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zY9zV}qqHNs9t>~<;D(Xj{W4pFSIWHStI(Wv$2L`fU(be^X6`j2Fg+2?4tRd?y|CYH z25W5~h^fdOu-L_f?wt2^K|4)^^K|x|e(POjdv6I;f#eRw`+0FQc4cooqv3GEW-76d0VXv{%glg7ehTK4FF1@=l zx==Fir%7?Ap5Ev2cNp|SHt~?pGIEN~}VI#f7 zG)@dD97lwy>p;aC0hpRMl6Y%W(Fd34?Fzj6GVP(0PNMqT`m()Jncv7%Y;b7x0m7Fx zbdXeg|EQ0mX*=MOeIO~d`pSuA13O46PW+iYt6q`G%D^q~1Dq}$Kh0e(rkhvHC(?$eFzkHW}KT(~4g;hzX)?gp`q!8(U$Wp_hQ3;A+P zud8KGw$1Evc>@)CXVc&d2>eYQ4f6b?(mXt(#1#(RL&`1C)(6u#CzFp{?BF=39-jP$ zToR2eSHu+#vuo29ozv%XT9AuO4GxNa4?jyNC{U$0pgkv(p5&s%9g1LQ78Zpj)|m7M qBQbu1d_ug(+Hw2eXA}oVJ$px2`=2R 0 \quad \exists N = N(\epsilon, x) > 0 \text{ s.d. } + |f_n(x) - f(x)| < \epsilon \quad \forall n \ge N + .\] +\end{definition} + +\begin{bsp} + \begin{enumerate}[(a)] + \item \begin{align*} + &f_n(x)\colon [0,2] \to \R \\ + &f_n(x) = \begin{cases} + n^2x & 0 \le x \le \frac{1}{n} \\ + 2n - n^2x & \frac{1}{n} \le x \le \frac{2}{n} \\ + 0 & \frac{2}{n} \le x \le 2 + \end{cases} + .\end{align*} + $(f_n(x))_{n\in\N}$ konvergiert punktweise gegen $f(x) \equiv 0$ $\forall x \in [0,2]$. + + $x=0$: $f_n(0) = 0 = f(0)$\\ + $0 < x \le 2$: $\forall n \ge \frac{2}{x}$, $f_n(x) = 0 = f(x)$ + \item $f_n(x) = x^{n}$, $f_n\colon [0,1] \to \R$. + \begin{figure}[h!] + \begin{tikzpicture} + \begin{axis}% + [grid=both, + minor tick num=4, + grid style={line width=.1pt, draw=gray!10}, + major grid style={line width=.2pt,draw=gray!50}, + axis lines=middle, + enlargelimits={abs=0.2}, + ymax=1, + ymin=0 + ] + \addplot[domain=0:1,samples=50,smooth,red] {x^1}; + \addplot[domain=0:1,samples=50,smooth,red] {x^2}; + \addplot[domain=0:1,samples=50,smooth,red] {x^3}; + \addplot[domain=0:1,samples=50,smooth,red] {x^4}; + \end{axis} + \end{tikzpicture} + \begin{tikzpicture} + \begin{axis}% + [grid=both, + minor tick num=4, + grid style={line width=.1pt, draw=gray!10}, + major grid style={line width=.2pt,draw=gray!50}, + axis lines=middle, + enlargelimits={abs=0.2}, + ymax=1, + ymin=0 + ] + \addplot[domain=0:1.1,samples=1000,smooth,red] {(and(x>=0 , x<1) * 0 + (and(x>=1, x< 10) * 1}; + \end{axis} + \end{tikzpicture} + \caption{$f_n(x) = x^{n}$ und ihre Grenzfunktion} + \end{figure} + \begin{align*} + (f_n(x))_{n\in\N} \xrightarrow[\text{punktweise}]{n \to \infty} f(x) + = \begin{cases} + 1 & \text{ falls } x = 1 \\ + 0 & \text{ falls } 0 \le x < 1 + \end{cases} + .\end{align*} + \end{enumerate} + \label{bsp:punktweisekonvergenz} +\end{bsp} + +\begin{bem} + Punktweiser Limes stetiger Funktionen muss nicht stetig sein. +\end{bem} + +\begin{definition} + Die Folge $(f_n)_{n\in\N}$ konvergiert gleichmäßig gegen $f\colon D \to \R$ falls gilt + \[ + \forall \epsilon > 0 \quad \exists N = N(\epsilon) \text{ s.d. } + |f_n(x) - f(x)| < \epsilon \quad \forall x \in D \text{ und } n \ge N + .\] +\end{definition} + +\begin{bem} + $\forall \epsilon > 0$, $\exists N = N(\epsilon)$, $\forall n \ge N$ gilt + \[ + \text{graph}(f_n) \subset \epsilon\text{-Umgebung von Graphen von } f + := \{ (x, y) \in D \times \R \mid | y - f(x)| < \epsilon\} + .\] + \begin{figure}[h!] + \begin{tikzpicture} + \begin{axis}% + [grid=both, + minor tick num=4, + grid style={line width=.1pt, draw=gray!10}, + major grid style={line width=.2pt,draw=gray!50}, + axis lines=middle, + enlargelimits={abs=0.2}, + ymax=1, + ymin=0 + ] + \addplot[domain=0:1,samples=50,smooth,dashed, blue] {0.3*sin(deg(8*x)) + 0.2*x + 0.7}; + \addplot[domain=0:1,samples=50,smooth,red] {0.3*sin(deg(8*x)) + 0.2*x + 0.5}; + \addplot[domain=0:1,samples=50,smooth,blue] {0.1 * sin(deg(40*x)) + 0.3*sin(deg(8*x)) + 0.2*x + 0.5}; + \addplot[domain=0:1,samples=50,smooth,dashed, blue] {0.3*sin(deg(8*x)) + 0.2*x + 0.3}; + \end{axis} + \end{tikzpicture} + \begin{tikzpicture} + \begin{axis}% + [grid=both, + minor tick num=4, + grid style={line width=.1pt, draw=gray!10}, + major grid style={line width=.2pt,draw=gray!50}, + axis lines=middle, + enlargelimits={abs=0.2}, + ymax=1, + ymin=-0.4, + ] + \addplot[domain=0:1.1,samples=50,smooth,dashed, blue] {0.3}; + \addplot[domain=0:1.001,samples=1000,smooth,red] {(and(x>=0 , x<1) * 0 + (and(x>=1, x< 10) * 1}; + \addplot[domain=0:1,samples=50,smooth,blue] {x^8}; + \addplot[domain=0:1.1,samples=50,smooth,dashed, blue] {-0.3}; + \end{axis} + \end{tikzpicture} + \caption{Links: ,,$\epsilon$-Schlauch'', Rechts: \ref{bsp:punktweisekonvergenz} (b) nicht gleichmäßig konvergent} + \end{figure} + Beispiele \ref{bsp:punktweisekonvergenz} (a) und (b) nicht gleichmäßig konvergent, zu (a): + Für $\epsilon = \frac{1}{2}$ gilt + $\left|f_n\left(\frac{1}{n}\right)- f\left(\frac{1}{n}\right)\right| = n > \frac{1}{2}$ +\end{bem} + +\begin{bsp} + $f_n\colon [0,2] \to \R$, $f_n(x) := \frac{1}{n} \sin \left( 2\pi n \cdot x \right) $ konvergiert + gleichmäßig gegen $f(x) = 0$. + \[ + | \sin(2 \pi n \cdot x) | \le 1 \quad \forall x \in R \implies + f_n(x) \xrightarrow{n \to \infty} 0 \quad \forall x \in [0,2] \implies \text{punktweise Konvergenz} + .\] Sei nun $\epsilon > 0$, dann $\exists N \in \N$ mit $\frac{1}{N} < \epsilon$. Damit folgt + \[ + \forall n \ge N \quad |f_n(x)| \le \frac{1}{N} < \epsilon \quad \forall x + \implies \text{gleichmäßige Konvergenz} + .\] +\end{bsp} +\begin{bem} + Konvergiert $f_n \colon D \to \R$ gleichmäßig gegen $f \colon D \to \R$, dann konvergiert $f_n$ punktweise + gegen $f$. Die Umkehrung gilt nicht, siehe \ref{bsp:punktweisekonvergenz} (a) und (b). +\end{bem} + +\begin{satz}[Gleichmäßiger Limes stetiger Funktionen ist stetig] + Es sei $D \subset \R$ und $f_n\colon D \to \R$ $\forall n \in \N$ stetig in $D$. + Sei $(f_n)_{n\in\N}$ gleichmäßig konvergent gegen $f\colon D \to \R$. Dann gilt: + $f$ ist stetig in $D$. +\end{satz} + +\begin{proof} + Seien $x_0 \in D$ und $\epsilon > 0$. + Zu zeigen: + $\exists \delta > 0$ $\forall x \in D\colon |x - x_0| < \delta \implies |f(x) - f(x_0)| < \epsilon$. + \[ + (f_n)_{n\in\N} \xrightarrow[\text{gleichmäßig}]{n \to \infty} f + \implies \exists N \in \N \text{ s.d. } \forall n \ge N \quad \forall x \in D + \text{ gilt } | f_n(x) - f(x)| < \frac{\epsilon}{3} + .\] + \[ + f_n \text{ stetig in } x_0 \implies \exists \delta \text{ s.d. } \forall x \in D \text{ gilt } + |x - x_0| < \delta \implies |f_n(x) - f_n(x_0)| < \frac{\epsilon}{3} + .\] + Zusammen: $\forall x$ mit $|x - x_0| < \delta $ gilt: + \begin{align*} + |f(x) - f(x_0)| &= |f(x) - f_n(x) + f_n(x) - f_n(x_0) + f_n(x_0) - f(x_0)| \\ + &\le |f(x) - f_n(x)| + |f_n(x) - f_n(x_0)| + |f_n(x_0) - f(x_0)| \\ + &\le \frac{\epsilon}{3} + \frac{\epsilon}{3} + \frac{\epsilon}{3} \\ + &= \epsilon + .\end{align*} +\end{proof} + +\subsection{Der Funktionenraum $C[a,b]$} + +\begin{definition}[Maximumnorm $\Vert\cdot \Vert_\infty$] + Es sei $D$ ein beschränktes, abgeschlossenes Intervall $I = [a,b]$ und + $f\colon [a,b] \to \R$ (oder $\mathbb{C}$) stetig. Dann + \[ + \Vert f \Vert_\infty := \max \{ |f(x)| \mid x \in [a,b]\} + .\] Das Maximum existiert wegen der Stetigkeit des Absolutbetrags und weil $[a,b]$ beschränkt und + abgeschlossen ist. +\end{definition} + +\begin{satz}[$\Vert \cdot \Vert_\infty$ und gleichmäßige Konvergenz] + Es sei $D = [a,b]$ und $f\colon [a,b] \to \R$ stetig. Dann gilt + \begin{enumerate}[(i)] + \item $(f_n)_{n\in\N}$ konvergiert gleichmäßig gegen $f \colon [a,b] \to \R + \iff \Vert f_n - f \Vert_\infty \to 0$, $n \to \infty$ + \item Cauchy-Kriterium: $(f_n)_{n\in\N}$ konvergiert gleichmäßig auf $[a,b] + \iff \forall \epsilon > 0$ $\exists N_0 \in \N$, s.d. $\forall n, m \ge N_0$ gilt + $\Vert f_n - f_m \Vert_{\infty} < \epsilon$ + \end{enumerate} +\end{satz} + +\begin{proof} + \begin{enumerate}[(i)] + \item ,,$\implies$'': Sei $\epsilon > 0$. Dann $\exists N \in \N$ s.d. + $|f_n(x) - f(x)| < \frac{\epsilon}{2}$ $\forall x \in [a,b]$ und $\forall n \ge N$. Damit folgt: + \[ + \max_{x \in [a,b]} |f_n(x) - f(x)| = \Vert f_n - f \Vert_{\infty} < \frac{\epsilon}{2} < \epsilon \implies \Vert f_n - f \Vert_{\infty} \xrightarrow{n \to \infty} 0 + .\] + ,,$\impliedby$'': Sei $\epsilon > 0$. Wegen + $\Vert f_n -f \Vert_{\infty} \xrightarrow{n \to \infty} 0$ $\exists N \in \N$ s.d. + $\Vert f_n - f \Vert_\infty < \epsilon$ $\forall n \ge N$. Damit folgt + $\forall n \ge N$ und $\forall x \in [a,b]$ + \[ + |f_n(x) - f(x)| \le \max_{x \in [a,b]} |f_n(x) - f(x)| = + \Vert f_n - f \Vert_{\infty} < \epsilon \implies \text{gleichmäßige Konvergenz} + .\] + \item ,,$\implies$'' $(f_n)_{n\in\N}$ konvergiert gleichmäßig auf $[a,b]$, d.h. + $\exists f: [a,b] \to \R$ s.d. $f_n \xrightarrow[\text{gleichmäßig}]{n \to \infty} f$. + + Sei $\epsilon > 0$, dann $\exists N_0 \in \N$ s.d. $|f_n(x) - f(x)| < \frac{\epsilon}{4}$ + $\forall n \ge N_0$ und $\forall x \in [a,b]$. + + Damit gilt $\forall n, m \ge N_0$ und $\forall x \in [a,b]$: + + \begin{align*} + &|f_n(x) - f_m(x)| \le |f_n(x) - f(x)| + |f(x) - f_m(x)| + \le \frac{\epsilon}{4} + \frac{\epsilon}{4} = \frac{\epsilon}{2} \\ + \implies &\max_{x \in [a,b]} |f_n(x) - f_m(x)| = \Vert f_n - f_m \Vert_{\infty} + \le \frac{\epsilon}{2} < \epsilon \quad \forall n, m \ge N_0 + .\end{align*} + + ,,$\impliedby$'': Sei $\epsilon > 0$, dann $\exists N_0 \in \N$, s.d. + \[ + |f_n(x) - f_m(x)| \le \Vert f_n - f_m \Vert_\infty < \frac{\epsilon}{2} \quad + \forall n,m \ge N_0 \quad \forall x \in [a,b] + .\] $\implies (f_n(x))_{n\in\N}$ ist eine Cauchy-Folge in $\R$.\\ + $\implies (f_n(x))_{n\in\N}$ konvergiert \\ + $\implies$ Definiere $f(x) := \lim_{n \to \infty} f_n(x)$, $x \in [a,b]$. + + Dann gilt $\forall n \ge N_0$, $\forall x \in [a,b]$: + \[ + |f_n(x) - f(x)| = \lim_{m \to \infty} |f_n(x) - f_m(x)| < \frac{\epsilon}{2} + \implies (f_n)_{n\in\N} \text{ konvergiert gleichmäßig gegen } f + .\] + \end{enumerate} +\end{proof} + +\begin{bem} + $\Vert \cdot \Vert_{\infty}$ erfüllt s.g. Normeigenschaften: + \begin{enumerate}[(N1)] + \item $\Vert f \Vert_\infty = 0 \implies f(x) = 0, x \in [a,b]$ (Definitheit) + \item $\Vert \alpha f \Vert_{\infty} = |\alpha| \cdot \Vert f \Vert_\infty$, $\alpha \in \R$ (Homogenität) + \item $\Vert f + g \Vert_{\infty} \le \Vert f \Vert_\infty + \Vert g \Vert_\infty$ (Dreiecksungleichung) + \end{enumerate} folgen direkt aus den Eigenschaften des Absolutbetrags. +\end{bem} + +\begin{definition} + Der Funktionenraum $C[a,b]$ definiert durch + \[ + C[a,b] := \{ f \colon [a,b] \to \R \mid f \text{ stetig }\} + ,\] ist mit $\Vert f \Vert_\infty$ ein normierter Vektorraum. +\end{definition} + +\begin{satz}[Vollständigkeit] + Der Raum $C[a,b]$ ist vollständig bezüglich gleichmäßiger Konvergenz, d.h. jede + Cauchy-Folge von Funktionen aus $C[a,b]$ besitzt einen Limes in $C[a,b]$ +\end{satz} + +\begin{proof} + Rannacher +\end{proof} + +\subsection{Integration und Grenzübergänge} + +Wichtige Frage: Wenn $f_n \to f$, gilt dann auch $\int_{a}^{b} f_n \to \int_{a}^{b} f$? + +\begin{satz} + Seien $f_n \colon [a,b] \to R$ stetige Riemann-integrierbare Funktionen und $f\colon [a,b] \to \R$ + mit $\Vert f_n - f \Vert_\infty \xrightarrow{n \to \infty} 0$ (gleichmäßige Konvergenz). Dann gilt + $f$ stetig und Riemann-integrierbar und + \[ + \lim_{n \to \infty} \int_{a}^{b} f_n(x) dx = \int_{a}^{b} f(x) dx + = \int_{a}^{b} \lim_{n \to \infty} f_n(x) dx + .\] +\end{satz} + +\begin{proof} + $f_n \xrightarrow[\text{gleichmäßig}]{n \to \infty} f \implies f$ stetig $\implies f$ Riemann-integrierbar. + + Es gilt + \[ + \left| \int_{a}^{b} f_n(x) dx - \int_{a}^{b} f(x) dx\right| = \left| \int_{a}^{b} (f_n(x) - f(x))dx\right| + \le \int_{a}^{b} |f_n(x) - f(x)| dx \le \underbrace{\max_{x \in [a,b]} |f_n(x) - f(x)|}_{= \underbrace{\Vert f_n - f \Vert_\infty}_{\xrightarrow{n \to \infty} 0}\underbrace{(b-a)}_{\text{beschränkt}}} + .\] +\end{proof} + +\begin{satz} + Seien $f_n \colon [a,b] \to \R$ stetige Funktionen $(n \in \N)$ und die Reihe + $\sum_{n=0}^{\infty} f_n$ konvergiere gleichmäßig auf $[a,b]$, d.h. die Folge der Partialsummen + $(\sum_{n=0}^{N} f_n)_{n\in\N}$ sei gleichmäßig konvergent. Dann gilt: + \[ + f(x) := \sum_{n=0}^{\infty} f_n(x) \quad \forall x \in [a,b] + .\] ist stetig und Riemann-integrierbar und + \begin{align*} + \int_{a}^{b} f(x) dx &= \sum_{n=0}^{\infty} \int_{a}^{b} f_n(x) dx \\ + \int_{a}^{b} \sum_{n=0}^{\infty} f_n(x) dx &= \sum_{n=0}^{\infty} \int_{a}^{b} f_n(x) dx + ,\end{align*} d.h. die Reihe wird gliedweise integriert. +\end{satz} + +\begin{proof} + $f_n$ sind stetig $\implies \sum_{n=0}^{N} f_n(x)$ stetig und Riemann-integrierbar. + + Die Folge der Partialsummen $(\sum_{n=0}^{N} f_n)_{n\in\N}$ konvergiert gleichmäßig, d.h. + \[ + f(x) = \sum_{n=0}^{\infty} f_n(x) \text{ stetig } = \lim_{n \to \infty} \left( \sum_{n=0}^{N} f_n(x) \right) \text{ gleichmäßiger Limes} + .\] Es gilt + \[ + \int_{a}^{b} \sum_{n=0}^{\infty} f_n(x) dx = + \sum_{n=0}^{N} \int_{a}^{b} f_n(x) dx + \int_{a}^{b} \sum_{n=N+1}^{\infty} f_n(x) dx + .\] Die Reihe konvergiert gleichmäßig, d.h. $\forall \epsilon > 0$ $\exists N_{\epsilon} \in \N$, s.d. + $\forall N \ge N_{\epsilon}$ gilt + \begin{align*} + &\left| \int_{a}^{b} \sum_{n=N+1}^{\infty} f_n(x) dx \right| \le \epsilon \cdot (b-a) \\ + \implies &\left| \int_{a}^{b} \sum_{n=0}^{\infty} f_n(x) dx - \sum_{n=0}^{N} \int_{a}^{b} f_n(x) dx \right| \le \epsilon \\ + \implies& \int_{a}^{b} \sum_{n=0}^{\infty} f_n(x) dx = \lim_{N \to \infty} \sum_{n=0}^{\infty} \int_{a}^{b} f_n(x) dx = \sum_{n=0}^{\infty} \int_{a}^{b} f_n(x) dx + .\end{align*} +\end{proof} + +\begin{korrolar}[Integration von Potenzreihen] + Es sei $\sum_{n=0}^{\infty} a_n(x - x_0)^{n}$ eine reelle Potenzreihe mit Konvergenzradius $\rho > 0$. + Dann konvergiert $\sum_{n=0}^{\infty} a_n (x - x_0)^{n}$ in jedem Intervall + $[x_0 - r, x_0 + r]$ für $0 < r < \rho$ gleichmäßig und für $[a,b] \subset \;]x_0 - \rho, x_0 + \rho[$ gilt + \[ + \int_{a}^{b} \sum_{n=0}^{\infty} a_n(x - x_0)^{n} dx = \sum_{n=0}^{\infty} \frac{a_n}{n+1}(x-x_0)^{n+1} + \Big|_{a}^{b} + .\] +\end{korrolar} + +\begin{proof} + Nur die gleichmäßige Konvergenz für $| x - x_0| \le r$ ist zu beweisen: Für $|x - x_0| \le r$, $r < \rho$ gilt: + \begin{align*} + \left\Vert \sum_{n=0}^{\infty} a_n(x-x_0)^{n} - \sum_{n=0}^{N} a_n(x-x_0)^{n} \right\Vert_{\infty} + &= \left\Vert \sum_{n=N+1}^{\infty} a_n(x - x_0)^{n}\right\Vert_\infty \\ + &\stackrel{|x - x_0| < r}{\le} \quad \sum_{n=N+1}^{\infty} |a_n| \cdot r^{n} \\ + &\stackrel{(*)}{\le} \sum_{n=N+1}^{\infty} \left(\frac{1}{\rho - \epsilon}\right)^{n} \cdot r^{n} \\ + &= \sum_{n=N+1}^{\infty} \underbrace{\left( \frac{r}{\rho-\epsilon} \right)^{n}}_{< 1} + \xrightarrow[\text{geometrische Reihe}]{N \to \infty} 0 + .\end{align*} + $(*)$: $\rho = \frac{1}{\limsup \sqrt[n]{|a_n|} }$, $r < \rho - \epsilon$ für ein $\epsilon > 0 \implies + \exists N_0 \in \N$, s.d. $\sqrt[n]{|a_n|} < \frac{1}{\rho - \epsilon} $ $\forall n \ge N_0$ +\end{proof} + +\end{document} diff --git a/analysisII.pdf b/analysisII.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f6002c02a8d092eff041ab7b64c33ba883a93829 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