Commit 2e4460b0 authored by Michael Kohlhase's avatar Michael Kohlhase

more_preloading

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:
......
Markdown is supported
0% or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment