Commit 59555b95 by 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} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
 \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} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
 \begin{modsig}[creators=miko]{maxnorm} \guse{pnorm} \gimport[smglom/arithmetics]{minmax} \symdef{maxnorm}[1]{\left\|#1\right\|_\infty} \symtest{maxnorm}{\maxnorm{x}} \end{modsig} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
 \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} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
 \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} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
source/pnorm.tex 0 → 100644
 \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} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
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