From 1e456fd461a73ac48317b21afca7318810239189 Mon Sep 17 00:00:00 2001 From: flavis Date: Fri, 15 May 2020 21:43:35 +0200 Subject: [PATCH] update ana4 --- sose2020/ana/lectures | 2 +- sose2020/ana/uebungen/ana4.pdf | Bin 0 -> 134238 bytes sose2020/ana/uebungen/ana4.tex | 211 +++++++++++++++++++++++++++++++++ 3 files changed, 212 insertions(+), 1 deletion(-) create mode 100644 sose2020/ana/uebungen/ana4.pdf create mode 100644 sose2020/ana/uebungen/ana4.tex diff --git a/sose2020/ana/lectures b/sose2020/ana/lectures index 3ad08e7..52281b1 160000 --- a/sose2020/ana/lectures +++ b/sose2020/ana/lectures @@ -1 +1 @@ -Subproject commit 3ad08e7437884b145c8cf14e3a5140269cf71bfd +Subproject commit 52281b15256442bfc7b1b8083034dd7e6b028adc diff --git a/sose2020/ana/uebungen/ana4.pdf b/sose2020/ana/uebungen/ana4.pdf new file mode 100644 index 0000000000000000000000000000000000000000..74c00a0324c61c83c1a6d150e2446ee6fed3c814 GIT binary patch literal 134238 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b/sose2020/ana/uebungen/ana4.tex @@ -0,0 +1,211 @@ +\documentclass[uebung]{../../../lecture} + +\begin{document} + +\title{Analysis II: Übungsblatt 4} +\author{Leon Burgard, Christian Merten} + +\punkte + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: Mit $d(x, y) = \sqrt{|x - y|}$ und $\varphi = \sqrt{x}$ ist $\rho$ keine Norm. + \begin{proof} + Aus dem letzten Zettel, folgt, dass $d(x,y) = \sqrt{|x-y|} $ eine Metrik + auf $\R$ ist. Es ist außerdem $\forall x, y \in \R$ + \[ + d(x, y) = \sqrt{|x - y|} = \sqrt{\Vert x - y \Vert_2} = \varphi(\Vert x - y\Vert_2) + = \rho(x - y) + .\] + Dann ist + \[ + \rho(4\cdot 4) = \sqrt{\Vert 16 \Vert_2} = 4 \neq 8 = 4 \cdot 2 = 4 \cdot \rho(4) + .\] + \end{proof} + \item Beh.: Teilmenge $O \subseteq \mathbb{K}^{n}$ g.d. offen, wenn sie keinen ihrer Randpunkte enthält. + \begin{proof} + ,,$\implies$``: + Sei $O \subseteq \mathbb{K}^{n}$ offen und $a \in \mathbb{K}^{n}$ ein Randpunkt von $O$. + Ang.: $a \in O \implies \exists \epsilon > 0$ mit $K_{\epsilon}(a) \subseteq O$. Also existiert + eine Umgebung von $a$, die ganz in $O$ liegt $\contr$. + + ,,$\impliedby$``: + Sei $O \subseteq \mathbb{K}^{n}$ nicht offen. Dann ex. $a \in O$, s.d. + $\forall \epsilon > 0$ gilt: $\exists y \in \mathbb{K}^{n} \setminus O$ mit + $y \in K_{\epsilon}(a)$. Da außerdem $a \in O$, ist $a$ Randpunkt von $O$, der + in $O$ liegt. + \end{proof} + \item Beh.: $\partial M$ von $M \subseteq \mathbb{K}^{n}$ ist abgeschlossen. + \begin{proof} + Sei $M \subseteq \mathbb{K}^{n}$. Dann folgt nach VL: $\partial M$ abgeschlossen. + \end{proof} + \item Beh.: Die Aussage ist falsch. + + \begin{proof} + Für $\mathbb{K}^{n} = \R$ und $M := [0,1]$ gilt $\partial M = \{0, 1\} $. Damit folgt + \[ + \left( \overline{M} \right)^{\circ} + = (\overline{[0,1]})^{\circ} = \left( [0,1] \right)^{\circ} = (0, 1) + \neq [0,1] = \overline{\left( (0,1) \right) } = \overline{\left( [0,1]^{\circ} \right) } + = \overline{\left( M^{\circ} \right) } + .\] + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: $\partial M = \{ x \in \R^{n} \mid \Vert x \Vert_{\infty} \le 1\} $, + $\overline{M} = \partial M$ und $M^{\circ} = \emptyset$. + \begin{proof} + Da $\Q$ dicht in $\R$, folgt $\forall a \in K_1(0)$, liegen in jeder Umgebung + von $a$ Punkte aus $K_1(0) \cap \Q^{n} = M$ und + Punkte aus $K_1(0) \cap (\R^{n} \setminus \Q^{n}) \subseteq M^{C}$. + + Außerdem sind alle Punkte der Einheitssphäre Randpunkte, damit folgt + \[ + \partial M = K_1(0) \cup \{ x \in R^{n} \mid \Vert x \Vert_\infty = 1\} = + \{x \in \R^{n} \mid \Vert x \Vert_\infty \le 1\} + .\] Da $M \subseteq \partial M$, folgt daraus + \begin{align*} + \overline{M} &= M \cup \partial M = \partial M \\ + M^{\circ} &= M \setminus \partial M = \emptyset + .\end{align*} + \end{proof} + \item Beh.: $\partial M = M$, $M^{\circ} = \emptyset$, $\overline{M} = M$. + \begin{proof} + Es gilt $\forall a \in M$, dass $\forall \epsilon > 0$ + Punkte $x \in M^{C}$ existieren mit + $x_1 \neq 0$ und $\Vert a - x \Vert_2 < \epsilon$. + Außerdem gilt für alle Randpunkte $a \in \partial M$, $a_1 = 0$, denn + angenommen, $a_1 = \delta \neq 0$. Dann ist + $K_{\frac{\delta}{2}}(a) \cap M = \emptyset$. Weiter + gilt $\Vert x \Vert_2 \le 1$, denn angenommen, $\exists \delta > 0$, s.d. + $\Vert x \Vert_2 = 1 + \delta > 1$, dann folgt analog + $K_{\frac{\delta}{2}}(a) \cap M = \emptyset$. + + Damit folgt + \begin{align*} + \overline{M} &= M \cup \partial M = M \cup M = M \\ + M^{\circ} &= M \setminus \partial M = M \setminus M = \emptyset + .\end{align*} + \end{proof} + \item Beh.: $\partial M = \{-1, 1\}^{n}$, $\overline{M} = M$, + $M^{\circ} = \{x \in \R^{n} \mid x \not\in [-1,1]^{n}\} $ + \begin{proof} + Es ist + \[ + M = \{ x \in \R^{n} \mid f(x) < 1\} = \{ x \in \R^{n} \mid x \not\in (-1,1)^{n}\} + .\] Als Randpunkte ergeben sich damit durch komponentenweise Anwendung + des eindimensionalen Falls: + \[ + \partial M = \{-1, 1\}^{n} + .\] Damit ist + \begin{align*} + M^{\circ} &= M \setminus \partial M = \{ x \in \R^{n} \mid x \not\in [-1, 1]^{n}\} \\ + \overline{M} &= M \cup \partial M = \{x \in \R^{n} \mid x \not\in (-1,1)^{n}\} = M + .\end{align*} + \end{proof} + \item Beh.: $\partial M = \{ x \in \R^{n} \mid g(x) \le 1\} $, + $M^{\circ} = M$, $M = [-1,1]^{n}$. + \begin{proof} + Es ist + \[ + g(x) = \frac{3}{2} - f(x) = \begin{cases} + \frac{1}{2} & x \in (-1,1)^{n} \\ + \frac{3}{2} & \text{sonst} + \end{cases} + .\] Also folgt + \[ + M = \{ x \in \R^{n} \mid x \in (-1,1)^{n}\} = (-1,1)^{n} + .\] Analog zu (c) folgt damit direkt + \begin{align*} + \partial M &= \{x \in \R^{n} \mid x \in \{-1,1\}^{n}\} \\ + M^{\circ} &= (-1,1)^{n} \setminus \{-1, 1\}^{n} = (-1,1)^{n} = M \\ + \overline{M} &= (-1,1)^{n} \cup \{-1, 1\}^{n} = [-1,1]^{n} + .\end{align*} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Seien $a < b \in \R$ und $\tilde{V}$, $V$ wie gegeben. + \begin{enumerate}[(a)] + \item Beh.: $(\cdot , \cdot )$ ist kein Skalarprodukt auf $\tilde{V}$. + \begin{proof} + Für $a = 0$, $b = 1$ und $f = 1 \in \tilde{V}$, folgt $f' = 0$ und + $f \neq 0$, aber + \[ + (f,f) = \int_{0}^{1} f'(x)f'(x) \d x = \int_{0}^{1} 0 \d x = 0 + .\] Damit ist (S1) nicht erfüllt. + \end{proof} + \item Beh.: $(\cdot , \cdot )$ ist ein Skalarprodukt auf $V$. + \begin{proof} + Seien $f, g \in V$ beliebig. + + Zunächst ist $(f,g)$ wohldefiniert, da $f$ und $g$ stückweise stetig differenzierbar, + sind $f'$ und $g'$ stückweise stetig, also stückweise R.-integrierbar. + \begin{enumerate}[(S1)] + \item Wegen der Monotonie des R.-Integrals + ist + \[ + (f,f) = \int_{a}^{b} \underbrace{(f'(x))^2}_{\ge 0} \d x \ge 0 + .\] + Sei nun + \[ + (f, f) = \int_{a}^{b} f'(x)^2 \d x = 0 + .\] Wegen $f \in \tilde{V}$, ex. eine Zerlegung $\Delta $ von $[a,b]$, s.d. + \[ + (f,f) = \sum_{[\tilde{a}, \tilde{b}] \in \Delta } + \int_{\tilde{a}}^{\tilde{b}} \underbrace{f'(x)^2}_{\ge 0} \d x + .\] Da die einzelnen Summanden alle nicht negativ sind, folgt + $\forall [\tilde{a}, \tilde{b}] \in \Delta $ mit + der Definitheit des R.-Integrals: + \[ + \int_{\tilde{a}}^{\tilde{b}} \underbrace{f'(x)^2}_{\ge 0} \d x = 0 + \qquad \quad \stackrel{f' \text{ stetig auf } [\tilde{a}, \tilde{b}]}{\implies} + \qquad \quad + f'(x)^2 = 0 \quad \forall x \in [\tilde{a}, \tilde{b}] + .\] Da dies für alle Teilintervalle gilt, folgt + \[ + f'(x)^2 = 0 \quad \forall x \in [a,b] \implies f' \equiv 0 + \implies \exists h \in \R: f(x) = h + .\] + Wegen $f(a) = f(b) = 0 \implies f(a) = h = 0 \implies h = 0$. Also $f \equiv 0$. + \item Folgt aus der Symmetrie des R.-Integrals. + \item Folgt aus der Linearität des R.-Integrals und der Linearität der Ableitung. + \end{enumerate} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Mit Gram-Schmidt-Verfahren folgt + \begin{align*} + b_1 &= \frac{1}{\sqrt{(1,1)}} = \frac{1}{\int_{0}^{1} 1 \d t } = 1 \\ + \widetilde{b_2} &= t - (1, t) = t - \int_{0}^{1} t \d t = t - \frac{1}{2} + \intertext{Es ist weiter} + (\widetilde{b_2}, \widetilde{b_2}) + &= \int_{0}^{1} \left(t-\frac{1}{2}\right)\left( t - \frac{1}{2} \right) \d x = \frac{1}{12} \\ + b_2 &= \sqrt{12} \left( t - \frac{1}{2} \right) + \intertext{Damit folgt} + \widetilde{b_3} &= t^2 - (t, t^2) - 12 \left( t - \frac{1}{2}, t^2 \right) t = t^2 - t + \frac{1}{6}\\ + (\widetilde{b_3}, \widetilde{b_3}) &= \int_{0}^{1} \left( t^2 - t + \frac{1}{6} \right)^2 \d x + = \frac{1}{180} \\ + b_3 &= \sqrt{180} \left( t^2 - t + \frac{1}{6} \right) \\ + \widetilde{b_4} &= t^3 - (1, t ^{3}) - 12 \left( t - \frac{1}{2}, t ^{3} \right) + \left( t - \frac{1}{2} \right) - 180 \left( t^2 - t + \frac{1}{6} \right)\left( t^2 - 1 + \frac{1}{6} \right) \\ + &= t ^{3} - \frac{3}{2} t^2 + \frac{3}{5} t - \frac{1}{20} \\ + (\widetilde{b_4}, \widetilde{b_4}) &= + \int_{0}^{1} \left( t ^{3} - \frac{3}{2} t^2 + \frac{3}{5} t - \frac{1}{20} \right)^2 \d x + = \frac{1}{2800} \\ + b_4 &= \sqrt{2800} \left( t ^{3} - \frac{3}{2}t^2 + \frac{3}{5}t - \frac{1}{20}\right) + \intertext{Also ist} + \mathcal{B} &= \left\{1, \sqrt{12}\left( t - \frac{1}{2} \right), + \sqrt{180}\left( t^2 - t + \frac{1}{6} \right), \sqrt{2800} + \left( t ^{3} - \frac{3}{2} t^2 + \frac{3}{5} t - \frac{1}{20} \right) \right\} + .\end{align*} + eine Orthonormalbasis auf $C([0,1])$ mit dem gegebenen Skalarprodukt. +\end{aufgabe} + +\end{document}