Commit 0465a504 authored by Michael Kohlhase's avatar Michael Kohlhase

power

parent 2bfe123f
Pipeline #305 skipped
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{metric-space}{norm-induced-metric}
Gegeben einen \mtrefii[norm?normed-vector-space]{normierten}{Vektorraum} mit Grundmenge
$V$ und \mtrefii[norm-induced-metric?induced-metric]{induzierter}{Metrik} $d$, dann ist
$\mvstructure{V,d}$ \mtrefii[metric-space?metric-space]{metrischer}{Raum}.
$V$ und \mtrefii[norm-induced-metric?induced-metric]{induzierter}{Metrik} $\ametricOp$,
dann ist $\mvstructure{V,\ametricOp}$
\mtrefii[metric-space?metric-space]{metrischer}{Raum}.
\end{gviewnl}
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\begin{gviewnl}[creators=miko,srccite=Rudin:fa73,fromrepos=smglom/calculus]{norm-metric}{en}
{metric-space}{norm-induced-metric}
Given a \trefiii[norm]{normed}{vector}{space} with base set $V$ and
\trefii[norm-induced-metric]{induced}{metric} $d$, then $\mvstructure{V,d}$ is a
\trefii[metric-space]{metric}{space}.
{metric-space}{norm-induced-metric} Given a \trefiii[norm]{normed}{vector}{space} with
base set $V$ and \trefii[norm-induced-metric]{induced}{metric} $\ametricOp$, then
$\mvstructure{V,\ametricOp}$ is a \trefii[metric-space]{metric}{space}.
\end{gviewnl}
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\mtrefi[vector-space?vector-space]{Vektorraums}
\mtrefi[vector-space?vector-addition]{Vektoraddition} $\vaddOp$
\defi[invariant]{translationsinvariant}, wenn
$\nappa{d}{v,w}=\nappa{d}{\vadd{a,v},\vadd{a,w}}$ f"ur alle $\minset{a,v,w}V$.
$\ametric{v}w=\ametric{\vadd{a,v}}{\vadd{a,w}}$ f"ur alle $\minset{a,v,w}V$.
\end{definition}
\end{mhmodnl}
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\vardef{vaddOp}{+}
\vardef[assocarg=1]{vadd}[1]{\assoc[p=500]\vaddOp{#1}}
\begin{definition}
We call a \mtrefi[metric-space?distance-function]{metric} $d$ on the
We call a \mtrefi[metric-space?distance-function]{metric} $\ametricOp$ on the
\trefii[vector-space]{base}{set} $V$ of a \trefii[vector-space]{vector}{space} with
\trefii[vector-space]{vector}{addition} $\vaddOp$ \defii{translation}{invariant}, iff
$\nappa{d}{v,w}=\nappa{d}{\vadd{a,v},\vadd{a,w}}$ for all $\minset{a,v,w}V$.
$\ametric{v}w=\ametric{\vadd{a,v}}{\vadd{a,w}}$ for all $\minset{a,v,w}V$.
\end{definition}
\end{mhmodnl}
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