19 files changed, 999 insertions, 6 deletions

diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc
index ad1081e..6616f56 100644
--- a/vorlesungen/slides/4/Makefile.inc
+++ b/vorlesungen/slides/4/Makefile.inc
@@ -17,6 +17,10 @@ chapter4 =								\
 	../slides/4/euklidpoly.tex					\
 	../slides/4/polynomefp.tex					\
 	../slides/4/schieberegister.tex					\
+	../slides/4/charakteristik.tex					\
+	../slides/4/char2.tex						\
+	../slides/4/frobenius.tex					\
+	../slides/4/qundr.tex						\
 	../slides/4/alpha.tex						\
 	../slides/4/chapter.tex
 
diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex
index a10712a..6872018 100644
--- a/vorlesungen/slides/4/chapter.tex
+++ b/vorlesungen/slides/4/chapter.tex
@@ -16,3 +16,7 @@
 \folie{4/polynomefp.tex}
 \folie{4/alpha.tex}
 \folie{4/schieberegister.tex}
+\folie{4/charakteristik.tex}
+\folie{4/char2.tex}
+\folie{4/frobenius.tex}
+\folie{4/qundr.tex}
diff --git a/vorlesungen/slides/4/char2.tex b/vorlesungen/slides/4/char2.tex
new file mode 100644
index 0000000..2b5709a
--- /dev/null
+++ b/vorlesungen/slides/4/char2.tex
@@ -0,0 +1,48 @@
+%
+% char2.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Charakteristik 2}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Plus und Minus}
+\[
+x+x = 2x = 0
+\uncover<2->{\Rightarrow
+-x=x}
+\]
+\end{block}
+\uncover<3->{%
+\begin{block}{Quadrieren}
+In $\mathbb{F}_2$ ist $2=0$, d.h
+\[
+(x+y)^2
+=
+x^2 + 2xy + y^2
+\uncover<4->{=
+x^2 + y^2}
+\]
+für alle $x,y\in\Bbbk$
+\end{block}}
+\uncover<6->{%
+\begin{block}{Frobenius-Automorphismus}
+\[
+(x+y)^{2^n} = x^{2^n}+y^{2^n}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{Pascal-Dreieck}
+\begin{center}
+\includegraphics[width=\textwidth]{../../buch/chapters/30-endlichekoerper/images/binomial2.pdf}
+\end{center}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/charakteristik.tex b/vorlesungen/slides/4/charakteristik.tex
new file mode 100644
index 0000000..a0d6d3e
--- /dev/null
+++ b/vorlesungen/slides/4/charakteristik.tex
@@ -0,0 +1,71 @@
+%
+% charakteristisk.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Primkörper und Charakteristik}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Primkörper}
+$1\in\Bbbk$
+\begin{enumerate}
+\item<2->
+$n\cdot 1\ne 0\;\forall n\in\mathbb{N}$\uncover<3->{:
+$\Rightarrow$
+$\mathbb{Z}\subset \Bbbk$}
+\uncover<4->{%
+$\Rightarrow$
+$\mathbb{Q}\subset \Bbbk$}
+\item<5->
+$\{n\mathbb{Z}\;|\;
+\text{$n\cdot 1 = 0$ in $\Bbbk$}\}
+=
+p\mathbb{Z}$
+\uncover<6->{
+$\Rightarrow$
+$\mathbb{F}_p\subset \Bbbk$}
+\end{enumerate}
+\end{block}
+\uncover<7->{%
+\begin{block}{Primkörper}
+Der Primkörper $\operatorname{Prim}(\Bbbk)$
+eines Körpers $\Bbbk$ ist der kleinste in $\Bbbk$
+enthaltene Körper
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<8->{%
+\begin{block}{Charakteristik}
+\vspace{-10pt}
+\[
+\operatorname{char}(\Bbbk)
+=
+\begin{cases}
+\uncover<9->{p&\qquad \operatorname{Prim}(\Bbbk) = \mathbb{F}_p}\\
+\uncover<10->{0&\qquad \operatorname{Prim}(\Bbbk) = \mathbb{Q}}
+\end{cases}
+\]
+\vspace{-10pt}
+\end{block}}
+\uncover<11->{%
+\begin{block}{Vektorraum}
+$\Bbbk$ ist ein Vektorraum über $\operatorname{Prim}(\Bbbk)$
+durch Einschränkung der Multiplikation auf $\operatorname{Prim}(\Bbbk)$
+(Körperstruktur vergessen)
+\end{block}}
+\uncover<12->{%
+\begin{block}{Endliche Körper}
+\begin{itemize}
+\item<13->
+Endliche Körper haben immer Charakteristik $p\ne 0$
+\item<14->
+$\Bbbk$ ist eine endlichdimensionaler $\mathbb{F}_p$-Vektorraum
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/euklidmatrix.tex b/vorlesungen/slides/4/euklidmatrix.tex
index be5b3ca..c63afec 100644
--- a/vorlesungen/slides/4/euklidmatrix.tex
+++ b/vorlesungen/slides/4/euklidmatrix.tex
@@ -18,7 +18,7 @@ a_k = b_kq_k + r_k
 \;\Rightarrow\;
 \left\{
 \begin{aligned}
-a_{k+1} &= b_k = \phantom{a_k-q_k}\llap{$-\mathstrut$}b_k \\
+a_{k+1} &= b_k = \phantom{a_k-q_k}b_k \\
 b_{k+1} &= \phantom{b_k}\llap{$r_k$} = a_k - q_kb_k 
 \end{aligned}
 \right.}
diff --git a/vorlesungen/slides/4/frobenius.tex b/vorlesungen/slides/4/frobenius.tex
new file mode 100644
index 0000000..56fd78f
--- /dev/null
+++ b/vorlesungen/slides/4/frobenius.tex
@@ -0,0 +1,54 @@
+%
+% frobenius.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Frobenius-Automorphismus}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+$\operatorname{Prim}(\Bbbk) = \mathbb{F}_p$
+\uncover<2->{%
+\begin{block}{Binomial-Koeffizienten}
+\vspace{-10pt}
+\begin{align*}
+\binom{p}{k}
+&=
+\frac{
+{\color{red}p}\cdot(p-1)\cdot(p-2)\cdot\dots\cdot (p-k+1)
+}{
+1\cdot2\cdot3\cdot\dots\cdot k
+}
+\intertext{{\color{red}$p$} wird nicht gekürzt wegen}
+\uncover<3->{1&\not\equiv 0 \mod p}\\
+\uncover<3->{2&\not\equiv 0 \mod p}\\
+\uncover<3->{ &\phantom{a}\vdots}\\
+\uncover<3->{k&\not\equiv 0 \mod p}
+\end{align*}
+\vspace{-10pt}
+\end{block}}
+\vspace{-5pt}
+\uncover<4->{%
+\begin{block}{Frobenius-Authomorphismus}
+\vspace{-10pt}
+\begin{align*}
+\uncover<5->{(x+y)^{p\phantom{\mathstrut^n}}
+&=
+x^{p\phantom{\mathstrut}^n}+y^{p\phantom{mathstrut^n}}}
+\\
+\uncover<6->{(x+y)^{p^n} &= x^{p^n}+y^{p^n}}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Pascal-Dreieck}
+\begin{center}
+\includegraphics[width=\textwidth]{../../buch/chapters/30-endlichekoerper/images/binomial5.pdf}
+\end{center}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/qundr.tex b/vorlesungen/slides/4/qundr.tex
new file mode 100644
index 0000000..a6f89bd
--- /dev/null
+++ b/vorlesungen/slides/4/qundr.tex
@@ -0,0 +1,138 @@
+%
+% qundr.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\definecolor{darkred}{rgb}{0.8,0,0}
+\definecolor{darkblue}{rgb}{0,0,0.8}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (ll) at (-6,-3.6);
+\coordinate (lr) at (6,-3.6);
+\coordinate (ur) at (6,3.6);
+\coordinate (ul) at (-6,3.6);
+
+\def\d{0.6}
+\def\D{0.5}
+
+\coordinate (q) at (0,{-2.25+\d});
+\coordinate (r) at (-1.5,{\d+\D});
+\coordinate (a) at (1.5,{\d-\D});
+\coordinate (c) at (0,{2.25+\d});
+
+\coordinate (m1) at ($0.5*(q)+0.5*(r)$);
+\coordinate (m2) at ($0.5*(q)+0.5*(a)$);
+\coordinate (m3) at ($0.5*(c)+0.5*(r)$);
+\coordinate (m4) at ($0.5*(c)+0.5*(a)$);
+
+\def\t{1.5}
+\coordinate (M1) at ($(m1)+\t*(m1)-\t*(m4)$);
+\coordinate (M2) at ($(m2)+\t*(m2)-\t*(m3)$);
+\coordinate (M4) at ($(m4)+\t*(m4)-\t*(m1)$);
+\coordinate (M3) at ($(m3)+\t*(m3)-\t*(m2)$);
+
+\begin{scope}
+\clip (ll) rectangle (ur);
+
+\uncover<3->{
+	\fill[color=blue!30]
+		($0.9*(m1)+0.1*(M1)+(-6,0)$) -- ($0.9*(m1)+0.1*(M1)$)
+		-- (M4) -- (ul) -- cycle;
+}
+
+\uncover<4->{
+	\fill[color=red!60,opacity=0.5]
+		($0.9*(m2)+0.1*(M2)$) -- ($0.9*(m2)+0.1*(M2)+(6,0)$)
+		-- (ur) -- (M3) -- cycle;
+}
+
+\uncover<2->{
+	\fill[color=darkgreen!60,opacity=0.5]
+		($1.09*(m3)-0.09*(M3)$) -- ($1.09*(m3)-0.09*(M3)+(-6,0)$)
+		-- (ll) -- (M2) -- cycle;
+}
+
+\uncover<6->{
+	\fill[color=gray,opacity=0.5]
+		({6-0.1},{\d+0.22}) rectangle ({6-2.4},{\d+0.62});
+	\node[color=yellow] at (6,\d) [above left] {überabzählbar\strut};
+
+	\fill[color=gray,opacity=0.5]
+		({-6+0.1},{\d-0.15}) rectangle ({-6+1.75},{\d-0.55});
+	\node[color=yellow] at (-6,\d) [below right] {abzählbar\strut};
+
+	\draw[color=yellow,line width=2pt] (-7,\d) -- (7,\d);
+}
+
+\end{scope}
+
+\node at (q) {$\mathbb{Q}$\strut};
+\node at ($(q)+(0,-0.2)$) [below] {Primkörper};
+
+\uncover<3->{
+	\node at (r) {$\mathbb{R}$\strut};
+	\node at (r) [left] {$\text{reelle Zahlen}=\mathstrut$};
+	\draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (q) -- (r);
+	\node at ($0.5*(q)+0.5*(r)$)
+		[below,rotate={atan((-2.25-\D)/1.5)}] {index $\infty$};
+	\node[color=blue] at (ul)
+		[above right] {topologische Vervollständigung};
+}
+
+\uncover<4->{
+	\node at (a) {$\mathbb{A}$\strut};
+	\node at (a) [right] {$\mathstrut = \text{algebraische Zahlen}$};
+	\draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (q) -- (a);
+	\node at ($0.5*(q)+0.5*(a)$)
+		[below,rotate={atan((2.25-\D)/1.5)}] {index $\infty$};
+	\node[color=red] at (ur)
+		[above left] {algebraische Vervollständigung};
+}
+
+\uncover<5->{
+	\node at (c) {$\mathbb{C}$\strut};
+	\draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (r) -- (c);
+	\draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (a) -- (c);
+	\node at ($(c)+(0,0.2)$) [above] {komplexe Zahlen};
+	\node at ($0.5*(r)+0.5*(c)$)
+		[above,rotate={atan((2.25-\D)/1.5)}] {index 2};
+	\node at ($0.5*(a)+0.5*(c)$)
+		[above,rotate={atan((-2.25-\D)/1.5)}] {index $\infty$};
+}
+
+\uncover<3->{
+	\node[color=darkblue] at (ul) [below right]
+	{\begin{minipage}{0.3\textwidth}\raggedright
+	Grenzwerte von Cauchy-Folgen in $\mathbb{Q}$ hinzufügen
+	\end{minipage}};
+}
+
+\uncover<4->{
+	\node[color=darkred] at (ur) [below left]
+	{\begin{minipage}{0.3\textwidth}\raggedleft
+	Nullstellen von Polynomen in $\mathbb{Q}[X]$ hinzufügen
+	\end{minipage}};
+}
+
+\uncover<2->{
+	\node[color=darkgreen] at (ll) [above right]
+	{\begin{minipage}{0.4\textwidth}\raggedright
+	\begin{block}{Archimedische Eigenschaft}
+	Für $a>b >0$ gibt es $n\in\mathbb{N}$ mit
+	$n\cdot b > a$
+	\end{block}
+	\end{minipage}};
+
+	\node[color=darkgreen] at (ll) [below right]
+		{geordneter Körper, nötig für die Definition von Cauchy-Folgen};
+}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc
index 4ca3de4..5b849ec 100644
--- a/vorlesungen/slides/5/Makefile.inc
+++ b/vorlesungen/slides/5/Makefile.inc
@@ -5,6 +5,8 @@
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
 chapter5 =								\
+	../slides/5/plan.tex						\
+	../slides/5/planbeispiele.tex					\
 	../slides/5/verzerrung.tex					\
 	../slides/5/motivation.tex					\
 	../slides/5/charpoly.tex					\
@@ -27,6 +29,8 @@ chapter5 =								\
 									\
 	../slides/5/spektrum.tex					\
 	../slides/5/normal.tex						\
+	../slides/5/normalbeispiel.tex					\
+	../slides/5/normalbeispiel34.tex				\
 	../slides/5/unitaer.tex						\
 									\
 	../slides/5/konvergenzradius.tex				\
@@ -36,9 +40,12 @@ chapter5 =								\
 	../slides/5/satzvongelfand.tex					\
 									\
 	../slides/5/stoneweierstrass.tex				\
+	../slides/5/swbeweis.tex					\
 	../slides/5/potenzreihenmethode.tex				\
 	../slides/5/logarithmusreihe.tex				\
 	../slides/5/exponentialfunktion.tex				\
 	../slides/5/hyperbolisch.tex					\
+									\
+	../slides/5/approximation.tex					\
 	../slides/5/chapter.tex
 
diff --git a/vorlesungen/slides/5/approximation.tex b/vorlesungen/slides/5/approximation.tex
new file mode 100644
index 0000000..a35bae7
--- /dev/null
+++ b/vorlesungen/slides/5/approximation.tex
@@ -0,0 +1,56 @@
+%
+% approximation.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+
+\begin{frame}[t]
+\frametitle{Approximation einer reellen Funktion}
+\vspace{-18pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.5\textwidth}
+\begin{block}{Gegeben}
+Eine stetige Funktion $f\colon[a,b]\to\mathbb{R}$
+\end{block}
+\end{column}
+\begin{column}{0.5\textwidth}
+\uncover<2->{%
+\begin{block}{Gesucht}
+Approximationspolynome $p_n\to f$ gleichmässig auf $[a,b]$
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<3->{%
+\begin{block}{Lösungsmöglichkeiten}
+\vspace{-3pt}
+\begin{center}
+\renewcommand{\arraystretch}{1.3}
+\begin{tabular}{|p{4.2cm}|l|}
+\hline
+Familie&Approximationspolynom für $[a,b]=[0,1]$
+\\
+\hline
+\uncover<4->{%
+\raggedright
+Lagrange-Interpolationspolynom}
+&\uncover<5->{%
+$\displaystyle\begin{aligned}
+l(x)&=(x-x_0)(x-x_1)\dots(x-x_n),\quad x_k = \frac{k}{n}
+\\
+p_n(x)&= \sum_{k=0}^n f(x_k)\frac{l(x)}{x-x_k}
+\end{aligned}$}
+\\
+\hline\uncover<6->{%
+\raggedright
+Approximation mit Bernstein-Polynomen}
+&\uncover<7->{$\displaystyle \begin{aligned}
+B_{k,n}(t) &= \frac{1}{(b-a)^n}\binom{n}{k}(t-a)^k(b-t)^{n-k}
+\\
+B_n(f)(t) &= \sum_{k=0}^n B_{k,n}(t) \cdot f\biggl(\frac{k}{n}\biggr)
+\end{aligned}$}
+\\
+\hline
+\end{tabular}
+\end{center}
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/5/beispiele/kombiniert.jpg b/vorlesungen/slides/5/beispiele/kombiniert.jpg
index 9cb789c..bebc36f 100644
--- a/vorlesungen/slides/5/beispiele/kombiniert.jpg
+++ b/vorlesungen/slides/5/beispiele/kombiniert.jpg
diff --git a/vorlesungen/slides/5/beispiele/kombiniert.pov b/vorlesungen/slides/5/beispiele/kombiniert.pov
index c187d08..d17adb7 100644
--- a/vorlesungen/slides/5/beispiele/kombiniert.pov
+++ b/vorlesungen/slides/5/beispiele/kombiniert.pov
@@ -18,5 +18,6 @@ ebene(k21, k22, gruen2)
 arrow(O, j11, at, orange1)
 arrow(O, j12, at, orange1)
 arrow(O, k11, at, gruen1)
+gerade(k11, gruen1)
 ebene(j11, j12, orange1)
 
diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex
index 96eea29..cdf2ea5 100644
--- a/vorlesungen/slides/5/chapter.tex
+++ b/vorlesungen/slides/5/chapter.tex
@@ -3,6 +3,8 @@
 %
 % (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
 %
+\folie{5/plan.tex}
+\folie{5/planbeispiele.tex}
 \folie{5/verzerrung.tex}
 \folie{5/motivation.tex}
 \folie{5/charpoly.tex}
@@ -28,9 +30,13 @@
 \folie{5/Aiteration.tex}
 \folie{5/satzvongelfand.tex}
 \folie{5/stoneweierstrass.tex}
+\folie{5/swbeweis.tex}
 \folie{5/potenzreihenmethode.tex}
 \folie{5/logarithmusreihe.tex}
 \folie{5/exponentialfunktion.tex}
 \folie{5/hyperbolisch.tex}
 \folie{5/spektrum.tex}
 \folie{5/normal.tex}
+\folie{5/normalbeispiel.tex}
+\folie{5/normalbeispiel34.tex}
+\folie{5/approximation.tex}
diff --git a/vorlesungen/slides/5/normalbeispiel.tex b/vorlesungen/slides/5/normalbeispiel.tex
new file mode 100644
index 0000000..e130c15
--- /dev/null
+++ b/vorlesungen/slides/5/normalbeispiel.tex
@@ -0,0 +1,108 @@
+%
+% normalbeispiel.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkred}{rgb}{0.8,0,0}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Beispiele für normale Matrizen}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.49\textwidth}
+\uncover<3->{%
+\begin{block}{Symmetrisch und Antisymmetrisch}
+$A\in M_n(\mathbb{C})$
+\begin{align*}
+A&=\pm A^t &&\Rightarrow &AA^* &=A\overline{A^t} =\pm A\overline{A}
+\\
+ &      &&            &     &=\pm\overline{A}A =\overline{A^t}A
+\\
+ &      &&            &     &=A^*A
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.49\textwidth}
+\uncover<4->{%
+\begin{block}{Orthogonal}
+$A\in M_n(\mathbb{R})\;\Rightarrow\; A^*=A^t$
+\begin{align*}
+AA^t&=I &&\Rightarrow& AA^*&=AA^t=I\\
+    &   &&           &     &=A^tA=A^*A
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.49\textwidth}
+\uncover<1->{%
+\begin{block}{Hermitesch und Antihermitesch}
+$A\in M_n(\mathbb{C})$
+\begin{align*}
+A&=\pm A^* &&\Rightarrow &AA^* &=\pm A^2=A^*A
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.49\textwidth}
+\uncover<2->{%
+\begin{block}{Unitär}
+$A\in M_n(\mathbb{C})$
+\begin{align*}
+AA^*&=I &&\Rightarrow& AA^*=I=A^*A
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+%\uncover<5->{%
+%\begin{block}{Weitere}
+%$N\in M_n(\mathbb{C})$ nilpotent, $N^k=0$\uncover<11->{
+%$\Rightarrow$
+%normal für $l=k-l\Rightarrow l=\frac{k}{2}$}
+%\uncover<6->{%
+%\[
+%\left.
+%\begin{aligned}
+%A  &=N^l+(N^t)^{k-l}
+%\\
+%A^t&=(N^t)^l+N^{k-1}
+%\end{aligned}
+%\right\}
+%\uncover<7->{%
+%\Rightarrow
+%\left\{
+%\begin{aligned}
+%\mathstrut
+%A^t A
+%&\only<8>{=
+%((N^t)^l+N^{k-l}) (N^l+(N^t)^{k-l})}
+%\uncover<9->{=
+%{\color<10>{darkgreen}(N^t)^lN^l}
+%\only<9>{+
+%{\color{orange}(N^t)^k}}
+%+
+%{\color<10>{darkred}N^{k-l}(N^t)^{k-l}}
+%\only<9>{+
+%{\color{orange}N^k}}}
+%\\
+%\mathstrut
+%A A^t
+%&\only<8>{= 
+%(N^l+(N^t)^{k-l})((N^t)^l+N^{k-l})}
+%\uncover<9->{=
+%{\color<10>{darkred}N^l(N^t)^l}
+%+
+%\only<9>{{\color{orange}N^k}
+%+
+%{\color{orange}(N^t)^k}
+%+}
+%{\color<10>{darkgreen}(N^t)^{k-l}N^{k-l}}}
+%\end{aligned}
+%\right.}
+%\hspace{20cm}
+%\]}
+%\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/5/normalbeispiel34.tex b/vorlesungen/slides/5/normalbeispiel34.tex
new file mode 100644
index 0000000..f2647b0
--- /dev/null
+++ b/vorlesungen/slides/5/normalbeispiel34.tex
@@ -0,0 +1,80 @@
+%
+% normalbeispiel34.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\definecolor{darkred}{rgb}{0.8,0,0}
+\begin{frame}[t]
+\frametitle{Beispiele normaler Matrizen für $n=3$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.49\textwidth}
+\begin{align*}
+A
+&=
+\begin{pmatrix}
+\alpha&\beta &   0  \\
+   0  &\alpha&\beta \\
+\beta &   0  &\alpha
+\end{pmatrix},
+\;
+A^t=
+\begin{pmatrix}
+\alpha&   0  &\beta \\
+\beta &\alpha&   0  \\
+   0  &\beta &\alpha
+\end{pmatrix}
+&
+\uncover<2->{%
+&\Rightarrow\left\{
+\begin{aligned}
+AA^t&=\begin{pmatrix}
+\alpha^2+\beta^2 & \alpha\beta      & \alpha\beta      \\
+\alpha\beta      & \alpha^2+\beta^2 & \alpha\beta      \\
+\alpha\beta      & \alpha\beta      & \alpha^2+\beta^2 
+\end{pmatrix}
+\\
+&\phantom{ooooooooooooooo}\|
+\\
+A^tA&=\begin{pmatrix}
+\alpha^2+\beta^2 & \alpha\beta      & \alpha\beta      \\
+\alpha\beta      & \alpha^2+\beta^2 & \alpha\beta      \\
+\alpha\beta      & \alpha\beta      & \alpha^2+\beta^2 
+\end{pmatrix}
+\end{aligned}\right.}
+\\
+\uncover<3->{
+A&=\alpha I + \beta O}\uncover<4->{, O=\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}\in \operatorname{O}(3)}
+&
+\uncover<5->{
+&\Rightarrow
+\left\{
+\begin{aligned}
+AA^*&= \alpha^2I^2 + \beta^2
+\ifthenelse{\boolean{presentation}}{ \only<6->{I} }{} \only<-5>{OO^*}
++ \alpha\beta(O+O^*)\\
+A^*A&= \alpha^2I^2 + \beta^2
+\ifthenelse{\boolean{presentation}}{ \only<6->{I} }{} \only<-5>{O^*O}
++ \alpha\beta(O^*+O)
+\end{aligned}
+\right.}
+\\
+\uncover<7->{A&=U+V^*,\text{normal}}\uncover<10->{\text{, }
+{\color{darkgreen}UV}={\color{darkgreen}VU}}
+&
+&\uncover<8->{\Rightarrow
+\left\{
+\begin{aligned}
+AA^* &= UU^* + {\color<9->{darkgreen}UV} + {\color<9->{darkred}V^*U^*} + V^*V
+\\
+A^*A &= U^*U + {\color<9->{darkred}U^*V^*} + {\color<9->{darkgreen}VU} + VV^*
+\end{aligned}
+\right.}
+\end{align*}
+\end{column}
+\begin{column}{0.49\textwidth}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/plan.tex b/vorlesungen/slides/5/plan.tex
new file mode 100644
index 0000000..23b1b93
--- /dev/null
+++ b/vorlesungen/slides/5/plan.tex
@@ -0,0 +1,198 @@
+%
+% plan.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.5,0}
+\definecolor{darkred}{rgb}{0.8,0.0,0}
+\begin{frame}[t]
+\frametitle{Was ist $f(A)$?}
+\vspace{-5pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\uncover<7->{
+	\fill[color=blue!20] (-1.5,0.7) rectangle (11.5,3.8);
+}
+
+\uncover<4->{
+	\fill[color=darkgreen!20] (-1.5,-0.7) rectangle (11.5,0.7);
+}
+
+\uncover<12->{
+	\fill[color=darkred!20] (-1.5,-0.7) rectangle (11.5,-3.8);
+}
+
+\begin{scope}[xshift=-1cm]
+\node at (0,0) [left] {$A$};
+\end{scope}
+
+%\foreach \x in {1,...,20}{
+%	\only<\x>{ \node at (-1,3) {\x}};
+%}
+
+%
+% Blauer Ast
+%
+
+\uncover<2->{
+	\draw[->,color=blue,shorten <= 0.3cm, shorten >= 0.0cm]
+		(-1.2,0) -- (0,1.3);
+
+	\begin{scope}[xshift=0cm,yshift=1.5cm]
+	\fill[color=white,opacity=0.7] (0,-0.6) rectangle (3.4,0.6);
+	\draw[color=blue] (0,-0.6) rectangle (3.4,0.6);
+	\node at (0,0) [right] {$\begin{aligned}
+	f&=p\in\mathbb{R}[X]\\
+	f(A)&=p(A)
+	\end{aligned}
+	$};
+	\end{scope}
+}
+
+\uncover<7->{
+	\draw[->,color=blue] (1.8,2.1) -- (3.6,3);
+
+	\begin{scope}[xshift=3.6cm,yshift=3cm]
+	\fill[color=white,opacity=0.7] (0,-0.6) rectangle (3.7,0.6);
+	\draw[color=blue] (0,-0.6) rectangle (3.7,0.6);
+	\node at (0,0) [right] {\begin{minipage}{3cm}\raggedright
+	$f$ durch $p_n\in\mathbb{R}[X]$\\
+	approximieren
+	\end{minipage}};
+	\end{scope}
+}
+
+\uncover<8->{
+	\draw[->,color=blue] (7.3,3) -- (9.5,1.9);
+
+	\begin{scope}[xshift=7.6cm,yshift=1.5cm]
+	\fill[color=white,opacity=0.7] (0,-0.35) rectangle (3.8,0.4);
+	\draw[color=blue] (0,-0.35) rectangle (3.8,0.4);
+	\node at (0,0) [right] {$\displaystyle f(A) = \lim_{n\to\infty}p_n(A)$};
+	\end{scope}
+}
+
+\uncover<9->{
+	\node[color=blue] at (3.6,1.6) [right] {\begin{minipage}{4cm}
+	\raggedright
+	Konvergenz $p_n\to f$\\
+	auf Spektrum $\operatorname{Sp}(A)\subset\mathbb{R}$
+	\end{minipage}};
+}
+
+\uncover<11->{
+	\node[color=blue] at (-1.5,3.8) [below right]
+		{$A$ symmetrisch: $A=A^*$};
+}
+\uncover<10->{
+	\node[color=blue] at (11.5,3.8) [below left] {$A$ diagonalisierbar};
+}
+
+%
+% Roter Ast
+%
+
+\uncover<12->{
+	\draw[->,color=darkred,shorten <= 0.3cm, shorten >= 0.0cm] (-1.2,0) -- (0,-1.3);
+
+	\begin{scope}[xshift=0cm,yshift=-1.5cm]
+	\fill[color=white,opacity=0.7] (0,-0.6) rectangle (3.4,0.6);
+	\draw[color=darkred] (0,-0.6) rectangle (3.4,0.6);
+	\node at (0,0) [right] {$\begin{aligned}
+	f&=p\in\mathbb{C}[Z,\overline{Z}]\\
+	f(A)&=p(A,A^*)
+	\end{aligned}$};
+	\end{scope}
+}
+
+\uncover<13->{
+	\node[color=darkred] at (1.7,-2.1) [below left]
+		{Für $|Z|^2 = Z\overline{Z}$};
+}
+
+\uncover<14->{
+	\draw[->,color=darkred] (1.8,-2.1) -- (3.6,-3);
+
+	\begin{scope}[xshift=3.6cm,yshift=-3cm]
+	\fill[color=white,opacity=0.7] (0,-0.6) rectangle (3.7,0.6);
+	\draw[color=darkred] (0,-0.6) rectangle (3.7,0.6);
+	\node at (0,0) [right] {\begin{minipage}{3.5cm}\raggedright
+	$f$ durch $q_n\in\mathbb{C}[Z,\overline{Z}]$\\
+	approximieren
+	\end{minipage}};
+	\end{scope}
+}
+
+\uncover<15->{
+	\draw[->,color=darkred] (7.3,-3) -- (9.5,-1.85);
+
+	\begin{scope}[xshift=7.6cm,yshift=-1.5cm]
+	\fill[color=white,opacity=0.7] (0,-0.35) rectangle (3.8,0.4);
+	\draw[color=darkred] (0,-0.35) rectangle (3.8,0.4);
+	\node at (0,0) [right]
+		{$\displaystyle f(A) = \lim_{n\to\infty}q_n(A,A^*)$};
+	\end{scope}
+}
+
+\uncover<16->{
+	\node[color=darkred] at (3.6,-1.8) [right] {\begin{minipage}{4cm}
+	\raggedright
+	Konvergenz $p_n\to f$\\
+	auf $\operatorname{Sp}(A)\cup\operatorname{Sp}(A^*)$
+	\end{minipage}};
+}
+
+\uncover<17->{
+	\node[color=darkred] at (11.5,-3.8) [above left] {%
+	\begin{minipage}{3.5cm}\raggedleft
+	nur sinnvoll definiert wenn 
+	$AA^*=A^*A$
+	\end{minipage}};
+}
+
+\uncover<18->{
+	\node[color=darkred] at (-1.5,-3.8) [above right]
+		{$A$ normal: $AA^*=A^*A$};
+}
+
+%
+% Grüner Ast
+%
+
+\uncover<3->{
+	\draw[->,color=darkgreen,shorten <= 0.0cm, shorten >= 0.0cm]
+		(-1,0) -- (0,0);
+
+	\begin{scope}[xshift=0cm,yshift=0cm]
+	\fill[color=white,opacity=0.7] (0,-0.6) rectangle (2.9,0.6);
+	\draw[color=darkgreen] (0,-0.6) rectangle (2.9,0.6);
+	\node at (0,0) [right] {$\displaystyle
+	f(z)=\sum_{k=0}^\infty a_kz^k$};
+	\end{scope}
+}
+
+\uncover<5->{
+	\node[color=darkgreen] at (5.9,0) [above] {$f(z)$ analytisch!};
+}
+\uncover<6->{
+	\node[color=darkgreen] at (5.9,0) [below]
+		{$\varrho(A)<\text{Konvergenzradius}$};
+}
+
+\uncover<4->{
+	\draw[->,color=darkgreen] (2.9,0) -- (8.5,0);
+
+	\begin{scope}[xshift=8.5cm]
+	\fill[color=white,opacity=0.7] (0,-0.6) rectangle (2.9,0.6);
+	\draw[color=darkgreen] (0,-0.6) rectangle (2.9,0.6);
+	\node at (0,0) [right] {$\displaystyle
+	 f(A)=\sum_{k=0}^\infty a_kA^k$};
+	\end{scope}
+}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/planbeispiele.tex b/vorlesungen/slides/5/planbeispiele.tex
new file mode 100644
index 0000000..7b98a95
--- /dev/null
+++ b/vorlesungen/slides/5/planbeispiele.tex
@@ -0,0 +1,103 @@
+%
+% planbeispiele.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\definecolor{darkred}{rgb}{0.8,0,0}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{frame}[t]
+\frametitle{Beispiele}
+\vspace{-15pt}
+\begin{columns}[t]
+\begin{column}{0.33\textwidth}
+\setbeamercolor{block body}{bg=blue!20}
+\setbeamercolor{block title}{bg=blue!20}
+\uncover<2->{%
+\begin{block}{$A$ diagonal, $\operatorname{Sp}(A)\subset\mathbb{R}$\strut}
+Beispiele:
+\begin{align*}
+f(x)
+&=
+x^k,
+\\
+f(x)&=
+\sqrt{x},
+\sqrt[k]{x}
+\\
+f(x)&=|x|
+\end{align*}
+\vspace{43pt}
+\end{block}}
+\end{column}
+\begin{column}{0.33\textwidth}
+\setbeamercolor{block body}{bg=darkgreen!20}
+\setbeamercolor{block title}{bg=darkgreen!20}
+\uncover<1->{%
+\begin{block}{$f(z)$ analytisch\strut}
+Beispiele:
+\begin{align*}
+e^z
+&=
+\sum_{k=0}^\infty \frac{z^k}{k!}
+\\
+\cos z
+&=
+\sum_{k=0}^\infty (-1)^k\frac{z^{2k}}{2k!}
+\\
+\sin z
+&=
+\sum_{k=0}^\infty (-1)^k\frac{z^{2k+1}}{(2k+1)!}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.33\textwidth}
+\setbeamercolor{block body}{bg=darkred!20}
+\setbeamercolor{block title}{bg=darkred!20}
+\uncover<3->{%
+\begin{block}{$A$ normal, $AA^*=A^*A$\strut}
+Beispiele:
+\begin{align*}
+f(z)&=\sqrt{z\overline{z}}=|z|
+\end{align*}
+\vspace{76pt}
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-10pt}
+\begin{columns}[t]
+\begin{column}{0.33\textwidth}
+\setbeamercolor{block body}{bg=blue!20}
+\setbeamercolor{block title}{bg=blue!20}
+\uncover<5->{%
+\begin{block}{}
+\vspace{-6pt}
+$f(A)$ wohldefiniert für {\color{blue}diagonalisierbare}
+Matrizen $A\in M_n(\mathbb{R})$
+\end{block}}
+\end{column}
+\begin{column}{0.33\textwidth}
+\setbeamercolor{block body}{bg=darkgreen!20}
+\setbeamercolor{block title}{bg=darkgreen!20}
+\uncover<4->{%
+\begin{block}{}
+\vspace{-6pt}
+$f(A)$ wohldefiniert für {\color{darkgreen}jedes} $A\in M_n(\mathbb{C})$
+\vspace{14pt}
+\end{block}}
+\end{column}
+\begin{column}{0.33\textwidth}
+\setbeamercolor{block body}{bg=darkred!20}
+\setbeamercolor{block title}{bg=darkred!20}
+\uncover<6->{%
+\begin{block}{}
+\vspace{-6pt}
+$f(A)$ wohldefiniert für {\color{darkred}normale}
+Matrizen $A\in M_n(\mathbb{C})$
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/stoneweierstrass.tex b/vorlesungen/slides/5/stoneweierstrass.tex
index 3f9cab5..e2e9e30 100644
--- a/vorlesungen/slides/5/stoneweierstrass.tex
+++ b/vorlesungen/slides/5/stoneweierstrass.tex
@@ -3,9 +3,64 @@
 %
 % (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
 \begin{frame}[t]
-\frametitle{Stone-Weierstrass}
-
-TODO XXX
-
+\frametitle{Allgemeiner Approximationssatz}
+\vspace{-20pt}
+\begin{columns}[t]
+\begin{column}{0.5\textwidth}
+\begin{theorem}[Stone-Weierstrass, $\mathbb{R}$]
+$A$ eine {\color{darkgreen}$\mathbb{R}$}-Algebra
+von stetigen Funktionen auf einem
+%abgeschlossenen und beschränkten
+kompakten
+Definitionsgebiet $D\subset {\color{darkgreen}\mathbb{R}}$, 
+\begin{itemize}
+\item<2-> konstante Funktion $c\in A$,
+\item<3-> für $d_1,d_2\in D$ gibt es ein $s\in A$ mit
+$s(d_1)\ne s(d_2)$.
+\end{itemize}
+\uncover<4->{%
+Dann lässt sich jede stetige Funktion durch Funktionen aus $A$
+approximieren}
+\end{theorem}
+\uncover<5->{
+\begin{block}{Anwendung}
+\uncover<6->{$A={\color{darkgreen}\mathbb{R}}[X]$}\uncover<7->{,
+$s(X)=X$}\uncover<8->{,
+jede stetige Funktion kann durch
+Polynome in $X$ approximiert werden}
+\end{block}}
+\end{column}
+\begin{column}{0.5\textwidth}
+\uncover<9->{%
+\begin{theorem}[Stone-Weierstrass, $\mathbb{C}$]
+$A$ eine {\color<10->{red}$\mathbb{C}$}-Algebra von stetigen Funktionen
+auf einem
+%abgeschlossenen und beschränkten
+kompakten
+Definitionsgebiet $D\subset {\color<10->{red}\mathbb{C}}$, 
+\begin{itemize}
+\item konstante Funktion $c\in A$,
+\item für $d_1,d_2\in D$ gibt es ein $s\in A$ mit
+$s(d_1)\ne s(d_2)$.
+\only<11->{
+\item {\color{red}$f\in A\Rightarrow \overline{f}\in A$}
+}
+\end{itemize}
+Dann lässt sich jede stetige Funktion durch Funktionen aus $A$
+approximieren
+\end{theorem}}
+\vspace{-5pt}
+\uncover<12->{%
+\begin{block}{Anwendung}
+$A={\color{red}\mathbb{C}}[Z,\overline{Z}]$\uncover<13->{,
+$s(Z{\color{red},\overline{Z}})=Z$}\uncover<14->{,
+jede stetige Funktion
+lässt sich durch Polynome in $Z{\color{red},\overline{Z}}$ approximieren}
+\end{block}}
+\end{column}
+\end{columns}
 \end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/swbeweis.tex b/vorlesungen/slides/5/swbeweis.tex
new file mode 100644
index 0000000..927322b
--- /dev/null
+++ b/vorlesungen/slides/5/swbeweis.tex
@@ -0,0 +1,56 @@
+%
+% swbeweis.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Beweisidee Stone-Weierstrass}
+\vspace{-15pt}
+\begin{columns}[t]
+\begin{column}{0.5\textwidth}
+\begin{enumerate}
+\item<1->
+$\exists$ eine monoton wachsende Folge von Polynomen $u_n(t)\to \sqrt{t}$
+gleichmässig auf $[0,1]\subset{\color{darkgreen}\mathbb{R}}$
+\item<2->
+$f\in A$, dann kann man $|f| = \sqrt{f^2}$ beliebig genau approximieren
+durch Funktionen
+in $A$
+\item<3->
+$f,g\in A$, dann kann
+\begin{align*}
+\max(a,b)&={\textstyle\frac12}(f+g+|f-g|)\\
+\min(a,b)&={\textstyle\frac12}(f+g-|f-g|)
+\end{align*}
+in $A$ beliebig genau approximiert werden.
+\end{enumerate}
+\end{column}
+\begin{column}{0.5\textwidth}
+\begin{enumerate}
+\setcounter{enumi}{3}
+\item<4->
+Für $x,y\in D$ und $\alpha,\beta\in\mathbb{R}$ gibt es $f\in A$ mit
+$f(x)=\alpha$ und $f(y)=\beta$
+\item<5->
+Zu
+$f\colon D\to\mathbb{R}$ stetig und $x\in D$ gibt es $g\in A$ mit $g(x)=f(x)$
+und $g(y) \le f(y)+\varepsilon$ für $y\ne x$
+\item<6->
+Für $f$ gibt es endlich viele Approximationen $g_i$ mit Punkten $x_i$
+wie in Schritt~4.
+Dann ist $\max_i g_i$ eine Approximation von $f$, die beliebig genau in
+$A$ approximiert werden kann.
+\end{enumerate}
+\end{column}
+\end{columns}
+
+\vspace{10pt}
+\uncover<7->{%
+Schritt~2 braucht in {\color{red}$\mathbb{C}$} die komplex Konjugierte:
+$|f|^2=f\overline{f}$}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex
index e4b9ad7..d079a05 100644
--- a/vorlesungen/slides/test.tex
+++ b/vorlesungen/slides/test.tex
@@ -4,14 +4,18 @@
 % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
 %\folie{5/verzerrung.tex}
+%\folie{5/plan.tex}
+%\folie{5/planbeispiele.tex}
+%\folie{5/approximation.tex}
  
 % XXX Visualisierung Cayley-Hamilton-Produkte
 % XXX \folie{5/chvisual.tex}
 
 % XXX stone weierstrass incomplete
 %\folie{5/stoneweierstrass.tex}
+%\folie{5/swbeweis.tex}
 
 % XXX polynome auf dem spektrum
 % XXX Motiviation für *-Operation
 %\folie{5/normal.tex}
-
+\folie{5/normalbeispiel34.tex}