92 lines
4.3 KiB
TeX
92 lines
4.3 KiB
TeX
\section{Projective Limits}
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\label{section:tvs-projective-limit}
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\begin{definition}[Projective Uniformity]
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\label{definition:tvs-initial}
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Let $E$ be a vector space over $K \in \RC$, $\seqi{F}$ be TVSs over $K$, and $\seqi{T}$ where $T_i \in \hom(E; F_i)$ for all $i \in I$, then there exists a uniformity $\fU$ on $E$ such that:
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\begin{enumerate}
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\item For each $i \in I$, $T_i \in L(E; F_i)$.
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\item[(U)] If $\mathfrak{V}$ is a uniformity on $E$ satisfying $(1)$, then $\mathfrak{V} \supset \fU$.
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\end{enumerate}
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Moreover,
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\begin{enumerate}
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\item[(3)] $\fU$ is translation-invariant.
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\item[(4)] $E$ equipped with the topology induced by $\fU$ is a topological vector space.
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\item[(5)] For any TVS $F$ over $K$ and linear map $T \in \hom(F; E)$, $T \in L(F; E)$ if and only if $T_i \circ T \in L(F; F_i)$ for all $i \in I$.
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\item[(6)] The collection
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\[
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\bracs{\bigcap_{j \in J}T_j^{-1}(U_j) \bigg | J \subset I \text{ finite}, U_j \in \cn_{F_j}(0)}
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\]
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is a fundamental system of neighbourhoods for $E$ at $0$.
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\end{enumerate}
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The uniformity $\fU$ and its topology are the \textbf{projective uniformity/topology} induced by $\seqi{T}$.
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\end{definition}
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\begin{proof}
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(1), (U): By \autoref{definition:initial-uniformity}.
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Let $U \in \fU$, then there exists $J \subset I$ finite and translation-invariant entourages $\seqj{U}$ such that
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\[
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U \subset V = \bigcap_{j \in J}(T_j \times T_j)^{-1}(U_j)
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\]
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(3): For each $j \in J$, $(x, y) \in (T_j \times T_j)^{-1}(U_j)$, and $z \in E$,
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\[
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(T_j \times T_j)(x + z, y + z) = (T_jx + T_jz, T_jy + T_jz) \in U_j
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\]
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so $(T_j \times T_j)^{-1}(U_j)$ is translation-invariant, and so is $V$.
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(4): By (TVS1) and (TVS2), for each $j \in J$, there exists an entourage $V_j$ of $F_j$ and $\eps_j > 0$ such that for any $(x, x'), (y, y') \in V_j$ and $\lambda, \lambda' \in K$ with $\abs{\lambda - \lambda'} < \eps_j$, $(x + y, x' + y'), (\lambda x, \lambda' x') \in U_j$.
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Therefore, for any $(x, x'), (y, y') \in \bigcap_{j \in J} T_j^{-1}(V_j)$ and $\lambda, \lambda' \in K$ with $\abs{\lambda - \lambda'} < \min_{j \in J}\eps$, $(x + y, x' + y'), (\lambda x, \lambda' x') \in V$.
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(5): By \autoref{definition:continuous-linear} and (4) of \autoref{definition:initial-uniformity}.
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(6): By \autoref{definition:initial-uniformity}.
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\end{proof}
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\begin{definition}[Projective Limit of Topological Vector Spaces]
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\label{definition:tvs-projective-limit}
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Let $(\seqi{E}, \bracsn{T^i_j|i, j \in I, i \lesssim j})$ be a downward-directed system of topological vector spaces over $K \in \RC$, then there exists $(E, \bracsn{T^E_i}_{i \in I})$ such that:
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\begin{enumerate}
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\item $E$ is a TVS over $K$.
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\item For each $i \in I$, $T^E_i \in L(E; E_i)$.
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\item For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:
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\[
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\xymatrix{
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E_i \ar@{->}[r]^{T^i_j} & E_j \\
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E \ar@{->}[u]^{T^E_i} \ar@{->}[ru]_{T^E_j} &
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}
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\]
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\item[(U)] For any pair $(F, \bracsn{S^F_i}_{i \in I})$ satisfying (1), (2), and (3), there exists a unique $S \in L(F; E)$ such that the following diagram commutes
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\[
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\xymatrix{
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& E_i \\
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F \ar@{->}[r]_{S} \ar@{->}[ru]^{S^F_i} & A \ar@{->}[u]_{T^E_i}
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}
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\]
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for all $i \in I$.
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\item For any TVS $F$ over $K$ and $S \in \hom(F; E)$, $S \in L(F; E)$ if and only if $T^E_i \circ S \in L(F; E_i)$ for all $i \in I$.
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\end{enumerate}
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The pair $(E, \bracsn{T^E_i}_{i \in I})$ is the \textbf{projective limit} of $(\seqi{E}, \bracsn{T^i_j|i, j \in I, i \lesssim j})$.
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\end{definition}
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\begin{proof}
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Let $(E, \bracsn{T^E_i}_{i \in I})$ be the inverse limit of $(\seqi{E}, \bracsn{T^i_j|i, j \in I, i \lesssim j})$ as $K$-vector spaces (\autoref{proposition:module-inverse-limit}).
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Equip $E$ with the projective topology generated by $\bracsn{T^E_i}_{i \in I}$, then $(E, \bracsn{T^E_i}_{i \in I})$ satisfies (1), (2), and (3).
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(5): By (5) of \autoref{definition:tvs-initial}.
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(U): By (U) of \autoref{proposition:module-inverse-limit}, there exists a unique $S \in \hom(F; E)$ such that the given diagram commutes. By (4), $S \in L(F; E)$.
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\end{proof}
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