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|y-z|} + |y-z| \\ + \stackrel{\sqrt{\cdot} \ge 0 }{\iff} & + d_2(x, z) = \sqrt{|x-z|} \le \sqrt{|x-y|} + \sqrt{|y-z|} + = d_2(x, y) + d_2(y, z) + .\end{align*} + \end{enumerate} + \item $d_3(x,y) = |x^2 - y^2|$ ist keine Metrik, denn + \[ + d(-1, 1) = |(-1)^2 - 1^2| = 0 + .\] aber $-1 \neq 1$ in $\R$. + \item $d_4(x,y) = |x - 2y|$ ist keine Metrik, denn + \[ + d_4(3, 2) = |3 - 2\cdot 2| = 1 \neq 4 = |2 - 2\cdot 3| = d_4(2,3) + .\] + \item + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Sei $(X, d)$ ein metrischer Vektorraum über $K \in \{\R, \mathbb{C}\} $. Die + Metrik $d\colon X \times X \to \R$ erfülle die gegeben Eigenschaften. + + Beh.: $\Vert x \Vert_d := d(x, 0)$ $\forall x \in X$ ist eine Norm auf $X$. + \begin{proof} + Seien $x, y \in X$ und $\alpha \in K$. + \begin{enumerate}[(N1)] + \item $\Vert x \Vert_d = d(x, 0) \quad \stackrel{d \text{ Metrik}}{\ge} \quad 0$ \\ + $x = 0 \; \stackrel{d \text{ Metrik}}{\iff} \; d(x,0) = 0 \iff \Vert x \Vert_d = 0$ + \item $\Vert \alpha x \Vert_d = d(\alpha x, 0) \stackrel{\text{E2}}{=} |\alpha| d(x, 0) + = |\alpha| \Vert x \Vert_d$ + \item Dreicksungleichung: + \begin{align*} + \Vert x + y \Vert \quad &= \quad d(x + y, 0) \\ + &\stackrel{\text{E1}}{=} \quad d(y, -x) \\ + &\stackrel{d \text{ Metrik}}= \quad d(-x, y) \\ + &\stackrel{d \text{ Metrik}}\le \quad d(-x, 0) + d(0, y) \\ + &\stackrel{\text{E2}}{=} \quad d(x, 0) + d(y, 0) \\ + &= \quad \Vert x \Vert_d + \Vert y \Vert_d + .\end{align*} + \end{enumerate} + \end{proof} +\end{aufgabe} + +\begin{aufgabe} + Seien $p, q \in \R$ mit $p, q > 1$, $\frac{1}{p} + \frac{1}{q} = 1$ + \begin{enumerate}[(a)] + \item Definiere + \begin{align*} + f(t) &:= (a^{p})^{t} \cdot (b^{q})^{1 - t} + \intertext{Es folgt} + f''(t) &= \underbrace{(p \ln a - q \ln b)^2}_{\ge 0} \cdot \underbrace{f(t)}_{\ge 0} + .\end{align*} + Also ist $f(t)$ konvex. + Es ist mit $\frac{1}{p} + \frac{1}{q} = 1 \implies \frac{1}{q} = 1 - \frac{1}{p}$. + Mit $\lambda = \frac{1}{p}, x = 1$ und $y = 0$ folgt damit + \[ + ab = + a^{\frac{1}{p} \cdot p}b^{q \cdot \left( 1-\frac{1}{p}\right) } + = f\left(\frac{1}{p}\right) + \le \frac{1}{p} f(1) + \left( 1 - \frac{1}{p} \right)f(0) + = \frac{a^{p}}{p} + \frac{b^{q}}{q} + .\] + \item + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Sei + \[ + B_1 := \{ f \in \mathcal{C}([0,1]) \mid \Vert f \Vert_{\infty} \le 1\} + \] die abgeschlossene Einheitskugel. + \begin{enumerate}[(a)] + \item Beh.: Die Folge $(f_n)_{n \in \N}$ mit + \[ + f_n(x) := \begin{cases} + - 2^{2n+4} \left( x - \frac{3}{2^{n+2}} \right)^2 + 1 + & x \in \left[ \frac{1}{2^{n+1}}, \frac{1}{2^{n}} \right] \\ + 0 & \text{sonst} + \end{cases} + .\] erfüllt die Eigenschaften. + \begin{proof} + Sei $n \in \N$ beliebig. + Zunächst ist $f_n$ stetig, denn: + $\forall x \in \left( \frac{1}{2^{n+1}}, \frac{1}{2^{n}} \right) $ + ist $f_n$ als Polynom stetig. + Für $x \in [0,1] \setminus \left[ \frac{1}{2^{n+1}}, \frac{1}{2n} \right] $ ist + $f_n$ als konstante Funktion stetig. Außerdem gilt + \[ + \lim_{x \nearrow \frac{1}{2^{n+1}}} f_n(x) = 0 + = - \frac{2^{2n+4}}{2^{2n+4}} + 1 + = - 2^{2n+4}\left( \frac{1}{2^{n+1}} - \frac{3}{2^{n+2}} \right)^2 + 1 + = f_n\left(\frac{1}{2^{n+1}}\right) + = \lim_{x \searrow \frac{1}{2^{n+1}}} f_n(x) + .\] Analog für $x = \frac{1}{2^{n}}$. + + Weiter ist $\Vert f_n \Vert_{\infty} = 1$, denn für + $x \in [0, 1] \setminus \left( \frac{1}{2^{n+1}}, \frac{1}{2^{n}} \right) $ ist + $f_n(x) = 0$ und für $x \in \left[ \frac{1}{2^{n+1}}, \frac{1}{2^{n}} \right] $ + ist der Scheitelpunkt der Parabel mit $x_{max} = \frac{3}{2^{n+2}}$ als + Maximum mit $f(x_{max}) = 1$ gegeben. + + Sei nun $m \in \N$ mit $m \neq n$ und $x \in [0,1]$. O.E. sei $m > n$. + Falls $ x \in \left( \frac{1}{2^{n+1}}, \frac{1}{2^{n}} \right) $, dann ist + $f_m(x) = 0$, denn $ m > n \implies m \ge n+1 \implies \frac{1}{2^{m}} \le \frac{1}{2^{n+1}}$.\\ + Falls $x \in \left( \frac{1}{2^{m+1}}, \frac{1}{2^{m}} \right) $, ist analog + $f_n(x) = 0$.\\ + Sonst gilt $f_n(x) = f_m(x) = 0$. + + Damit folgt $f_n(x) f_m(x) = 0$. + \end{proof} + \item Beh.: Eine Folge $(f_n)_{n\in\N}$ mit den in (a) gegebenen Eigenschaften, hat keine + konvergente Teilfolge. + \begin{proof} + Sei $(f_n)_{n\in\N} \subseteq B_1$ mit $\Vert f_n \Vert_{\infty} = 1$ $\forall n \in \N$ + und $f_n(x) f_m(x) = 0$ $\forall x \in [0,1], n, m \in \N, n \neq m$. + + Sei weiter $f \in \mathcal{C}([0,1])$ beliebige Funktion + und $(f_{n_k})_{k\in\N} \subseteq (f_n)_{n\in\N}$ Teilfolge. + + Ang.: $\Vert f_{n_k} - f \Vert_{\infty} \xrightarrow{k \to \infty} 0$ + + Ang.: $\Vert f \Vert_{\infty} \neq 1$. Dann sei + $\epsilon = \frac{|1 - \Vert f \Vert_{\infty}|}{2}$. Dann gilt + $\forall k \in \N$ $\Vert f_{n_k} \Vert_{\infty} = 1$, also folgt + \[ + \Vert f_{n_k} - f \Vert \ge | 1 - \Vert f \Vert_{\infty} | > \epsilon \quad \contr + .\] + Damit ist $\Vert f \Vert_{\infty} = 1$. Sei nun $x_{max} \in [0,1]$ beliebig + mit $f(x_{max}) = 1$. + + Falls $\forall k \in \N$ gilt $f_{n_k}(x_{max}) = 0 \implies \Vert f_{n_k} - f \Vert \ge 1$ + $\forall k \in \N$.\\ + Falls $\exists k \in \N$ mit $f_{n_k}(x_{max}) = 1$: Dann gilt $\forall m > k$: + \[ + f_{n_k}(x_{max}) \cdot f_{n_m}(x_{max}) = 0 + \qquad + \stackrel{\R \text{ nullteilerfrei}}{\implies} + \qquad + f_{n_m}(x_{max}) = 0 \implies \Vert f - f_{n_m} \Vert_{\infty} \ge 1 \quad + .\] Also konvergiert $f_{n_k}$ nicht gegen $f$. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document}