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zfBaUeHGk6IH%cw;rT=rS?hAuTZSK!1O6~4%-zjytfBvP^>Hd9Esp$S^M5)U&|DaUz ztWZ_z_N)|2J)XT-R_gWakg69XL# PH!v_XG72RnMNdWwt1q+} diff --git a/ana4.tex b/ana4.tex index ce2601a..a3159c8 100644 --- a/ana4.tex +++ b/ana4.tex @@ -88,8 +88,8 @@ Das heißt $\lim_{n \to \infty} x_{i}^{(k)} = x_{i}$ (komponentenweise Konvergenz in $\R$ oder $\C$) \end{bem} -\begin{satz} - (Satz von Cauchy und Satz von Bolzano-Weierstraß) +\begin{satz}[Satz von Cauchy und Satz von Bolzano-Weierstraß] + \label{satz:bolzano} \begin{enumerate}[1)] \item Jede Cauchy-Folge in $K^{n}$ konvergiert, d.h. der normierte Raum $(K^{n}, \norm{\cdot}_{\infty})$ ist vollständig. Ein vollständiger normierter Raum wird Banach-Raum genannt. \item Jede beschränkte Folge in $K^{n}$ besitzt eine konvergente Teilfolge. @@ -166,8 +166,8 @@ Bezeichnung: $\norm{\cdot}$ irgendeine Norm. \end{enumerate} \end{bsp} -\begin{satz} - (Eigenschaften offener Mengen) Es gilt: +\begin{satz}[Eigenschaften offener Mengen] + Es gilt: \begin{enumerate}[(1)] \item Sind $U$ und $V (\subseteq \K^{n})$ offen, dann ist $U \cap V$ offen. \item Sei $U_{i} \subset \K^{n}, i \in I$ eine Familie offener Teilmengen. Dann ist auch $\underset{i \in I}{\bigcup} U_{i}$ offen. @@ -188,8 +188,8 @@ Bezeichnung: $\norm{\cdot}$ irgendeine Norm. \end{enumerate} \end{korrolar} -\begin{definition} - (Abgeschlossene Menge) Eine Teilmenge $A \subset \K^{n}$ heißt abgeschlossen, wenn ihr Komplement $A^{c} \coloneqq \K^{n} \setminus A$ offen ist. +\begin{definition}[Abgeschlossene Menge] + Eine Teilmenge $A \subset \K^{n}$ heißt abgeschlossen, wenn ihr Komplement $A^{c} \coloneqq \K^{n} \setminus A$ offen ist. \end{definition} \begin{bsp} @@ -202,8 +202,7 @@ Bezeichnung: $\norm{\cdot}$ irgendeine Norm. \end{enumerate} \end{bsp} -\begin{satz} - (Eigenschaften abgeschlossener Mengen) +\begin{satz}[Eigenschaften abgeschlossener Mengen] \begin{enumerate}[(1)] \item Sind $V, U$ ($V,U \subset \K^{n}$) abgeschlossen, dann ist $U \cup V \subset \K^{n}$ auch abgeschlossen. \item Sind $U_{i}, (i \in I)$ abgeschlossene Menge in $\K^{n}$. Dann ist $\underset{i \in I}{\bigcap} U_{i}$ auch abgeschlossen. @@ -226,15 +225,4 @@ Bezeichnung: $\norm{\cdot}$ irgendeine Norm. \end{enumerate} \end{bsp} - - - - - - - - - - - -\end{document} \ No newline at end of file +\end{document}