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ANAG and LAAG into 1. Semester folder
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@ -164,3 +164,4 @@ TSWLatexianTemp*
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*.thm
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*.gz
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*.toc
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Vorlesung LAAG.tex
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@ -3,8 +3,9 @@
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\item[Frage:] Frage: algebraische Gleichung $a_0+a_1x+\dots+a_x^k=0\;(a_j\in \whole)$\\
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i.A nur für $k=1$ lösbar (d.h. lin. Gl.)
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\end{description}
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\begin{exmpn}
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$x^2 - 2 = 0$ keine Lösung in $\ratio$. Angenommen es existiert eine Lösung $x = \frac{m}{n} \in \ratio$, o.B.d.A. höchstens eine der Zahlen $m,n$ gerade $\Rightarrow \frac{m^2}{n^2} = 2 \Rightarrow m^2 = 2n^2 \Rightarrow m$ gerade $\overset{m=2k}{\Rightarrow} 4k^2 = 2n^2 \Rightarrow 2n^2 \Rightarrow 2k^2 = n^2 \Rightarrow n$ gerade $\Rightarrow \lightning$.
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$x^2 - 2 = 0$ keine Lösung in $\ratio$. Angenommen es existiert eine Lösung $x = \frac{m}{n} \in \ratio$, o.B.d.A. höchstens eine der Zahlen $m,n$ gerade $\Rightarrow \frac{m^2}{n^2} = 2 \Rightarrow m^2 = 2n^2 \Rightarrow m$ gerade $\overset{m=2k}{\Rightarrow} 4k^2 = 2n^2 \Rightarrow 2n^2 \Rightarrow 2k^2 = n^2 \Rightarrow n$ gerade $\Rightarrow \lightning$.\QEDA
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\end{exmpn}
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\noindent Offenbar $1,4^2 < 2 < 1,5^2,\; 1,41^2 < 2 < 1,42^2,\;\dots,$ falls es $\sqrt{2}$ gibt, kann diese in $\ratio$ beliebig genau approximiert werden. Es folgt, dass $\ratio$ anscheinend "`Lücken"' hat.
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@ -18,7 +19,7 @@
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$\field$ sei ein (bel.) Körper mit bel. Elementen $0, 1$ bzw. $0_K, 1_K$.
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\begin{satz}
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Sei $\field$ Körper. Dann gilt $\forall a,b \in \field$:
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\begin{enumerate}[label=[1), nolistsep]
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\begin{enumerate}[label={\arabic*)}, nolistsep]
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\item $0,1, (-a), b^{-1}$ sind eindeutig bestimmt
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\item $(-0) = 0$, $1^{-1} = 1$
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\item $-(-a) = a$, $(b^{-1})^{-1} = b$ $(b \neq 0)$
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@ -31,7 +32,7 @@ $\field$ sei ein (bel.) Körper mit bel. Elementen $0, 1$ bzw. $0_K, 1_K$.
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\end{satz}
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\begin{proof}
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\begin{enumerate}[label=[zu 1)]
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\begin{enumerate}[label={\arabic*)}]
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\item vgl. lin. Algebra
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\item betrachte $0 + 0 = 0$ bzw. $1 \cdot 1 = 1$
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\item $(-a) + a = 0 \overset{komm}{\Rightarrow} a = -(-a)$ Rest analog
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@ -44,12 +45,12 @@ $\field$ sei ein (bel.) Körper mit bel. Elementen $0, 1$ bzw. $0_K, 1_K$.
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\end{enumerate}
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\end{proof}
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\noindent Setze für alle $a, \dots a_k \in \field,n\in \natur_{\geq 1}$
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\begin{description}
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Setze für alle $a, \dots a_k \in \field,n\in \natur_{\geq 1}$
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\begin{itemize}
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\item[Vielfache] $n\cdot a$ (kein Produkt in $\field$!)
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\item[Potenzen] $a^n=\prod_{k=1}^{n} a_k \text{für } n \in N_{\geq 1}$ damit $(-n)a:=n(-a) \text{, } 0_{\natur}a=0_{\natur} \text{ für } n\in\natur_{\geq1}\\
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a^{-n}=(a^-1)^n \text{, }a^{0_{\natur}}:=1_{\field} \text{ für } n \in \natur_{\geq 1}, a \neq 0\\
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beachte: 0^0 = (0_\natur)^{0_{\natur}} \text{ \underline{nicht} definiert!}$
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beachte: 0^0 = (0_\natur)^{0_{\natur}} \text{ \emph{nicht} definiert!}$
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\item[Rechenregeln] $\forall\;a,b\in \field\text{, } m,n\in \whole \text{ (sofern Potenz definiert) } $
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\end{description}
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\end{itemize}
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%TODO
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@ -39,7 +39,7 @@ Man hat $d(x,y) = 0 \forall x,y \in X$, dann
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\begin{exmpn}[induzierte Metrik]
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Sei $(X,d)$ metrischer Raum, $Y \subset X$\\
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$\Rightarrow (Y,d)$ ist metrischer Raum mit \underline{induzierter Metrik} $\tilde{d}(x,y):=d(x,y)\forall x,y \in Y$
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$\Rightarrow (Y,d)$ ist metrischer Raum mit \emph{induzierter Metrik} $\tilde{d}(x,y):=d(x,y)\forall x,y \in Y$
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\end{exmpn}
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\section{Normierte Räume}
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@ -48,13 +48,13 @@ wichtiger Spezialfall: normierte Vektorraum(VR)
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\begin{mydefn}[Norm]
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Sei $X$ Vektorraum über $K=\real$ oder $K=\comp$.\\
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Abbildung $\Vert \cdot \Vert: X \to \real$ heißt \underline{Norm} auf $X$ falls $\forall x,y \in X, \forall \lambda \in \real$ gilt:
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Abbildung $\Vert \cdot \Vert: X \to \real$ heißt \emph{Norm} auf $X$ falls $\forall x,y \in X, \forall \lambda \in \real$ gilt:
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\begin{enumerate}[label={\alph*)}]
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\item $\Vert x\Vert = \Leftrightarrow x=0$
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\item $\Vert \lambda x\Vert = \vert \lambda \vert \Vert x\Vert$ (Homogenität)
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\item $\Vert x+y\Vert \leq \Vert x\Vert + \Vert y\Vert$ ($\Delta$-Ungleichung)
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\end{enumerate}
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$(X,\Vert \cdot\Vert)$ heißt \underline{normierter Raum}.
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$(X,\Vert \cdot\Vert)$ heißt \emph{normierter Raum}.
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\end{mydefn}
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\begin{align*}
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@ -66,7 +66,7 @@ Analog Satz 5.5 folgt\\
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\begin{align}
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\vert \Vert x \Vert - \Vert y \Vert\vert &\leq \Vert x-y\Vert \forall x,y \in X
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\end{align}
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$\Vert \cdot\Vert: X \to \real_{\geq0}$ heißt \underline{Halbraum} falls nur b), c) gelten analog Beispiel \ref{8_1_exmp_metrik} folgt.
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$\Vert \cdot\Vert: X \to \real_{\geq0}$ heißt \emph{Halbraum} falls nur b), c) gelten analog Beispiel \ref{8_1_exmp_metrik} folgt.
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\begin{satz}
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Sei $(X,\Vert\cdot \Vert)$ normierter Raum, dann $X$ metrischer Raum mit Metrik $d(x,y):=\Vert x-y \Vert\forall x,y \in X$.
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@ -84,11 +84,11 @@ $\Vert \cdot\Vert: X \to \real_{\geq0}$ heißt \underline{Halbraum} falls nur b)
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\end{cases*}
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\end{align*}
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Standardnorm in $\real^n$:
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$\vert \cdot \vert = \vert \cdot \vert_{p=2}$ heißt \underline{eukldische Norm}.\\
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$\vert \cdot \vert = \vert \cdot \vert_{p=2}$ heißt \emph{eukldische Norm}.\\
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\end{exmpn}
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\begin{mydefn}[Skalarprodukt]
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$\langle x,y \rangle = \sum_{i=1}^{n}$ heißt \underline{Skalarprodukt} (inneres Produkt) von $x,y \in \real^n$ offenbar $\langle x,y \rangle = \vert x \vert_2 \forall x \in comp$ nur für euklidische Räume gibt es Skalarprodukt (nur für euklische Norm!).\\
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$\langle x,y \rangle = \sum_{i=1}^{n}$ heißt \emph{Skalarprodukt} (inneres Produkt) von $x,y \in \real^n$ offenbar $\langle x,y \rangle = \vert x \vert_2 \forall x \in comp$ nur für euklidische Räume gibt es Skalarprodukt (nur für euklische Norm!).\\
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Man hat $\vert \langle x,y\rangle \vert \leq \vert x \vert_2 \cdot \vert y \vert_2 \forall x,y \in \real^n$ Cauchy-Schwarsche Ungleichung (CSU), denn
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\begin{align*}
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\vert \langle x,z \rangle \vert &= \vert \sum_{i=1}^{n} x_i y_i \vert \leq \sum_{i=1}^{n}\vert x_i y_i\vert & \Delta\text{-Ungleichung in } \real\\
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@ -99,19 +99,19 @@ $\Vert \cdot\Vert: X \to \real_{\geq0}$ heißt \underline{Halbraum} falls nur b)
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\begin{exmpn}
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$X=\comp^n$ ist Vektorraum über $\comp$, $x=(x_1,\dots,x_n) \in\comp^n, x_i \in \comp$\\
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analog zum Bsp. \ref{8_5_exmp_Norm} sind $\vert \cdot \vert_{p} \text{ und } \vert \cdot \vert_{\infty}$ Normen auf $\comp^n$\\
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$\langle x,y\rangle = \sum_{i=1}^{n} \bar{x}_i y_i\forall x_i, y_i \in \comp$ heißt \underline{Skalarprodukt} von $x,y \in \comp^n$ (beachte $\langle x,y\rangle \in \comp, \langle x,x \rangle=\vert x \vert^2$) \\
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$\langle x,y\rangle = \sum_{i=1}^{n} \bar{x}_i y_i\forall x_i, y_i \in \comp$ heißt \emph{Skalarprodukt} von $x,y \in \comp^n$ (beachte $\langle x,y\rangle \in \comp, \langle x,x \rangle=\vert x \vert^2$) \\
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$\overset{\text{wie oben}}{\Rightarrow} \vert \langle x,y\rangle \vert \leq \vert x \vert\cdot \vert y \vert \forall x,y \in \comp^n$
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\end{exmpn}
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\begin{mydefn}[Orthogonalität]
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$x,y \in \real^n(\comp^n)$ heißen \underline{orthogonal} falls $\langle x,y\rangle =0$
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$x,y \in \real^n(\comp^n)$ heißen \emph{orthogonal} falls $\langle x,y\rangle =0$
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\end{mydefn}
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\begin{exmpn}
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Sei $M$ beliebige Menge, $f: M \to \real$\\
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$\Vert f\Vert:= \sup\{\vert f(x) \vert \mid x\in M\}$. Dann ist \\
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\[\mathcal{B}(M):= \{f: M \to \real \mid \Vert f\Vert < \infty\}\]
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\underline{Menge der beschränkte Funktionen} auf $M$\\
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\emph{Menge der beschränkte Funktionen} auf $M$\\
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$\mathcal{B}(M)$ ist Vektorraum auf $\real$
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\begin{enumerate}[label={\alph*)}]
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\item $((f+g)(x) = f(x) + g(x)$
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@ -159,26 +159,26 @@ $\Vert \cdot\Vert: X \to \real_{\geq0}$ heißt \underline{Halbraum} falls nur b)
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Sei $(X,d)$ metrischer Raum.
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\begin{itemize}
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\item $B_r(a):= \{ a \in X \mid d(a,x) <r \}$ heißt offene \underline{Kugel} um $a$ mit Radius $r>0$
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\item $B_r[a]:= \overline{B}_r(a) = \{ a \in X \mid d(a,x) \leq r \}$ heißt abgeschlossene \underline{Kugel} um $a$ mit Radius $r>0$
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\item $B_r[a]:= \overline{B}_r(a) = \{ a \in X \mid d(a,x) \leq r \}$ heißt abgeschlossene \emph{Kugel} um $a$ mit Radius $r>0$
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\end{itemize}
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\end{mydefn}
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Hinweis: muss keine übliche Kugel sein z.B. $\{x\in \real^n \mid d(0,x) < 1\}$ ist Quadrat $B_r(0)$.
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\begin{mydefn}
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\begin{itemize}
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\item Menge $M\subset X$ \underline{offen} falls $\forall x \in M\;\exists \epsilon > 0\; B_{\epsilon}(x) \subset M$
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\item Menge $M\subset X$ \emph{offen} falls $\forall x \in M\;\exists \epsilon > 0\; B_{\epsilon}(x) \subset M$
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\item Menge $M$ offen falls $X\setminus M$ abgeschlossen
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\item $U \subset X$ Umgebung von $M \subset X$ falls $\exists V \subset X$ offen mit $M \subset V \subset U$
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\item $x \in M$ \underline{innerer Punkt} von $M$ falls $\exists \epsilon >0\colon B_{\epsilon}(x) \subset M$
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\item $x \in M$ \underline{äußerer Punkt} von $M$ falls $\exists \epsilon >0\colon B_{\epsilon}(x) \subset X\setminus M$
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\item $x \in X$ \underline{Randpunkt} von $M$ falls $x$ weder innerer noch äußerer Punkt ist
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\item $\inter M:=$ Menge der \underline{inneren} Punkte von $M$ heißen inneres von $M$
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\item $\ext M:=$ Menge der \underline{äußeren} Punkte von $M$ heißen äußeres von $M$
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\item $\partial M:=$ Menge der Randpunkte von $M$ heißt \underline{Rand} von $M$
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\item $x \in M$ \emph{innerer Punkt} von $M$ falls $\exists \epsilon >0\colon B_{\epsilon}(x) \subset M$
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\item $x \in M$ \emph{äußerer Punkt} von $M$ falls $\exists \epsilon >0\colon B_{\epsilon}(x) \subset X\setminus M$
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\item $x \in X$ \emph{Randpunkt} von $M$ falls $x$ weder innerer noch äußerer Punkt ist
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\item $\inter M:=$ Menge der \emph{inneren} Punkte von $M$ heißen inneres von $M$
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\item $\ext M:=$ Menge der \emph{äußeren} Punkte von $M$ heißen äußeres von $M$
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\item $\partial M:=$ Menge der Randpunkte von $M$ heißt \emph{Rand} von $M$
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\item $\cl M:= \overline{M}:=\overline{\inter M} \cup \partial M$ heißt Abschluss von $M$ (closure)
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\item $M \subset X$ \underline{beschränkt} falls $\exists a \in X, r >0\; M \subset B_r(a)$
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\item $x \in X$ \underline{Häufungskt (Hp)} von $M$ falls $\forall \epsilon > 0$ enhält \underline{$B_{\epsilon}(x)$ unendlich viele} Elemente aus $M$
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\item $x \in M$ \underline{isolierter} Punkt von $M$ falls $x$ kein Hp von $M$
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\item $M \subset X$ \emph{beschränkt} falls $\exists a \in X, r >0\; M \subset B_r(a)$
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\item $x \in X$ \emph{Häufungskt (Hp)} von $M$ falls $\forall \epsilon > 0$ enhält \emph{$B_{\epsilon}(x)$ unendlich viele} Elemente aus $M$
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\item $x \in M$ \emph{isolierter} Punkt von $M$ falls $x$ kein Hp von $M$
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\end{itemize}
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\end{mydefn}
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23
1. Semester/ANAG/TeX_files/chapter09_konvergenz.tex
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23
1. Semester/ANAG/TeX_files/chapter09_konvergenz.tex
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@ -0,0 +1,23 @@
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\chapter{Konvergenz}
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Sei $(X,d)$ metrischer Raum.
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\textbf{Ab jetzt alles ohne Bweise, folgen später.}
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\begin{mydef}[konvergente Folge, Grenzwert]
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Folge $\{a_n\}_{n\in\natur}$ (d.h. $a_n \in X$) heißt konvergent falls $a\in X$ existiert mit $\forall \epsilon > 0\exists n_0 \in \natur\colon d(a_n,a) <\epsilon \quad \forall n \geq n_0$. Dann heißt $a$ Grenzwert (Limes).\\ Schreibe $a = \lim_{n\to \infty} a_n$ bzw. $a_n \longrightarrow a$ für $n \longrightarrow \infty$ oder $a_n \overset{n \to \infty}{\longrightarrow} a$.
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\end{mydef}
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Sprich: ``'Für jede Kugel um Grenzwert befinden sich ab einem gewissen Index fasst alle FOlgenglieder innerhalb der Kugel.'' Folge $\{a_n\}$ heißt divergent, falls sie nicht konvergent ist.
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\begin{folg}
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Für Folge $\{a_n\}$ gilt: $\forall > 0\quad a = \lim_{n\to \infty} a_n \Leftrightarrow$ jede Kugel $B_{\epsilon}(a)$ enthält fast alle Folgeglieder $a_n$, das heißt alle $a_n$ bis auf endlich viele.
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\end{folg}
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\begin{exmp}[Konstante Folge]
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Sei $\{a_n\} = \{a\}_{n\in \natur}$ (das heißt $a_n = a \forall n$) $\Rightarrow d(a_n,a) = d(a,a) = 0 < \epsilon \forall \epsilon > 0, n \in \natur \Rightarrow a = \lim_{n\to \infty} a_n$.
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\end{exmp}
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\begin{exmp}
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$\forall \epsilon > 0 \exists n_0 \in \natur\colon \frac{1}{n} = \vert \frac{1}{n} - 0 \vert = d(\frac{1}{n},0)<\epsilon \forall n \geq n_0 \Rightarrow \lim_{n\to \infty} \frac{1}{n} = 0$.
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\end{exmp}
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\relax
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\@writefile{lof}{\ttl@tocsep }
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\@writefile{toc}{\contentsline {part}{I\hspace {1em}Grundlagen der Mathematik}{1}{part.1}}
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||||
}
|
||||
Loading…
Add table
Reference in a new issue