diff options
Diffstat (limited to 'vorlesungen/slides/2/hilbertraum/rieszbeispiel.tex')
| -rw-r--r-- | vorlesungen/slides/2/hilbertraum/rieszbeispiel.tex | 27 |
1 files changed, 19 insertions, 8 deletions
diff --git a/vorlesungen/slides/2/hilbertraum/rieszbeispiel.tex b/vorlesungen/slides/2/hilbertraum/rieszbeispiel.tex index e2c26f5..de9383f 100644 --- a/vorlesungen/slides/2/hilbertraum/rieszbeispiel.tex +++ b/vorlesungen/slides/2/hilbertraum/rieszbeispiel.tex @@ -19,30 +19,33 @@ f({\color{blue}x}) &= \begin{pmatrix}f_1&f_2&\dots&f_n\end{pmatrix} {\color{blue}x} \\ +\uncover<2->{ {\color{red}v}&= \rlap{$ \begin{pmatrix} \overline{f}_1&\overline{f}_2&\dots&\overline{f}_n \end{pmatrix}^t -\;\Rightarrow\; -f({\color{blue}x})=\langle {\color{red}v},{\color{blue}x}\rangle -$} +\uncover<3->{\;\Rightarrow\; +f({\color{blue}x})=\langle {\color{red}v},{\color{blue}x}\rangle} +$}} \end{align*} \end{block} \end{column} \begin{column}{0.48\textwidth} +\uncover<4->{% \begin{block}{Linearform auf $L^2([a,b])$} \begin{align*} {\color{red}x}&\in L^2([a,b]) \\ +\uncover<5->{ f&\colon L^2([a,b]) \to \mathbb{C} -: {\color{red}x} \mapsto f({\color{red}x}) -\intertext{Riesz-Darstellungssatz: $\exists {\color{blue}v}\in L^2([a,b])$} -f({\color{red}x}) +: {\color{red}x} \mapsto f({\color{red}x})} +\intertext{\uncover<6->{Riesz-Darstellungssatz: $\exists {\color{blue}v}\in L^2([a,b])$}} +\uncover<7->{f({\color{red}x}) &= -\int_a^b {\color{blue}\overline{v}(t)}{\color{red}x(t)}\,dt +\int_a^b {\color{blue}\overline{v}(t)}{\color{red}x(t)}\,dt} \end{align*} -\end{block} +\end{block}} \end{column} \end{columns} \begin{center} @@ -50,10 +53,12 @@ f({\color{red}x}) \begin{scope}[xshift=-3.5cm] \def\s{0.058} \foreach \n in {0,...,5}{ +\uncover<3->{ \draw[color=red,line width=3pt] ({\n+\s},{1/(\n+0.5)}) -- ({\n+\s},0); \node[color=red] at ({\n},{-0.2+1/(\n+0.5)}) [above right] {$v_\n\mathstrut$}; +} \draw[color=blue,line width=3pt] ({\n-\s},{0.4+0.55*sin(200*\n)+0.25*\n}) -- ({\n-\s},0); \node[color=blue] at ({\n},{-0.2+0.4+0.55*sin(200*\n)+0.25*\n}) @@ -67,17 +72,22 @@ f({\color{red}x}) } \node at (5.6,0) [below] {$\cdots$\strut}; \end{scope} +\uncover<4->{ \begin{scope}[xshift=3.5cm] +\uncover<7->{ \fill[color=red!40,opacity=0.5] plot[domain=0:5,samples=100] (\x,{1/(\x+0.5)}) -- (5,0) -- (0,0) -- cycle; +} \fill[color=blue!40,opacity=0.5] plot[domain=0:5,samples=100] (\x,{0.4+0.55*sin(200*\x)+0.25*\x}) -- (5,0) -- (0,0) -- cycle; +\uncover<7->{ \draw[color=red,line width=1.4pt] plot[domain=0:5,samples=100] (\x,{1/(\x+0.5)}); \node[color=red] at (0,2) [right] {$x(t)$}; +} \draw[color=blue,line width=1.4pt] plot[domain=0:5,samples=100] (\x,{0.4+0.55*sin(200*\x)+0.25*\x}); @@ -90,6 +100,7 @@ f({\color{red}x}) \draw (5.0,-0.1) -- (5.0,0.1); \node at (5.0,0) [below] {$b$\strut}; \end{scope} +} \end{tikzpicture} \end{center} \end{frame} |