diff --git a/.gitignore b/.gitignore index 73c93ee..0584950 100644 --- a/.gitignore +++ b/.gitignore @@ -164,3 +164,4 @@ TSWLatexianTemp* *.thm *.gz *.toc +Vorlesung LAAG.tex diff --git a/ANAG/README.md b/1. Semester/ANAG/README.md similarity index 100% rename from ANAG/README.md rename to 1. Semester/ANAG/README.md diff --git a/ANAG/TeX_files/chapter01_Grundbegriffe_aus_Mengenlehre_und_Logik.tex b/1. Semester/ANAG/TeX_files/chapter01_Grundbegriffe_aus_Mengenlehre_und_Logik.tex similarity index 100% rename from ANAG/TeX_files/chapter01_Grundbegriffe_aus_Mengenlehre_und_Logik.tex rename to 1. Semester/ANAG/TeX_files/chapter01_Grundbegriffe_aus_Mengenlehre_und_Logik.tex diff --git a/ANAG/TeX_files/chapter02_Aufbau_einer_math_Theorie.tex b/1. Semester/ANAG/TeX_files/chapter02_Aufbau_einer_math_Theorie.tex similarity index 100% rename from ANAG/TeX_files/chapter02_Aufbau_einer_math_Theorie.tex rename to 1. 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Semester/ANAG/TeX_files/chapter05_reelle_Zahlen.tex @@ -3,8 +3,9 @@ \item[Frage:] Frage: algebraische Gleichung $a_0+a_1x+\dots+a_x^k=0\;(a_j\in \whole)$\\ i.A nur für $k=1$ lösbar (d.h. lin. Gl.) \end{description} + \begin{exmpn} - $x^2 - 2 = 0$ 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 $\Rightarrow \lightning$. + $x^2 - 2 = 0$ 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 \end{exmpn} \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. @@ -18,7 +19,7 @@ $\field$ sei ein (bel.) Körper mit bel. Elementen $0, 1$ bzw. $0_K, 1_K$. \begin{satz} Sei $\field$ Körper. Dann gilt $\forall a,b \in \field$: - \begin{enumerate}[label=[1), nolistsep] + \begin{enumerate}[label={\arabic*)}, nolistsep] \item $0,1, (-a), b^{-1}$ sind eindeutig bestimmt \item $(-0) = 0$, $1^{-1} = 1$ \item $-(-a) = a$, $(b^{-1})^{-1} = b$ $(b \neq 0)$ @@ -31,7 +32,7 @@ $\field$ sei ein (bel.) Körper mit bel. Elementen $0, 1$ bzw. $0_K, 1_K$. \end{satz} \begin{proof} - \begin{enumerate}[label=[zu 1)] + \begin{enumerate}[label={\arabic*)}] \item vgl. lin. Algebra \item betrachte $0 + 0 = 0$ bzw. $1 \cdot 1 = 1$ \item $(-a) + a = 0 \overset{komm}{\Rightarrow} a = -(-a)$ Rest analog @@ -44,12 +45,12 @@ $\field$ sei ein (bel.) Körper mit bel. Elementen $0, 1$ bzw. $0_K, 1_K$. \end{enumerate} \end{proof} -\noindent Setze für alle $a, \dots a_k \in \field,n\in \natur_{\geq 1}$ -\begin{description} +Setze für alle $a, \dots a_k \in \field,n\in \natur_{\geq 1}$ +\begin{itemize} \item[Vielfache] $n\cdot a$ (kein Produkt in $\field$!) \item[Potenzen] $a^n=\prod_{k=1}^{n} a_k \text{für } n \in N_{\geq 1}$ damit $(-n)a:=n(-a) \text{, } 0_{\natur}a=0_{\natur} \text{ für } n\in\natur_{\geq1}\\ a^{-n}=(a^-1)^n \text{, }a^{0_{\natur}}:=1_{\field} \text{ für } n \in \natur_{\geq 1}, a \neq 0\\ - beachte: 0^0 = (0_\natur)^{0_{\natur}} \text{ \underline{nicht} definiert!}$ + beachte: 0^0 = (0_\natur)^{0_{\natur}} \text{ \emph{nicht} definiert!}$ \item[Rechenregeln] $\forall\;a,b\in \field\text{, } m,n\in \whole \text{ (sofern Potenz definiert) } $ -\end{description} +\end{itemize} %TODO \ No newline at end of file diff --git a/ANAG/TeX_files/chapter06_komplexe_zahlen.tex b/1. 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Semester/ANAG/TeX_files/chapter08_metr_raeume.tex @@ -39,7 +39,7 @@ Man hat $d(x,y) = 0 \forall x,y \in X$, dann \begin{exmpn}[induzierte Metrik] Sei $(X,d)$ metrischer Raum, $Y \subset X$\\ - $\Rightarrow (Y,d)$ ist metrischer Raum mit \underline{induzierter Metrik} $\tilde{d}(x,y):=d(x,y)\forall x,y \in Y$ + $\Rightarrow (Y,d)$ ist metrischer Raum mit \emph{induzierter Metrik} $\tilde{d}(x,y):=d(x,y)\forall x,y \in Y$ \end{exmpn} \section{Normierte Räume} @@ -48,13 +48,13 @@ wichtiger Spezialfall: normierte Vektorraum(VR) \begin{mydefn}[Norm] Sei $X$ Vektorraum über $K=\real$ oder $K=\comp$.\\ - Abbildung $\Vert \cdot \Vert: X \to \real$ heißt \underline{Norm} auf $X$ falls $\forall x,y \in X, \forall \lambda \in \real$ gilt: + Abbildung $\Vert \cdot \Vert: X \to \real$ heißt \emph{Norm} auf $X$ falls $\forall x,y \in X, \forall \lambda \in \real$ gilt: \begin{enumerate}[label={\alph*)}] \item $\Vert x\Vert = \Leftrightarrow x=0$ \item $\Vert \lambda x\Vert = \vert \lambda \vert \Vert x\Vert$ (Homogenität) \item $\Vert x+y\Vert \leq \Vert x\Vert + \Vert y\Vert$ ($\Delta$-Ungleichung) \end{enumerate} - $(X,\Vert \cdot\Vert)$ heißt \underline{normierter Raum}. + $(X,\Vert \cdot\Vert)$ heißt \emph{normierter Raum}. \end{mydefn} \begin{align*} @@ -66,7 +66,7 @@ Analog Satz 5.5 folgt\\ \begin{align} \vert \Vert x \Vert - \Vert y \Vert\vert &\leq \Vert x-y\Vert \forall x,y \in X \end{align} -$\Vert \cdot\Vert: X \to \real_{\geq0}$ heißt \underline{Halbraum} falls nur b), c) gelten analog Beispiel \ref{8_1_exmp_metrik} folgt. +$\Vert \cdot\Vert: X \to \real_{\geq0}$ heißt \emph{Halbraum} falls nur b), c) gelten analog Beispiel \ref{8_1_exmp_metrik} folgt. \begin{satz} Sei $(X,\Vert\cdot \Vert)$ normierter Raum, dann $X$ metrischer Raum mit Metrik $d(x,y):=\Vert x-y \Vert\forall x,y \in X$. @@ -84,11 +84,11 @@ $\Vert \cdot\Vert: X \to \real_{\geq0}$ heißt \underline{Halbraum} falls nur b) \end{cases*} \end{align*} Standardnorm in $\real^n$: - $\vert \cdot \vert = \vert \cdot \vert_{p=2}$ heißt \underline{eukldische Norm}.\\ + $\vert \cdot \vert = \vert \cdot \vert_{p=2}$ heißt \emph{eukldische Norm}.\\ \end{exmpn} \begin{mydefn}[Skalarprodukt] - $\langle x,y \rangle = \sum_{i=1}^{n}$ 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!).\\ + $\langle x,y \rangle = \sum_{i=1}^{n}$ 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!).\\ 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 \begin{align*} \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\\ @@ -99,19 +99,19 @@ $\Vert \cdot\Vert: X \to \real_{\geq0}$ heißt \underline{Halbraum} falls nur b) \begin{exmpn} $X=\comp^n$ ist Vektorraum über $\comp$, $x=(x_1,\dots,x_n) \in\comp^n, x_i \in \comp$\\ analog zum Bsp. \ref{8_5_exmp_Norm} sind $\vert \cdot \vert_{p} \text{ und } \vert \cdot \vert_{\infty}$ Normen auf $\comp^n$\\ - $\langle x,y\rangle = \sum_{i=1}^{n} \bar{x}_i y_i\forall x_i, y_i \in \comp$ heißt \underline{Skalarprodukt} von $x,y \in \comp^n$ (beachte $\langle x,y\rangle \in \comp, \langle x,x \rangle=\vert x \vert^2$) \\ + $\langle x,y\rangle = \sum_{i=1}^{n} \bar{x}_i y_i\forall x_i, y_i \in \comp$ heißt \emph{Skalarprodukt} von $x,y \in \comp^n$ (beachte $\langle x,y\rangle \in \comp, \langle x,x \rangle=\vert x \vert^2$) \\ $\overset{\text{wie oben}}{\Rightarrow} \vert \langle x,y\rangle \vert \leq \vert x \vert\cdot \vert y \vert \forall x,y \in \comp^n$ \end{exmpn} \begin{mydefn}[Orthogonalität] - $x,y \in \real^n(\comp^n)$ heißen \underline{orthogonal} falls $\langle x,y\rangle =0$ + $x,y \in \real^n(\comp^n)$ heißen \emph{orthogonal} falls $\langle x,y\rangle =0$ \end{mydefn} \begin{exmpn} Sei $M$ beliebige Menge, $f: M \to \real$\\ $\Vert f\Vert:= \sup\{\vert f(x) \vert \mid x\in M\}$. Dann ist \\ \[\mathcal{B}(M):= \{f: M \to \real \mid \Vert f\Vert < \infty\}\] - \underline{Menge der beschränkte Funktionen} auf $M$\\ + \emph{Menge der beschränkte Funktionen} auf $M$\\ $\mathcal{B}(M)$ ist Vektorraum auf $\real$ \begin{enumerate}[label={\alph*)}] \item $((f+g)(x) = f(x) + g(x)$ @@ -159,26 +159,26 @@ $\Vert \cdot\Vert: X \to \real_{\geq0}$ heißt \underline{Halbraum} falls nur b) Sei $(X,d)$ metrischer Raum. \begin{itemize} \item $B_r(a):= \{ a \in X \mid d(a,x) 0$ - \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 $r>0$ + \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 $r>0$ \end{itemize} \end{mydefn} Hinweis: muss keine übliche Kugel sein z.B. $\{x\in \real^n \mid d(0,x) < 1\}$ ist Quadrat $B_r(0)$. \begin{mydefn} \begin{itemize} - \item Menge $M\subset X$ \underline{offen} falls $\forall x \in M\;\exists \epsilon > 0\; B_{\epsilon}(x) \subset M$ + \item Menge $M\subset X$ \emph{offen} falls $\forall x \in M\;\exists \epsilon > 0\; B_{\epsilon}(x) \subset M$ \item Menge $M$ offen falls $X\setminus M$ abgeschlossen \item $U \subset X$ Umgebung von $M \subset X$ falls $\exists V \subset X$ offen mit $M \subset V \subset U$ - \item $x \in M$ \underline{innerer Punkt} von $M$ falls $\exists \epsilon >0\colon B_{\epsilon}(x) \subset M$ - \item $x \in M$ \underline{äußerer Punkt} von $M$ falls $\exists \epsilon >0\colon B_{\epsilon}(x) \subset X\setminus M$ - \item $x \in X$ \underline{Randpunkt} von $M$ falls $x$ weder innerer noch äußerer Punkt ist - \item $\inter M:=$ Menge der \underline{inneren} Punkte von $M$ heißen inneres von $M$ - \item $\ext M:=$ Menge der \underline{äußeren} Punkte von $M$ heißen äußeres von $M$ - \item $\partial M:=$ Menge der Randpunkte von $M$ heißt \underline{Rand} von $M$ + \item $x \in M$ \emph{innerer Punkt} von $M$ falls $\exists \epsilon >0\colon B_{\epsilon}(x) \subset M$ + \item $x \in M$ \emph{äußerer Punkt} von $M$ falls $\exists \epsilon >0\colon B_{\epsilon}(x) \subset X\setminus M$ + \item $x \in X$ \emph{Randpunkt} von $M$ falls $x$ weder innerer noch äußerer Punkt ist + \item $\inter M:=$ Menge der \emph{inneren} Punkte von $M$ heißen inneres von $M$ + \item $\ext M:=$ Menge der \emph{äußeren} Punkte von $M$ heißen äußeres von $M$ + \item $\partial M:=$ Menge der Randpunkte von $M$ heißt \emph{Rand} von $M$ \item $\cl M:= \overline{M}:=\overline{\inter M} \cup \partial M$ heißt Abschluss von $M$ (closure) - \item $M \subset X$ \underline{beschränkt} falls $\exists a \in X, r >0\; M \subset B_r(a)$ - \item $x \in X$ \underline{Häufungskt (Hp)} von $M$ falls $\forall \epsilon > 0$ enhält \underline{$B_{\epsilon}(x)$ unendlich viele} Elemente aus $M$ - \item $x \in M$ \underline{isolierter} Punkt von $M$ falls $x$ kein Hp von $M$ + \item $M \subset X$ \emph{beschränkt} falls $\exists a \in X, r >0\; M \subset B_r(a)$ + \item $x \in X$ \emph{Häufungskt (Hp)} von $M$ falls $\forall \epsilon > 0$ enhält \emph{$B_{\epsilon}(x)$ unendlich viele} Elemente aus $M$ + \item $x \in M$ \emph{isolierter} Punkt von $M$ falls $x$ kein Hp von $M$ \end{itemize} \end{mydefn} diff --git a/1. Semester/ANAG/TeX_files/chapter09_konvergenz.tex b/1. Semester/ANAG/TeX_files/chapter09_konvergenz.tex new file mode 100644 index 0000000..bd34413 --- /dev/null +++ b/1. Semester/ANAG/TeX_files/chapter09_konvergenz.tex @@ -0,0 +1,23 @@ +\chapter{Konvergenz} +Sei $(X,d)$ metrischer Raum. + +\textbf{Ab jetzt alles ohne Bweise, folgen später.} + +\begin{mydef}[konvergente Folge, Grenzwert] + 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 $a_n \overset{n \to \infty}{\longrightarrow} a$. +\end{mydef} + +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. + +\begin{folg} + 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. +\end{folg} + +\begin{exmp}[Konstante Folge] + Sei $\{a_n\} = \{a\}_{n\in \natur}$ (das heißt $a_n = a \forall n$) $\Rightarrow d(a_n,a) = d(a,a) = 0 < \epsilon \forall \epsilon > 0, n \in \natur \Rightarrow a = \lim_{n\to \infty} a_n$. +\end{exmp} + +\begin{exmp} + + $\forall \epsilon > 0 \exists n_0 \in \natur\colon \frac{1}{n} = \vert \frac{1}{n} - 0 \vert = d(\frac{1}{n},0)<\epsilon \forall n \geq n_0 \Rightarrow \lim_{n\to \infty} \frac{1}{n} = 0$. +\end{exmp} \ No newline at end of file diff --git a/ANAG/TeX_files/chapter10_vollst.tex b/1. Semester/ANAG/TeX_files/chapter10_vollst.tex similarity index 100% rename from ANAG/TeX_files/chapter10_vollst.tex rename to 1. 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