From 38f326c35243942285c43310751025d2c16801dd Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andreas=20M=C3=BCller?= <andreas.mueller@ost.ch>
Date: Thu, 4 Mar 2021 20:30:33 +0100
Subject: Spektraltheorie 1. Teil

---
 vorlesungen/slides/5/injektiv.tex | 74 ++++++++++++++++++++++++++++++++++++++-
 1 file changed, 73 insertions(+), 1 deletion(-)

(limited to 'vorlesungen/slides/5/injektiv.tex')

diff --git a/vorlesungen/slides/5/injektiv.tex b/vorlesungen/slides/5/injektiv.tex
index e08cafd..90cbcd6 100644
--- a/vorlesungen/slides/5/injektiv.tex
+++ b/vorlesungen/slides/5/injektiv.tex
@@ -3,7 +3,79 @@
 %
 % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 %
+\bgroup
+\def\sx{1.05}
 \begin{frame}[t]
 \frametitle{$f$ injektiv auf $\mathcal{J}(f)$}
-XXX
+\setlength{\abovedisplayskip}{8pt}
+\setlength{\belowdisplayskip}{8pt}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.58\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\fill[color=orange!20]
+	({0*\sx},-3.0) -- ({1*\sx},-2.0) -- ({2*\sx},-1.5) --
+	({3*\sx},-1.1) -- ({4*\sx},-0.9) -- ({5*\sx},-0.8) --
+	({6*\sx},-0.8) --
+	({6*\sx},0.8) -- ({5*\sx},0.8) -- ({4*\sx},0.9) --
+	({3*\sx},1.1) -- ({2*\sx},1.5) -- ({1*\sx},2.0) --
+	({0*\sx},3.0) -- cycle;
+\fill[color=orange!40] (0,-0.8) rectangle ({6*\sx},0.8);
+
+\foreach \x in {0,...,6}{
+	\draw[color=gray,line width=3pt] ({\x*\sx},-3)--({\sx*\x},3);
+}
+\foreach \x in {0,1,2,3}{
+	\node at ({\sx*\x},-3) [below] {$\x$};
+}
+\node at ({\sx*5},-3) [below] {$l$};
+\node at ({\sx*6},-3) [below] {$l+1$};
+\draw[->] (-0.1,-3.5) -- ({6*\sx+0.4},-3.5) coordinate[label={below:$k$}];
+
+\draw[line width=3pt,color=orange] ({0*\sx},-3.0) -- ({0*\sx},3.0);
+\draw[line width=3pt,color=orange] ({1*\sx},-2.0) -- ({1*\sx},2.0);
+\draw[line width=3pt,color=orange] ({2*\sx},-1.5) -- ({2*\sx},1.5);
+\draw[line width=3pt,color=orange] ({3*\sx},-1.1) -- ({3*\sx},1.1);
+\draw[line width=3pt,color=orange] ({4*\sx},-0.9) -- ({4*\sx},0.9);
+\draw[line width=3pt,color=orange] ({5*\sx},-0.8) -- ({5*\sx},0.8);
+\draw[line width=3pt,color=orange] ({6*\sx},-0.8) -- ({6*\sx},0.8);
+
+\foreach \x in {0,1,2,3}{
+	\node at ({\x*\sx},0) [rotate=90] {$\mathcal{J}^{\x}(A)$};
+}
+\node at ({4*\sx},0) {$\cdots$};
+\node at ({5*\sx},0) [rotate=90] {$\mathcal{J}^{l}(A)$};
+\node at ({6*\sx},0) [rotate=90] {$\mathcal{J}^{l+1}(A)$};
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.38\textwidth}
+\begin{block}{stationär}
+$l$ der $k$-Wert, ab dem gilt
+\begin{align*}
+\mathcal{J}^l(A) &= \mathcal{J}^{l+1}(A) = A\mathcal{J}^l(A)
+\end{align*}
+\end{block}
+\vspace{-10pt}
+\uncover<2->{%
+\begin{block}{Dimension}
+\vspace{-10pt}
+\[
+\dim \mathcal{J}^l(A) = \dim\mathcal{J}^{l+1}(A)
+\]
+\uncover<3->{%
+d.~h.~$A$ ist bijektiv als Selbstabbildung von
+$\mathcal{J}(A)$}
+\uncover<4->{%
+\[
+\Downarrow
+\]
+$A|\mathcal{J}(A)$ ist injektiv}
+\end{block}}
+\end{column}
+\end{columns}
 \end{frame}
+\egroup
-- 
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