Commit 0995e666 authored by Michael Kohlhase's avatar 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}
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