ +\end_inset + + es base hilbertiana de +\begin_inset Formula $A^{2}(\Omega)$ +\end_inset + + y +\begin_inset Formula $f\in A^{2}(\Omega)$ +\end_inset + +, el desarrollo en serie de Fourier de +\begin_inset Formula $f$ +\end_inset + +, +\begin_inset Formula $\sum_{n}\langle f,\omega_{n}\rangle\omega_{n}$ +\end_inset + +, converge uniformemente a +\begin_inset Formula $f$ +\end_inset + + en compactos de +\begin_inset Formula $\Omega$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $\psi_{n}(z)\coloneqq(z-a)^{n}$ +\end_inset + +, +\begin_inset Formula $(\frac{\psi_{n}}{\Vert\psi_{n}\Vert})_{n}$ +\end_inset + + es una base hilbertiana de +\begin_inset Formula $A^{2}(D(a,r))$ +\end_inset + +, y el desarrollo en serie de potencias es el desarrollo en serie de Fourier + sobre esta base. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Como +\series bold +teorema +\series default +, si +\begin_inset Formula $\Omega\subsetneq\mathbb{C}$ +\end_inset + + es un abierto simplemente conexo y +\begin_inset Formula $f:\Omega\to D(0,1)$ +\end_inset + + es un isomorfismo, +\begin_inset Formula +\[ +\left(z\mapsto\sqrt{\frac{n}{\pi}}(f(z))^{n-1}\dot{f}(z)\right)_{n} +\] + +\end_inset + +es base hilbertiana de +\begin_inset Formula $A^{2}(\Omega)$ +\end_inset + +, y en particular para +\begin_inset Formula $R>0$ +\end_inset + +, +\begin_inset Formula +\[ +\left(z\mapsto\sqrt{\frac{n}{\pi}}R^{-n}z^{n-1}\right)_{n} +\] + +\end_inset + + es base hilbertiana de +\begin_inset Formula $A^{2}(D(0,R))$ +\end_inset + +. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +nproof +\end_layout + +\end_inset + + +\end_layout + \end_body \end_document |