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Bokuan Li
2026-03-06 14:06:15 -05:00
parent 173727665b
commit 5034bc4220
109 changed files with 1184 additions and 410 deletions
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17
src/fa/tvs/projective.tex
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@@ -17,31 +17,34 @@
\[
\bracs{\bigcap_{j \in J}T_j^{-1}(U_j) \bigg | J \subset I \text{ finite}, U_j \in \cn_{F_j}(0)}
\]
is a fundamental system of neighbourhoods for $E$ at $0$.
\end{enumerate}
The uniformity $\fU$ and its topology are the \textbf{projective uniformity/topology} induced by $\seqi{T}$.
\end{definition}
\begin{proof}
(1), (U): By \ref{definition:initial-uniformity}.
(1), (U): By \autoref{definition:initial-uniformity}.
Let $U \in \fU$, then there exists $J \subset I$ finite and translation-invariant entourages $\seqj{U}$ such that
\[
U \subset V = \bigcap_{j \in J}(T_j \times T_j)^{-1}(U_j)
\]
(3): For each $j \in J$, $(x, y) \in (T_j \times T_j)^{-1}(U_j)$, and $z \in E$,
\[
(T_j \times T_j)(x + z, y + z) = (T_jx + T_jz, T_jy + T_jz) \in U_j
\]
so $(T_j \times T_j)^{-1}(U_j)$ is translation-invariant, and so is $V$.
(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 $\abs{\lambda - \lambda'} < \eps_j$, $(x + y, x' + y'), (\lambda x, \lambda' x') \in U_j$.
Therefore, for any $(x, x'), (y, y') \in \bigcap_{j \in J} T_j^{-1}(V_j)$ and $\lambda, \lambda' \in K$ with $\abs{\lambda - \lambda'} < \min_{j \in J}\eps$, $(x + y, x' + y'), (\lambda x, \lambda' x') \in V$.
(5): By \ref{definition:continuous-linear} and (4) of \ref{definition:initial-uniformity}.
(5): By \autoref{definition:continuous-linear} and (4) of \autoref{definition:initial-uniformity}.
(6): By \ref{definition:initial-uniformity}.
(6): By \autoref{definition:initial-uniformity}.
\end{proof}
\begin{definition}[Projective Limit of Topological Vector Spaces]
@@ -57,6 +60,7 @@
E \ar@{->}[u]^{T^E_i} \ar@{->}[ru]_{T^E_j} &
}
\]
\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
\[
@@ -66,17 +70,18 @@
}
\]
for all $i \in I$.
\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 $i \in I$.
\end{enumerate}
The pair $(E, \bracsn{T^E_i}_{i \in I})$ is the \textbf{projective limit} of $(\seqi{E}, \bracsn{T^i_j|i, j \in I, i \lesssim j})$.
\end{definition}
\begin{proof}
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 (\ref{proposition:module-inverse-limit}).
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}).
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).
(5): By (5) of \ref{definition:tvs-initial}.
(5): By (5) of \autoref{definition:tvs-initial}.
(U): By (U) of \ref{proposition:module-inverse-limit}, there exists a unique $S \in \hom(F; E)$ such that the given diagram commutes. By (4), $S \in L(F; E)$.
(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), $S \in L(F; E)$.
\end{proof}