Commit 59555b95 authored by Michael Kohlhase's avatar Michael Kohlhase

two more modules

parent 0b197657
\begin{mhmodnl}[creators=miko]{maxnorm}{de}
\begin{definition}\guse{pnorm}\guse[smglom/calculus]{functionlimit}\guse[smglom/arithmetics]{infinity}
Ist $\inset{\ntupli{x}1n}{\ndim\RealNumbers{n}}$, so nennen wir
$\fundefeq{x}{\maxnorm{x}}{\maxover{i}{\realabsval{x_i}}}$ die
\defi[name=maximum-norm]{Maximumsnorm} (oder \defi[name=maximum-norm]{Tschebyschew-Norm}).
Es gilt $\limfun{p}{\infinity}{\pnorm{p}x}=\maxnorm{x}$, daher nennen wir die
\mtrefii[?maximum-norm]{Maximumsnorm} auch \defi[name=maximum-norm]{$\ell_\infty$-Norm} oder
\defi[name=maximum-norm]{$L^\infty$-Norm}.
\end{definition}
\end{mhmodnl}
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\begin{mhmodnl}[creators=miko]{maxnorm}{en}
\begin{definition}\guse{pnorm}\guse[smglom/calculus]{functionlimit}\guse[smglom/arithmetics]{infinity}
Let $\inset{\ntupli{x}1n}{\ndim\RealNumbers{n}}$, then we call
$\fundefeq{x}{\maxnorm{x}}{\maxover{i}{\realabsval{x_i}}}$ its \defii{maximum}{norm}
(or \defii[name=maximum-norm]{uniform}{norm}). As
$\limfun{p}{\infinity}{\pnorm{p}x}=\maxnorm{x}$, we also call the
\trefii{maximum}{norm} the \defi[name=maximum-norm]{$\ell_\infty$-norm} or the
\defi[name=maximum-norm]{$L^\infty$-norm}.
\end{definition}
\end{mhmodnl}
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\begin{modsig}[creators=miko]{maxnorm}
\guse{pnorm}
\gimport[smglom/arithmetics]{minmax}
\symdef{maxnorm}[1]{\left\|#1\right\|_\infty}
\symtest{maxnorm}{\maxnorm{x}}
\end{modsig}
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\begin{mhmodnl}[creators=miko]{pnorm}{de}
\begin{definition}
Ist $\realmorethan{p}1$ eine \mtrefii[realnumbers?real-number]{reelle}{Zahl} und
$\inset{\ntupli{x}1n}{\ndim\RealNumbers{n}}$, so nennen wir
\[\fundefeq{x}{\pnorm{p}x}{\realpower[basebrack]{\Sumfromto{i}1n{\realabsval{x_i}}}{\frac1p}}\]
seine \defi[name=pnorm]{$p$-Norm} (auch \defi[name=pnorm]{$\ell_p$-Norm} oder \defi[name=pnorm]{$L^p$-Norm}).
\end{definition}
\begin{definition}
Wir nennen die \mtrefi[?pnorm]{1-Norm} auch \defi[name=taxicab-norm]{Summennorm}
oder \defi[name=taxicab-norm]{Betragssummennorm}, und die \mtrefi[?pnorm]{2-Norm} the
\defii{Euklidsche}{Norm} oder \defii[name=Euclidean-norm]{Euklidsche}{L"ange}.
\end{definition}
\end{mhmodnl}
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\begin{mhmodnl}[creators=miko]{pnorm}{en}
\begin{definition}
Let $\realmorethan{p}1$ be a \trefii[realnumbers]{real}{number} and
$\inset{\ntupli{x}1n}{\ndim\RealNumbers{n}}$, then we call
\[\fundefeq{x}{\pnorm{p}x}{\realpower[basebrack]{\Sumfromto{i}1n{\realabsval{x_i}}}{\frac1p}}\]
its \defi[name=pnorm]{$p$-norm} (also \defi[name=pnorm]{$\ell_p$-norm} or \defi[name=pnorm]{$L^p$-norm}).
\end{definition}
\begin{definition}
The \mtrefi[?pnorm]{1-norm} is called the \defii{taxicab}{norm}, and the
\mtrefi[?pnorm]{2-norm} the \defii{Euclidean}{norm} or the
\defii[name=Euclidean-norm]{Euclidean}{length}.
\end{definition}
\end{mhmodnl}
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\begin{modsig}[creators=miko]{pnorm}
\gimport[smglom/arithmetics]{realarith}
\gimport[smglom/linear-algebra]{vector-space}
\gimport[smglom/sets]{cartesian-space}
\gimport[smglom/arithmetics]{sum}
\symdef{pnorm}[2]{\left\|#2\right\|_{#1}}
\symtest{pnorm}{\pnorm{p}x}
\end{modsig}
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