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zB5U3?$7#H8ZN1t44hTLQP2G6X&R5pc12B#gejn{l)8GiWAi->pL`^+2n9~^uZqHHr zSA132_cb|t=LdoetRsk-yOM9}Jtt{8I;iQUApURNvS=8G*nQqTs^}Oa1m4{(inre* zyukFedtC5Xvp~>;x(DcJd)$&#UT%Nb7!C?IW$q(ajV@$09x9|Ogeo%oe@#-=!5p<+6DN)2-Vv;b-*TQwy pn7<+2=kXynFaF<`y}g63or8 0, a \in \mathbb{K}^{n}$ wähle $\delta := \frac{\epsilon}{\Vert A \Vert}$. + Sei nun $x \in \mathbb{K}^{n}$ beliebig mit $\Vert x - a\Vert < \delta $. Dann ist + \[ + | \Vert Ax \Vert - \Vert A a \Vert | \le \Vert Ax - Aa \Vert + = \Vert A (x-a) \Vert + \qquad \stackrel{\text{verträgl.}}{\le } \qquad + \Vert A \Vert \Vert x -a \Vert < \Vert A \Vert \delta = \epsilon + .\] + + $f$ nimmt als stetige Funktion auf kompakten Mengen ihr Maximum an, d.h. + $\exists x_{max} \in M$, s.d. + \[ + \Vert Ax_{max} \Vert = \sup_{x \in M} \Vert Ax \Vert = \max_{x \in M} \Vert Ax \Vert + .\] + \end{proof} + \item Beh.: Die Frobenius-Norm ist mit der euklidschen Vektornorm verträglich und submultiplikativ. + \begin{proof} + Seien $A \in \mathbb{K}^{n \times n}$ und $x \in \mathbb{K}^{n}$ beliebig. + $A_i$ bezeichne die $i$-te Zeile der Matrix $A$. + Dann gilt + \begin{salign*} + \Vert A x \Vert_{2}^2 &= \sum_{i=1}^{n} \left( \sum_{j=1}^{n} |a_{ij}x_j| \right)^2 \\ + &= \sum_{i=1}^{n} (A_i, x)_2^2 \\ + &\stackrel{\text{C.S.U.}}{\le } \sum_{i=1}^{n} \Vert A_i \Vert_2^2 \Vert x \Vert_2^2 \\ + &= \Vert x \Vert_2^2 \sum_{i=1}^{n} \Vert A_i \Vert_{2}^2 \\ + &= \Vert x \Vert_2^2 \sum_{i=1}^{n} \sum_{j=1}^{n} |a_i|^2 \\ + &= \Vert x \Vert_2^2 \Vert A \Vert_F^2 + .\end{salign*} + Also $\Vert \cdot \Vert_F$ verträglich. + + Seien $A \in \mathbb{K}^{n \times n}$ und $x \in \mathbb{K}^{n}$ beliebig. + $A_i$ bezeichne die $i$-te Zeile der Matrix $A$, $B_j$ die $j$-te Spalte (!) + der Matrix $B$. + Dann gilt + \begin{salign*} + \Vert A B \Vert_{F}^2 &= \sum_{i,j=1}^{n} \left| \sum_{k=1}^{n} a_{ik}b_{kj} \right|^2 \\ + &= \sum_{i,j=1}^{n} (A_i, B_j)_2^2 \\ + &\stackrel{\text{C.S.U.}}{\le } \sum_{i,j=1}^{n} \Vert A_i \Vert_2^2 \Vert B_j \Vert_2^2 \\ + &= \sum_{i=1}^{n} \Vert A_i \Vert_2^2 \sum_{j=1}^{n} \Vert B_j \Vert_{2}^2 \\ + &= \Vert A \Vert_F \Vert B \Vert_F + .\end{salign*} + Also $\Vert \cdot \Vert_F$ submultiplikativ. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: Die Menge $M$ + \[ + M := \{ A \in \mathbb{K}^{n \times n} \mid A \text{ regulär }\} + \] ist offen. + \begin{proof} + Sei $A \in M$ beliebig. Da $A$ regulär ist $A \neq 0$ und $\exists A^{-1} \neq 0$. Wähle + $\epsilon := \frac{1}{\Vert A^{-1} \Vert}$. + + Sei $B \in K_{\epsilon}(A)$. Dann ist $\Vert A - B \Vert < \epsilon$, also + $\Vert A - B \Vert < \frac{1}{\Vert A^{-1} \Vert}$. Mit dem Störungssatz folgt + damit, dass $B$ regulär ist, also $B \in M$. D.h. $K_{\epsilon}(A) \subseteq M$ und damit + $M$ offen. + \end{proof} + \item Beh.: Die Resolventenabbildung + \[ + R\colon \text{Res}(A) \subseteq \mathbb{C} \to M \subseteq \mathbb{K}^{n \times n} + .\] ist stetig. + \begin{proof} + Sei $(z_k)_{k\in\N} \subseteq \text{Res}(A)$ Folge mit $z_k \xrightarrow{k \to \infty} a$. + Dann ist + \begin{align*} + R(z_k) &= (A - z_k I)^{-1} \\ + &= (A - aI + aI - z_kI)^{-1} \\ + &= (A - aI + (a-z_k) I)^{-1} \\ + &= ((A - aI)(I + (a-z_k) (A-aI)^{-1}))^{-1} \\ + &= \underbrace{(I + (\overbrace{a - z_k}^{\xrightarrow{k \to \infty} 0})R(a))^{-1}}_{\xrightarrow{k \to \infty} I} R(a) \\ + &\xrightarrow{k \to \infty} R(a) + .\end{align*} + Damit ist $R$ stetig auf $\text{Res}(A)$. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe}[] + Sei $f \colon \mathbb{K}^{n} \to \R$ stetig, $c \in \R$ und + \[ + M := \{ x \in \mathbb{K}^{n} \mid f(x) = c\} + .\] Beh.: $M$ ist abgeschlossen. + \begin{proof} + Z.z.: $M^{C} = \mathbb{K}^{n} \setminus M$ ist offen. Sei $a \in M^{C}$, d.h. + $f(a) \neq c$. Wähle $\epsilon := \frac{|f(a) - c|}{2}$. Da $f$ stetig, ex. $\delta > 0$, s.d. + $\forall x \in \mathbb{K}^{n}$ mit $\Vert x - a \Vert < \delta $: $|f(x) - f(a)| < \epsilon$. + + Sei $x \in K_{\delta}(a)$. Dann ist $\Vert x - a \Vert < \delta$. + Ang.: $f(x) = c \implies |f(x) - f(a)| = |c - f(a)| = 2 \epsilon > \epsilon$ $\contr$. + + Also ist $f(x) \neq c \implies x \in M^{C}$. Damit ist $M^{C}$ offen und $M$ abgeschlossen. + \end{proof} + Beh.: $M$ ist i.A. nicht kompakt. + \begin{proof} + Für $n = 1$ und $f(x) = \sin x$, $c = 0$ ist + $M = \{ k \pi \mid k \in \N\}$ + nicht beschränkt, also insbesondere nicht kompakt. + \end{proof} +\end{aufgabe} + +\begin{aufgabe}[] + Definiere + \[ + S\colon C([0, \pi]) \to \R, \quad f \mapsto S(f) := \int_{0}^{\pi} \cos(f(x)) \d x + .\] Beh.: $S$ stetig. + \begin{proof} + Seien $a, b \in \R$. O.E.: $a \le b$. $\cos(x)$ auf $\R$ differenzierbar, also insbesondere + auf $[a,b]$. Dann folgt mit dem Mittelwertsatz der Differentialrechnung: $\exists \xi \in (a,b)$ + mit + \[ + - \sin(\xi) = \frac{\cos(b)- \cos(a)}{b - a} + .\] Wegen $|\sin x| \le 1$ $\forall x \in \R$ folgt + \[ + | \cos(b)- \cos(a)| = | b -a | | \sin(\xi)| \le | b - a| \qquad (*) + .\] + + Sei $\epsilon > 0$ beliebig und $f \in C([0, \pi])$. Dann gilt $\forall g \in C([0, \pi])$ mit + $\Vert f - g \Vert_{\infty} < \delta := \frac{\epsilon}{\pi}$: + \begin{salign*} + |S(f) - S(g)| &= \left| \int_{0}^{\pi} \cos(g(x) \d x - \int_{0}^{\pi} \cos(f(x)) \d x \right| \\ + &\le \int_{0}^{\pi} | \cos(g(x)) - \cos(f(x)) | \d x \\ + &\stackrel{(*)}{\le } \int_{0}^{\pi} |g(x) - f(x)| \d x \\ + &\le \int_{0}^{\pi} \Vert g - f \Vert_{\infty} \d x \\ + &< \int_{0}^{\pi} \delta \d x \\ + &= \pi \delta \\ + &= \epsilon + .\end{salign*} + \vspace*{-10mm} + \end{proof} +\end{aufgabe} + +\end{document}