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author | Lukaszogg <82384106+Lukaszogg@users.noreply.github.com> | 2021-07-08 20:10:11 +0200 |
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committer | Lukaszogg <82384106+Lukaszogg@users.noreply.github.com> | 2021-07-08 20:10:11 +0200 |
commit | 14033ca595b5c933caea3b214d2246529e6845b8 (patch) | |
tree | 0d6d2b2eb34e5ef5df3c517be5c1c9d803fa066c /vorlesungen/slides/2/hilbertraum/l2beispiel.tex | |
parent | Update teil1.tex (diff) | |
parent | Only include buch.ind if it exists. (diff) | |
download | SeminarMatrizen-14033ca595b5c933caea3b214d2246529e6845b8.tar.gz SeminarMatrizen-14033ca595b5c933caea3b214d2246529e6845b8.zip |
Merge remote-tracking branch 'upstream/master'
Diffstat (limited to 'vorlesungen/slides/2/hilbertraum/l2beispiel.tex')
-rw-r--r-- | vorlesungen/slides/2/hilbertraum/l2beispiel.tex | 82 |
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diff --git a/vorlesungen/slides/2/hilbertraum/l2beispiel.tex b/vorlesungen/slides/2/hilbertraum/l2beispiel.tex new file mode 100644 index 0000000..3ae44af --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/l2beispiel.tex @@ -0,0 +1,82 @@ +% +% l2beispiel.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Beispiele: $\mathbb{R},\mathbb{R}^2,\dots,\mathbb{R}^n,\dots,l^2$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +\begin{itemize} +\item<2-> Quadratsummierbare Folgen von komplexen Zahlen +\[ +l^2 += +\biggl\{ +(x_k)_{k\in\mathbb{N}}\,\bigg|\, \sum_{k=0}^\infty |x_k|^2 < \infty +\biggr\} +\] +\item<3-> Skalarprodukt: +\begin{align*} +\langle x,y\rangle +&= +\sum_{k=0}^\infty \overline{x}_ky_k, +& +\uncover<4->{\|x\|^2 = \sum_{k=0}^\infty |x_k|^2} +\end{align*} +\item<5-> Vollständigkeit, +Konvergenz: Cauchy-Schwarz-Ungleichung +\[ +\biggl| +\sum_{k=0}^\infty \overline{x}_ky_k +\biggr| +\le +\sum_{k=0}^\infty |x_k|^2 +\sum_{l=0}^\infty |y_l|^2 +\] +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Standardbasisvektoren} +\begin{align*} +e_i +&= +(0,\dots,0,\underset{\underset{\textstyle i}{\textstyle\uparrow}}{1},0,\dots) +\\ +\uncover<7->{(e_i)_k &= \delta_{ik}} +\end{align*} +\uncover<8->{sind orthonormiert: +\begin{align*} +\langle e_i,e_j\rangle +&= +\sum_k \overline{\delta}_{ik}\delta_{jk} +\uncover<9->{= +\delta_{ij}} +\end{align*}} +\end{block}} +\vspace{-16pt} +\uncover<10->{% +\begin{block}{Analyse} +$x_k$ kann mit Skalarprodukten gefunden werden: +\begin{align*} +\hat{x}_i += +\langle e_i,x\rangle +&\uncover<11->{= +\sum_{k=0}^\infty \overline{\delta}_{ik} x_k} +\uncover<12->{= +x_i} +\end{align*} +\uncover<13->{(Fourier-Koeffizienten)} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup |