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\end{enumerate} \end{definition} @@ -121,8 +121,8 @@ Wobei $M \coloneqq \sum_{k=1}^{n} \norm{e^{(k)}}$. \\ Setze $$S_{1} \coloneqq \{ x \in \K^{n} \ | \ \norm{x}_{\infty} = 1\}, \ \ m \coloneqq \inf \{ \norm{x} \ | \ x \in S_{1} \} \geq 0.$$ \\ Es gzz.: $m > 0$. Annahme $m=0$. Dann existiert eine Folge $(x^{(k)})_{k \in \N}$, $x^{(k)} \in S_{1}$, sodass $\norm{ x^{(k)} } \overset{k \to \infty}{\longrightarrow} 0$. Aus $x^{(k)} \in S_{1}$ folgt $(x^{(k)})_{k \in \N}$ ist beschränkt in der $\ell_{\infty}$-Norm. Dann impliziert der Satz von Bolzano-Weierstraß: es existiert eine konvergente Teilfolge, o.B.d.A. $(x^{(k)}) \overset{k \to \infty}{\longrightarrow} x$ in der $\ell_{\infty}$-Norm, dann: \begin{align*} - & \ \ \ \underbrace{\left| \underbrace{\norm{x^{(k)}}_{\infty}}_{=1} - \norm{x}_{\infty} \right|}_{=|1-\norm{x}_{\infty}} \leq \underbrace{\norm{x^{(k)} -x}_{\infty}}_{\overset{k \to \infty}{\longrightarrow} 0} - \implies |1-\norm{x}_{\infty} \overset{k \to \infty}{\longrightarrow} 0 + & \ \ \ \underbrace{\left| \underbrace{\norm{x^{(k)}}_{\infty}}_{=1} - \norm{x}_{\infty} \right|}_{=|1-\norm{x}_{\infty}|} \leq \underbrace{\norm{x^{(k)} -x}_{\infty}}_{\overset{k \to \infty}{\longrightarrow} 0} + \implies |1-\norm{x}_{\infty}| \overset{k \to \infty}{\longrightarrow} 0 \implies \norm{x}_{\infty} = 1 \implies x\in S_{1}. \end{align*} Anderseits: @@ -134,7 +134,7 @@ \end{proof} \begin{korrolar} - Auf $K^{n}$ sind alle Komvergenzen in irgendeiner Norm äquivalent zur Konvergenz in der $\ell_{\infty}$-Norm. (= komponentenweiser Konvergenz) + Auf $K^{n}$ sind alle Konvergenzen in irgendeiner Norm äquivalent zur Konvergenz in der $\ell_{\infty}$-Norm. (= komponentenweiser Konvergenz) \end{korrolar} \begin{bem} @@ -160,9 +160,9 @@ Bezeichnung: $\norm{\cdot}$ irgendeine Norm. \begin{bsp} \begin{enumerate}[(1)] - \item $]a,b[ \subseteq \R$ ist offen ($a 0$, die $a 0$, da $a7fWFmjW%%Ym=vFoeFI>)pYgP&$2L_@-uunmzJ8v89$4rRS0 zks>vp{rr@>=nZp)JR_G;i31dqX%7pRef9ze0yi+15g!98f8|y^Z`&{s-SaCrQiB+N zf1q32BuKLqhNjvXoSKn^C{iKEDUe^^qbMnsEw^ZkHbuHf1bIA?_ujj^(`@#Up?q^f zQI06O82)dwlhbo)5SPSQ!%%jM7&U|`hP32FJBhLtn&ZoAy(}L#j|q3!H&4F!97gM^ z*d*842XJALeh7Vat6;!6wG@kZAX7UBGD+Z~DW^m#*PTk>Tt+zuH|=Sio7?^hqQ3$R^0Ix9 z!Rz~lka8i6DRLjzt`cs;mEijrOnd|TpORD=g$ev9e|@i1B%zrTbyw zU0B}xCO&W+pJute?;^~0Bg_mD)PcL!&AVu_3%Vgzh<<%O;er#RZEvuA1J}?7-yAw( z0*1p$*AsACBFy0Fr7`&9fU5Cq$`oiznSESEhwEia)d!z9%Yf*J=@MTwKwjXtya7Gg zK;}UBe=3mOD*1}y#GwKXi&(}OK_%%3C!8C<7z@#sc?{G-CmaELsKy%4ZpoBi&e3S?L*fy98DdZQ28s5 z7F*!1qk@Nv6e_Bg(6h{)i>sK_lnFzY5 ztjc;}b`K7gjj$aKFV+n>U^EhKq=A*hLipG3SN)jjnyG$B8cR6B2bflObyikBzs$}j z$1i`O;_T{37H5Sl{=NDA{`lz03*!qVg|*5Q$ESrO!I}%pPFQD)Wk2s_WJH 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