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diff --git a/vorlesungen/slides/2/linearformnormen.tex b/vorlesungen/slides/2/linearformnormen.tex new file mode 100644 index 0000000..8993f66 --- /dev/null +++ b/vorlesungen/slides/2/linearformnormen.tex @@ -0,0 +1,76 @@ +% +% linearformnormen.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Linearformen} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Linearformen $\varphi\colon L^1\to\mathbb{R}$} +Beispiel: $g\in C([a,b])$ +\[ +\varphi(f) += +\int_a^b g(x)f(x)\,dx +\] +\uncover<2->{% +erfüllt +\begin{align*} +|\varphi(f)| +&= +\biggl|\int_a^b g(x)f(x)\,dx\biggr| +\\ +\uncover<3->{ +&\le \|g\|_\infty\cdot \|f\|_1 +} +\end{align*}} +\uncover<4->{% +und hat daher die Operatornorm +\[ +\|\varphi\|_{C([a,b])^*} += +\|g\|_\infty +\]} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Linearformen $\varphi\colon L^2\to\mathbb{R}$} +\uncover<5->{% +Darstellungssatz von Riesz: $\exists g\in L^2$ +\[ +\varphi(f) = \langle g,f\rangle +\]} +\uncover<6->{% +erfüllt Cauchy-Schwarz} +\begin{align*} +\uncover<7->{ +|\varphi(f)| +&= +|\langle g,f\rangle|} +\\ +\uncover<8->{ +&\le +\|g\|_2 \cdot \|f\|_2 +} +\end{align*} +\uncover<9->{% +und hat daher die Operatornorm +\[ +\|\varphi\|_{L^2([a,b])^*} += \|g\|_2 +\]} +\end{block} +\end{column} +\end{columns} + +\vspace{8pt} +{\usebeamercolor[fg]{title} +\uncover<10->{% +$\Rightarrow$ +Operatornorm hängt von den Vektorraumnormen ab} +} +\end{frame} |