diff options
Diffstat (limited to 'af/n1.lyx')
| -rw-r--r-- | af/n1.lyx | 132 |
1 files changed, 116 insertions, 16 deletions
@@ -3710,24 +3710,23 @@ Demostración: \end_inset es -\begin_inset Note Note -status open - -\begin_layout Plain Layout -TODO -\end_layout +\begin_inset Formula +\begin{align*} +\Vert x-y\Vert & =\left\Vert \frac{x_{0}-y_{0}}{\Vert x_{0}-y_{0}\Vert}-y\right\Vert =\frac{1}{\Vert x_{0}-y_{0}\Vert}\left\Vert x_{0}-y_{0}-\Vert x_{0}-y_{0}\Vert y\right\Vert =\\ + & =\frac{\left\Vert x_{0}-(y_{0}+\Vert x_{0}-y_{0}\Vert y)\right\Vert }{\Vert x_{0}-y_{0}\Vert}\geq\frac{d}{\Vert x_{0}-y_{0}\Vert}>1-\varepsilon, +\end{align*} \end_inset - -\begin_inset Formula -\[ -\Vert x-y\Vert=\left\Vert \frac{x_{0}-y_{0}}{\Vert x_{0}-y_{0}\Vert}-y\right\Vert =\frac{1}{\Vert x_{0}-y_{0}\Vert}\left\Vert x_{0}-y_{0}-\Vert x_{0}-y_{0}\Vert y\right\Vert \geq -\] - +donde usamos que +\begin_inset Formula $y_{0}+\Vert x_{0}-y_{0}\Vert y\in Y$ \end_inset +, y entonces +\begin_inset Formula $d(x,Y)\geq1-\varepsilon$ +\end_inset +. \end_layout \begin_layout Standard @@ -3748,15 +3747,15 @@ Si \end_inset y una sucesión de vectores unitarios -\begin_inset Formula $\{y_{n}\}_{n}\subseteq X$ +\begin_inset Formula $\{x_{n}\}_{n}\subseteq X$ \end_inset con cada -\begin_inset Formula $y_{n}\in M_{n}$ +\begin_inset Formula $x_{n}\in M_{n}$ \end_inset y -\begin_inset Formula $d(M_{n},y_{n+1})\geq\frac{1}{2}$ +\begin_inset Formula $d(M_{n},x_{n+1})\geq\frac{1}{2}$ \end_inset . @@ -3764,7 +3763,108 @@ Si \series bold Demostración: \series default - + Tomamos +\begin_inset Formula $x_{1}\in X$ +\end_inset + + unitario y por inducción, para +\begin_inset Formula $n\geq1$ +\end_inset + +, +\begin_inset Formula $M_{n}\coloneqq\text{span}\{x_{1},\dots,x_{n}\}\neq X$ +\end_inset + + por ser +\begin_inset Formula $X$ +\end_inset + + de dimensión infinita, luego por el lema de Riesz existe +\begin_inset Formula $x_{n+1}\in X$ +\end_inset + + unitario con +\begin_inset Formula $d(x_{n+1},M_{n})\geq\frac{1}{2}$ +\end_inset + +. +\end_layout + +\begin_layout Standard + +\series bold +Teorema de Riesz: +\series default + Un espacio normado +\begin_inset Formula $X$ +\end_inset + + es de dimensión finita si y sólo si todo cerrado y acotado de +\begin_inset Formula $X$ +\end_inset + + es compacto, si y sólo si +\begin_inset Formula $B_{X}$ +\end_inset + + es compacta. +\end_layout + +\begin_layout Description +\begin_inset Formula $1\implies2]$ +\end_inset + + Visto. +\end_layout + +\begin_layout Description +\begin_inset Formula $2\implies3]$ +\end_inset + + Obvio. +\end_layout + +\begin_layout Description +\begin_inset Formula $3\implies1]$ +\end_inset + + Si +\begin_inset Formula $X$ +\end_inset + + tuviera dimensión infinita, habría una sucesión +\begin_inset Formula $\{y_{n}\}_{n}\in S_{X}\subseteq B_{X}$ +\end_inset + + con +\begin_inset Formula $\Vert y_{n}-y_{m}\Vert\geq\frac{1}{2}$ +\end_inset + + para cada +\begin_inset Formula $n\neq m$ +\end_inset + + y por tanto no hay puntos de acumulación. +\begin_inset Formula $\#$ +\end_inset + + +\end_layout + +\begin_layout Standard +Dado un espacio normado +\begin_inset Formula $X$ +\end_inset + +: +\end_layout + +\begin_layout Enumerate +Si +\begin_inset Formula $Y\leq X$ +\end_inset + + tiene dimensión finita \begin_inset Note Note status open |