68 lines
3.4 KiB
TeX
68 lines
3.4 KiB
TeX
\section{Barreled Spaces}
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\label{section:barrel}
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\begin{definition}[Barrel]
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\label{definition:barrel}
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Let $E$ be a TVS over $K \in \RC$ and $D \subset E$, then $D$ is a \textbf{barrel} if it is convex, circled, radial, and closed.
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\end{definition}
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\begin{definition}[Barreled Space]
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\label{definition:barreled-space}
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Let $E$ be a locally convex space over $K \in \RC$, then the following are equivalent:
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\begin{enumerate}
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\item The barrels of $E$ forms a fundamental system of neighbourhoods at $0$.
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\item Every barrel in $E$ is a neighbourhood of $0$.
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\item Every lower semicontinuous seminorm on $E$ is continuous.
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\end{enumerate}
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\end{definition}
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\begin{proof}
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$(2) \Rightarrow (1)$: Let $\fB \subset \cn_E(0)$ be a fundamental system of neighbourhoods at $0$ consisting of convex, circled, and radial sets, then $\ol{\fB} = \bracsn{\ol U|U \in \fB}$ is a fundamental system of neighbourhoods at $0$ consisting of barrels.
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$(2) \Rightarrow (3)$: Let $\rho: E \to [0, \infty)$ be a lower semicontinuous seminorm, then $\bracs{\rho > 1}$ is open and $\bracs{\rho \le 1}$ is a Barrel. In which case, $\rho$ is continuous by (4) of \autoref{lemma:continuous-seminorm}.
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$(3) \Rightarrow (2)$: Let $D \subset E$ be a barrel and $\rho: E \to [0, \infty)$ be its gauge. By (4) of \autoref{definition:gauge}, $D = \bracs{\rho \le 1}$, so $\bracs{\rho > 1}$ is open, and $\rho$ is semicontinuous. By assumption, $\rho$ is continuous, so $D \in \cn_E(0)$ by (5) of \autoref{lemma:continuous-seminorm}.
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\end{proof}
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\begin{summary}
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\label{summary:barreled-space}
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The following types of locally convex spaces are barreled:
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\begin{enumerate}
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\item Every locally convex space with the Baire property.
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\item Every Banach space and every Fréchet space.
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\item Inductive limits of barreled spaces.
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\item Spaces of type (LB) and (LF).
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\item The locally convex direct sum of barreled spaces.
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\item Products of barreled spaces.
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\end{enumerate}
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\end{summary}
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\begin{proof}
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(1), (2): \autoref{proposition:baire-barrel}.
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(3), (4), (5): \autoref{proposition:barrel-limit}.
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(6): TODO.
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\end{proof}
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\begin{proposition}
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\label{proposition:baire-barrel}
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Let $E$ be a locally convex space over $K \in \RC$. If $E$ is a Baire space, then $E$ is barreled.
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\end{proposition}
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\begin{proof}[Proof, {{\cite[II.7.1]{SchaeferWolff}}}. ]
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Let $D \subset E$ be a Barrel, then $E = \bigcup_{n \in \natp}nD$ is a countable union of closed sets. Since $E$ is Baire, there exists $n \in \natp$, $U \in \cn_E(0)$ circled, and $x \in E$ such that $x + U \in nB$. In which case,
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\[
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U \subset (x + U) - (x + U) \subset nB - nB = 2nB
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\]
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so $2nB$ and thus $B$ is a neighbourhood of $0$.
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\end{proof}
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\begin{proposition}
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\label{proposition:barrel-limit}
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Let $\seqi{E}$ be locally convex spaces over $K \in \RC$, $E$ be a vector space over $K$, and $\seqi{T}$ such that $T_i \in \hom(E_i; E)$ for all $i \in I$, then the inductive locally convex topology on $E$ induced by $\seqi{T}$ is barreled.
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\end{proposition}
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\begin{proof}[Proof, {{\cite[II.7.2]{SchaeferWolff}}}. ]
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Let $D \subset E$ be a barrel, then for each $i \in I$, $T_i^{-1}(D) \subset E_i$ is also a barrel, and thus a neighbourhood of $0$ in $E_i$. By (5) of \autoref{definition:lc-inductive}, $D$ is a neighbourhood of $0$ in $E$, so $E$ is barreled.
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\end{proof}
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