Commit 0995e666 by Michael Kohlhase

### tweaks

parent e26e89a7
 ... ... @@ -2,7 +2,7 @@ \begin{definition} We say that a \trefiii [topological-vectorspace]{topological}{vector}{space} has the \defii{Heine-Borel}{property}, iff every \trefi[open-set-topology]{closed} and \trefi[bounded-topvr]{bounded} subset of its \trefii[topological-vectorspace]{base}{set} \trefi[bounded-topvr]{bounded} \trefi[subsupset]{subset} of its \trefii[topological-vectorspace]{base}{set} is \trefi[compact]{compact}. \end{definition} \end{mhmodnl} ... ...
 ... ... @@ -7,7 +7,7 @@ \mtrefii[topological-vectorspace?topological-vector-space]{topologischer}{Vektorraum} mit \mtrefi[vector-space?base-set]{Grundmenge} $V$ und \mtrefi[vector-space?zero-vector]{Nullvektor} $\vzero$ "uber einem \mtrefi[partial-order?ordered]{geordnetem} \mtrefi[field?field]{K"orper} \mtrefi[ordered-ring?ordered-ring]{geordnetem} \mtrefi[field?field]{K"orper} $\mvstructure{\cF,\poleOp}$ mit \mtrefi[field?base-set]{Grundmenge} $F$. Dann nennen wir $\sseteq{E}V$ \defi[name=bounded]{beschr"ankt}, falls es f"ur jede \mtrefi[neighborhood?neighborhood]{Umgebung} $N$ von $\vzero$ in $V$, ein ... ...
 ... ... @@ -6,8 +6,8 @@ Let $\mvstructure{\cV,\cO}$ be a \trefiii[topological-vectorspace]{topological}{vector}{space} with \trefii[vector-space]{base}{set} $V$ and \trefii[vector-space]{zero}{vector} $\vzero$ over an \trefi[partial-order]{ordered} \trefi[field]{field} $\mvstructure{\cF,\poleOp}$ with base set $F$. The we call a set $\sseteq{E}V$ \defi{bounded}, if for every over an \mtrefi[ordered-ring?ordered-ring]{ordered} \trefi[field]{field} $\mvstructure{\cF,\poleOp}$ with \trefii[magma]{base}{set} $F$. The we call a set $\sseteq{E}V$ \defi{bounded}, if for every \trefi[neighborhood]{neighborhood} $N$ of $\vzero$ in $V$, there is an $\pole{a}\vzero$ in $F$, such that $\sseteq{E}{\smul{a}V}$. \end{definition} ... ...
 \begin{modsig}[creators=miko,srccite=Rudin:fa73]{bounded-topvr} \gimport{topological-vectorspace} \gimport[smglom/algebra]{ordered-ring} \symi{bounded} \end{modsig} ... ...
 \begin{mhmodnl}[creators=miko]{frechet-space}{de} \begin{definition} Wir nennen einen \mtrefi[fspace?F-space]{F-Raum} Wir nennen einen \mtrefi[fspace?fspace]{F-Raum} \defii[name=Frechet-space]{Fre\'echet}{Raum}, falls er \mtrefii[locally-convex?locally-convex]{lokal}{konvex} ist. \end{definition} ... ...
 \begin{mhmodnl}[creators=miko]{frechet-space}{en} \begin{definition} We call an \trefi[fspace]{F-space} a \defii[name=Frechet-space]{Fre\'echet}{space}, iff it is We call an \mtrefi[fspace?fspace]{F-space} a \defii[name=Frechet-space]{Fre\'echet}{space}, iff it is \trefii[locally-convex]{locally}{convex}. \end{definition} \end{mhmodnl} ... ...
 ... ... @@ -3,7 +3,7 @@ Ein \mtrefii[topological-vectorspace?topological-vector-space]{topologischer}{Vektorraum} $\mvstructure{X,\cO}$ hei"st \defii[name=locally-bounded]{lokal}{beschr"ankt}, wenn der \mtrefi[vector-space?zero]{Nullvektor} eine \mtrefi[bounded-topvr?bounded]{beschr"ankte} \mtrefi[vector-space?zero-vector]{Nullvektor} eine \mtrefi[bounded-topvr?bounded]{beschr"ankte} \mtrefi[neighborhood?neighborhood]{Umgebung} hat. \end{definition} \end{mhmodnl} ... ...
 ... ... @@ -18,7 +18,7 @@ \end{spfstep} \begin{spfstep} die \mtrefi[metric-space?triangle-inequality]{Dreiecksungleichung} for $d$ folgt aus der f"ur $\anormOp$. \mtrefi[norm?triangle-equality]{der f"ur $\anormOp$}. \end{spfstep} \end{sproof} \end{mhmodnl}
 ... ... @@ -8,14 +8,15 @@ \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 $\anormOp$ separates points), iff $x=y$. $\ametric{x}y=0$, iff $\anorm{x-y}=0$, iff $x-y=0$ (as $\anormOp$ \trefii[norm]{separates}{points}), iff $x=y$. \end{spfstep} \begin{spfstep} $\ametric{x}y=\anorm{x-y}=\anorm{y-x}=\ametric{y}x$ by absolute homogeneity of $\anormOp$. $\ametric{x}y=\anorm{x-y}=\anorm{y-x}=\ametric{y}x$ by \trefi[norm]{absolute}{homogeneity} of $\anormOp$. \end{spfstep} \begin{spfstep} the triangle inequality for $d$ follows from 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} ... ...
 ... ... @@ -3,8 +3,7 @@ Wir nennen einen \mtrefii[topological-vectorspace?topological-vector-space]{topologischen}{Vektorraum} $\cV$ \defi[name=normable]{normierbar}, wenn es eine \mtrefi[norm?norm]{Norm} gibt, so dass die \mtrefi[norm-induced-metric?induced-metric]{induzierte} \mtrefi[metric-space?distance-function]{Abstandsfunktion} auf der die \mtrefii[norm-induced-metric?induced-metric]{induzierte}{Abstandsfunktion} auf der \mtrefi[topological-vectorspace?base-set]{Grundmenge} von $\cV$ die Topologie von $\cV$ \mtrefi[metric-induced-topology?induced-topology]{induziert}. \end{definition} ... ...
 ... ... @@ -2,7 +2,7 @@ \begin{definition} We call a \trefiii[topological-vectorspace]{topological}{vector}{space} $\cV$ \defi{normable}, iff there is a \trefi[norm]{norm}, such that the \trefi[norm-induced-metric]{induced} \trefii[metric-space]{distance}{function} on the \trefii[norm-induced-metric]{induced}{metric} on the \trefii[topological-vectorspace]{base}{set} of $\cV$ \mtrefi[metric-induced-topology?induced-topology]{induces} the topology of $\cV$. \end{definition} ... ...
 \begin{modsig}[creators=miko]{translation-invariant-metric} \gimport[smglom/calculus]{metric-space} \gimport[smglom/linear-algebra]{transscale} \gimport[smglom/calculus]{metric-space} \symii{translation}{invariant} \end{modsig} %%% Local Variables: ... ...
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