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Fix typos

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Anton Mosich 2022-10-05 18:46:19 +02:00
parent 5c3d0f60a2
commit a20b7bff63
Signed by: Flugschwein
GPG Key ID: 9303E1C32E3A14A0
1 changed files with 11 additions and 11 deletions
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LinAlg2.tex
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@ -2617,15 +2617,15 @@ Behauptung: $\inner fp > 0$
\item $\inner{\alpha(v)}{w} = \inner{v}{\alpha^*(w)} = \overline{\inner{\alpha^*(w)}{v}} =$ \\
$\overline{\inner{w}{(\alpha^*)^{{}^*}(v)}} = \inner{(\alpha^*)^{{}^*}(v)}{w} \; \forall v \in V, w \in W$ \\
$\implies \inner{\alpha(v) - (\alpha^*)^{{}^*}(v)}{w} = 0 \; \forall v \in V, w \in W, \;
w:= \alpha(v) - (\alpha^*)^{{}^*}$ \\
$\implies \inner{\alpha(v) - (\alpha^*)^{{}^*}}{\alpha(v) - (\alpha^*)^{{}^*}} = 0 \iff
\norm{\alpha(v) - (\alpha^*)^{{}^*}} = 0 \implies \forall v \in V: \alpha(v) = (\alpha^*)^{{}^*}$
w:= \alpha(v) - (\alpha^*)^{{}^*}(v)$ \\
$\implies \inner{\alpha(v) - (\alpha^*)^{{}^*}(v)}{\alpha(v) - (\alpha^*)^{{}^*}(v)} = 0 \iff
\norm{\alpha(v) - (\alpha^*)^{{}^*}(v)} = 0 \implies \forall v \in V: \alpha(v) = (\alpha^*)^{{}^*}(v)$
\item
\begin{align*}
\inner{(\alpha + \beta)}{w} & = \inner{v}{(\alpha + \beta)^*(w)} = \inner{\alpha(v) + \beta(v)}{w} \\
& = \inner{\alpha(v)}{w} + \inner{\beta(v)}{w} = \inner{v}{\alpha^*(w)} +
\inner{v}{\beta^*(w)} \\
& = \inner{v}{\alpha^*(w) + \beta^*(w)}
\inner{(\alpha + \beta)(v)}{w} & = \inner{v}{(\alpha + \beta)^*(w)} = \inner{\alpha(v) + \beta(v)}{w} \\
& = \inner{\alpha(v)}{w} + \inner{\beta(v)}{w} = \inner{v}{\alpha^*(w)} +
\inner{v}{\beta^*(w)} \\
& = \inner{v}{\alpha^*(w) + \beta^*(w)}
\end{align*}
\item $ \inner{(\lambda \alpha)(v)}{w} = \inner{v}{(\lambda \alpha)^*(w)} $\\
$ = \lambda \inner{\alpha(v)}{w} = \lambda \inner{v}{\alpha^*(w)} = \inner{v}{\overline{\lambda} \alpha^*(w)}$
@ -3158,10 +3158,10 @@ Das sind genau die Längen- und Winkelerhaltenden Abbildungen.
\item[$\implies$:] Seien $v,w \in V$.
$\alpha^{-1}$ existiert wegen Korollar~\ref{theo:3.3.3}b)
\begin{align*}
\inner{v}{\alpha^*(w) - \alpha^{-1}} & = \inner{v}{\alpha^*(w)} - \inner{v}{\alpha^{-1}(w)} \\
& = \inner{\alpha(v)}{w} - \inner{v}{\alpha^{-1}(w)} \\
& = \inner{\alpha(v)}{w} - \inner{\alpha(v)}{\alpha(\alpha^{-1}(w))} \\
& = \inner{\alpha(v)}{w} - \inner{\alpha(v)}{w} = 0
\inner{v}{\alpha^*(w) - \alpha^{-1}(w)} & = \inner{v}{\alpha^*(w)} - \inner{v}{\alpha^{-1}(w)} \\
& = \inner{\alpha(v)}{w} - \inner{v}{\alpha^{-1}(w)} \\
& = \inner{\alpha(v)}{w} - \inner{\alpha(v)}{\alpha(\alpha^{-1}(w))} \\
& = \inner{\alpha(v)}{w} - \inner{\alpha(v)}{w} = 0
\end{align*}
\item[$\impliedby$:] Sei $\alpha^* = \alpha^{-1}, u,v,w\in V, v = \alpha(w)$
\[