Merge pull request #1 from AndreasFMueller/master

update

1 files changed, 94 insertions, 0 deletions

diff --git a/vorlesungen/slides/2/cauchyschwarz.tex b/vorlesungen/slides/2/cauchyschwarz.tex
new file mode 100644
index 0000000..a24ada8
--- /dev/null
+++ b/vorlesungen/slides/2/cauchyschwarz.tex
@@ -0,0 +1,94 @@
+%
+% cauchyschwarz.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.5,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Cauchy-Schwarz-Ungleichung}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Satz (Cauchy-Schwarz)}
+$\langle\;,\;\rangle$ eine positiv definite, hermitesche Sesquilinearform
+\[
+{\color{darkgreen}
+|\operatorname{Re}\langle u,v\rangle|
+\le
+|\langle u,v\rangle|
+\le
+\|u\|_2\cdot \|v\|_2
+}
+\]
+Gleichheit genau dann, wenn $u$ und $v$ linear abhängig sind
+\end{block}
+\begin{block}{Dreiecksungleichung}
+\vspace{-12pt}
+\begin{align*}
+\|u+v\|_2^2
+&=
+\|u\|_2^2 + 2\operatorname{Re}\langle u,v\rangle + \|v\|_2^2
+\\
+&\le
+\|u\|_2^2 + 2{\color{darkgreen}|\langle u,v\rangle|} + \|v\|_2^2
+\\
+&\le
+\|u\|_2^2 + 2{\color{darkgreen}\|u\|_2\cdot \|v\|_2} + \|v\|_2^2
+\\
+&=(\|u\|_2 + \|v\|_2)^2
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{proof}[Beweis]
+Die quadratische Funktion
+\begin{align*}
+Q(t)
+&=
+\langle u+tv,u+tv\rangle \ge 0
+\\
+\uncover<3->{
+Q(t)
+&=
+\|u\|_2^2 + 2t\operatorname{Re}\langle u,v\rangle + t^2\|v\|_2^2}
+\end{align*}
+\uncover<4->{hat ihr Minimum bei}%
+\begin{align*}
+\uncover<5->{
+t&=
+-\operatorname{Re}\langle u,v\rangle/\|v\|_2^2}
+\intertext{\uncover<6->{mit Wert}}
+\uncover<7->{
+Q(t)
+&=
+\|u\|_2^2
+-2\operatorname{Re}\langle u,v\rangle^2/\|v\|_2^2}
+\\
+\uncover<7->{
+&\qquad + \operatorname{Re}\langle u,v\rangle^2/\|v\|_2^2}
+\\
+\uncover<8->{
+0
+&\le
+\|u\|_2^2-\operatorname{Re}\langle u,v\rangle^2/\|v\|_2^2}
+\\
+\uncover<9->{
+\operatorname{Re}\langle u,v\rangle^2
+&\le
+\|u\|_2^2\cdot\|v\|_2^2}
+\\
+\uncover<10->{
+\operatorname{Re}\langle u,v\rangle
+&\le
+\|u\|_2\cdot\|v\|_2}
+\qedhere
+\end{align*}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup