Commit 0b197657 authored by Michael Kohlhase's avatar Michael Kohlhase

superseded by pnorm.tex

parent a154a1f8
\begin{mhmodnl}[creators=miko]{pnorm}{en}
\begin{definition}
Let $\cV$ be a \trefi[realnumbers]{real} \trefii[vector-space]{vector}{space} and
$\vbaseset$ its \trefii[vector-space]{base}{set}}, then we call
The \defii{Euclidean}{norm} on an \trefii[euclideanspace]{Euclidean}{space}
$\ndim{n}\RealNumbers$ is the function
\[\fundefeq{v}{\euclideannorm{\ntupli{v}1n}}{\sqrt{\Sumfromto{i}1n{\realpower{v_i}2}}}\]
The \trefii{Euclidean}{norm} is also called the \defii[name=Euclidean-norm]{Euclidean}{length},
\defii[name=Euclidean-norm]{$L^2$}{distance}, \defii[name=Euclidean-norm]{$L^2$}{norm},
\defii[name=Euclidean-norm]{$\ell^2$}{distance}, or \defii[name=Euclidean-norm]{$\ell^2$}{norm},
\end{definition}
\end{mhmodnl}
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\begin{modsig}[creators=miko]{euclidean-norm}
\gimport[smglom/topology]{euclideanspace}
\gimport[smglom/arithmetics]{realarith}
\gimport[smglom/arithmetics]{sum}
\symdef[name=Euclidean-norm]{euclideannorm}[1]{\left|#1\right|}
\symtest{euclideannorm}{\euclideannorm{v}}
\end{modsig}
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