Commit 2e4460b0 by Michael Kohlhase

parent 6f29dcda
 ... ... @@ -4,12 +4,12 @@ \trefiis[complexnumbers]{complex}{number}, $V$ a \trefii[vector-space]{vector}{space} over $F$, and $\fun\innerproductOp{V,V}F$ a function with \begin{enumerate} \item $\innerproduct{x}y=\compconjugate{\innerproduct{y}x}$ \item $\eq{\innerproduct{x}y,\compconjugate{\innerproduct{y}x}}$ (\defii{conjugate}{symmetry}) \item $\innerproduct{\smul{a}x}y=\comptimes{a{\innerproduct{x}y}}$ and $\innerproduct{\vadd{x,y}}z=\vadd{\innerproduct{x}z,\innerproduct{y}z}$ \item $\eq{\innerproduct{\smul{a}x}y,\comptimes{a{\innerproduct{x}y}}}$ and $\eq{\innerproduct{\vadd{x,y}}z,\vadd{\innerproduct{x}z,\innerproduct{y}z}}$ (\mtrefi[linear-map?linear]{linearity} in the first argument) \item $\realmethan{\innerproduct{x}x}0$ and $\eq{\innerproduct{x}x,0}$ iff $x=0$, \item $\realmethan{\innerproduct{x}x}0$ and $\eq{\innerproduct{x}x,0}$ iff $\eq{x,0}$, (\defii{positive}{definiteness}), \end{enumerate} then $\mvstructure{V,\innerproductOp}$ is called an \defiii{inner}{product}{space} ... ...
 ... ... @@ -9,11 +9,11 @@ \begin{sproof}[for=obl.norm-metric.de]{Wir zeigen die drei Eigenschaften einer \mtrefi[metric-space?distance-function]{Abstrandsfunktion}:} \begin{spfstep} $\ametric{x}y=0$, gdw. $\anorm{x-y}=0$, gdw. $x-y=0$ ($\anormOp$ ist \mtrefi[norm?separtes-points]{definit}), iff $x=y$. $\eq{\ametric{x}y,0}$, gdw. $\eq{\anorm{x-y},0}$, gdw. $\eq{x-y,0}$ ($\anormOp$ ist \mtrefi[norm?separtes-points]{definit}), iff $\eq{x,y}$. \end{spfstep} \begin{spfstep} $\ametric{x}y=\anorm{x-y}=\anorm{y-x}=\ametric{y}x$ wegen der $\eq{\ametric{x}y,\anorm{x-y},\anorm{y-x},\ametric{y}x}$ wegen der \mtrefii[norm?absolute-homogeneity]{absouten}{Homogenit"at} von $\anormOp$. \end{spfstep} \begin{spfstep} ... ...
 ... ... @@ -8,15 +8,16 @@ \end{definition} \begin{sproof}[for=obl.norm-metric.en]{we prove the three conditions for a distance function:} \begin{spfstep} $\ametric{x}y=0$, iff $\anorm{x-y}=0$, iff $x-y=0$ (as $\anormOp$ \trefii[norm]{separates}{points}), iff $x=y$. $\eq{\ametric{x}y,0}$, iff $\eq{\anorm{x-y},0}$, iff $\eq{x-y,0}$ (as $\anormOp$ \trefii[norm]{separates}{points}), iff $\eq{x,y}$. \end{spfstep} \begin{spfstep} $\ametric{x}y=\anorm{x-y}=\anorm{y-x}=\ametric{y}x$ by $\eq{\ametric{x}y,\anorm{x-y},\anorm{y-x},\ametric{y}x}$ by \trefi[norm]{absolute}{homogeneity} of $\anormOp$. \end{spfstep} \begin{spfstep} the \trefi[metric-space]{triangle}{inequality} for $d$ follows from \mtrefi[norm?triangle-equality]{that for $\anormOp$}. the \trefi[metric-space]{triangle}{inequality} for $d$ follows from \mtrefi[norm?triangle-equality]{that for $\anormOp$}. \end{spfstep} \end{sproof} \end{mhmodnl} ... ...
 ... ... @@ -7,11 +7,11 @@ $\fun\anormOp\vbaseset\RealNumbers$ eine \defi[name=norm]{Norm} auf $\cV$, wenn f"ur alle $\inset{a}F$ und $\minset{u,v}\vbaseset$ gilt: \begin{enumerate} \item $\anorm{\smul{a}v}=\realtimes{\realabsval{a},\anorm{v}}$ \item $\eq{\anorm{\smul{a}v},\realtimes{\realabsval{a},\anorm{v}}}$ (\defii[name=absolute-homogeneity]{absolute}{Homogenit"at}). \item $\reallethan{\anorm{\vadd{u,v}}}{\realplus{\anorm{u},\anorm{v}}}$ (\defi[name=triangle-inequality]{Dreiecksungleichung}). \item If $\anorm{v}=0$, then $v$ is the zero vector \item If $\eq{\anorm{v},0}$, then $v$ is the zero vector (\defi[name=separates-points]{Definitheit}). \end{enumerate} Wir nennen das Paar $\mvstructure{\cV,\anormOp}$ einen ... ...
 ... ... @@ -7,13 +7,13 @@ $\inset{a}F$ and $\minset{u,v}\vbaseset$ \begin{enumerate} \item \assdef[absolute-homogeneity]{$\anorm{\smul{a}v}=\realtimes{\realabsval{a},\anorm{v}}$} \assdef[absolute-homogeneity]{$\eq{\anorm{\smul{a}v},\realtimes{\realabsval{a},\anorm{v}}}$} (\defii{absolute}{homogeneity} or \defii[name=absolute-homogeneity]{absolute}{scalability}). \item \assdef[triangle-equality]{$\reallethan{\anorm{\vadd{u,v}}}{\realplus{\anorm{u},\anorm{v}}}$} (\defii{triangle}{inequality} or \defi[name=triangle-inequality]{subadditivity}). \item \assdef[separates-points]{If $\anorm{v}=0$, then $v$ is the zero vector} \item \assdef[separates-points]{If $\eq{\anorm{v},0}$, then $v$ is the zero vector} ($\anormOp$ \defii{separates}{points}). \end{enumerate} We call the pair $\mvstructure{\cV,\anormOp}$ a \defiii{normed}{vector}{space} with ... ...
 ... ... @@ -2,8 +2,9 @@ \begin{definition} Let $\realmorethan{p}1$ be a \trefii[realnumbers]{real}{number} and $\inset{\ntupli{x}1n}{\ndim\RealNumbers{n}}$, then we call $\fundefeq{p,x}{\pnorm{p}x}{\realpower[basebrack]{\Sumfromto{i}1n{\realabsval{x_i}}}{\frac1p}}$ its \defi[name=pnorm]{$p$-norm} (also \defi[name=pnorm]{$\ell_p$-norm} or \defi[name=pnorm]{$L^p$-norm}). $\fundefeq{p,x}{\pnorm{p}x}{\realpower[basebrack]{\Sumfromto{i}1n{\realabsval{\tupsel{x}i}}}{\ratdivide[frac]1p}}$ its \defi[name=pnorm]{$p$-norm} (also \defi[name=pnorm]{$\ell_p$-norm} or \defi[name=pnorm]{$L^p$-norm}). \end{definition} \begin{definition} ... ...
 \begin{mhmodnl}[creators=miko]{translation-invariant-metric}{de} \begin{definition} \vardef{vaddOp}{+} \vardef[assocarg=1]{vadd}[1]{\assoc[p=500]\vaddOp{#1}} Wir nennen eine \mtrefi[metric-space?distance-function]{Metrik} $d$ auf der \vardef{vaddOp}{+} \vardef[assocarg=1]{vadd}[1]{\assoc[p=500]\vaddOp{#1}} Wir nennen eine \mtrefi[metric-space?distance-function]{Metrik} $d$ auf der \mtrefi[vector-space?base-set]{Grundmenge} $V$ eines \mtrefi[vector-space?vector-space]{Vektorraums} \mtrefi[vector-space?vector-addition]{Vektoraddition} $\vaddOp$ \defi[name=translation-invariant]{translationsinvariant}, wenn $\ametric{v}w=\ametric{\vadd{a,v}}{\vadd{a,w}}$ f"ur alle $\minset{a,v,w}V$. $\eq{\ametric{v}w,\ametric{\vadd{a,v}}{\vadd{a,w}}}$ f"ur alle $\minset{a,v,w}V$. \end{definition} \end{mhmodnl} %%% Local Variables: ... ...
 ... ... @@ -5,7 +5,7 @@ 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 $\ametric{v}w=\ametric{\vadd{a,v}}{\vadd{a,w}}$ for all $\minset{a,v,w}V$. $\eq{\ametric{v}w,\ametric{\vadd{a,v}}{\vadd{a,w}}}$ for all $\minset{a,v,w}V$. \end{definition} \end{mhmodnl} %%% Local Variables: ... ...
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