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author | Andreas Müller <andreas.mueller@ost.ch> | 2021-08-03 07:37:42 +0200 |
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committer | GitHub <noreply@github.com> | 2021-08-03 07:37:42 +0200 |
commit | f31aca6129f3c84f1ed4f59378fd31cbdc58ec3b (patch) | |
tree | 97c32dbdcbcc888a9030d149f5a765f006fcd631 /vorlesungen/slides/2/hilbertraum/riesz.tex | |
parent | 1. Version Kapitel Rotation und Spiegelung (diff) | |
parent | Merge pull request #60 from Kuehnee/master (diff) | |
download | SeminarMatrizen-f31aca6129f3c84f1ed4f59378fd31cbdc58ec3b.tar.gz SeminarMatrizen-f31aca6129f3c84f1ed4f59378fd31cbdc58ec3b.zip |
Merge branch 'master' into master
Diffstat (limited to 'vorlesungen/slides/2/hilbertraum/riesz.tex')
-rw-r--r-- | vorlesungen/slides/2/hilbertraum/riesz.tex | 76 |
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diff --git a/vorlesungen/slides/2/hilbertraum/riesz.tex b/vorlesungen/slides/2/hilbertraum/riesz.tex new file mode 100644 index 0000000..437fb3c --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/riesz.tex @@ -0,0 +1,76 @@ +% +% riesz.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Darstellungssatz von Riesz} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Dualraum} +$V$ ein Vektorraum, $V^*$ der Raum aller Linearformen +\[ +f\colon V\to \mathbb{C} +\] +\end{block} +\uncover<3->{% +\begin{block}{Beispiel: $l^\infty$} +$l^\infty=\text{beschränkte Folgen in $\mathbb{C}$}$, +Linearformen: +\begin{align*} +\uncover<4->{ +f(x) +&= +\sum_{i=0}^\infty f_ix_i} +\\ +\uncover<5->{ +\|f\| +&= +\sup_{\|x\|_{\infty}\le 1} +|f(x)|} +\uncover<6->{= +\sum_{k\in\mathbb{N}} |f_k|} +\\ +\uncover<7->{ +\Rightarrow +l^{\infty*} +&= +l^1} +\uncover<9->{\qquad(\ne l^2)} +\\ +\uncover<8->{ +&=\{\text{summierbare Folgen in $\mathbb{C}$}\} +} +\end{align*} + +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Beispiel: $\mathbb{C}^n$} +${\mathbb{C}^n}^* = \mathbb{C}^n$ +\end{block}} +\uncover<10->{% +\begin{theorem}[Riesz] +Zu einer stetigen Linearform $f\colon H\to\mathbb{C}$ gibt es $v\in H$ mit +\[ +f(x) = \langle v,x\rangle +\quad\forall x\in H +\] +und $\|f\| = \|v\|$ +\end{theorem}} +\uncover<11->{% +\begin{block}{Dualraum von $H$} +$H^*=H$ +\end{block}}% +\uncover<12->{% +Der Hilbertraum ist die ``intuitiv richtige, unendlichdimensionale'' +Verallgemeinerung von $\mathbb{C}^n$} +\end{column} +\end{columns} +\end{frame} +\egroup |