From c0a76cc2f0dbd3078d71efe68829d05f329b6c39 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Thu, 29 Jan 2026 12:04:26 -0500 Subject: [PATCH] Added inductive limits of TVS. --- src/fa/tvs/index.tex | 1 + src/fa/tvs/inductive.tex | 70 +++++++++++++++++++++++++++++++++++++++ src/fa/tvs/projective.tex | 53 ++++------------------------- 3 files changed, 77 insertions(+), 47 deletions(-) create mode 100644 src/fa/tvs/inductive.tex diff --git a/src/fa/tvs/index.tex b/src/fa/tvs/index.tex index 725e1e0..39a9416 100644 --- a/src/fa/tvs/index.tex +++ b/src/fa/tvs/index.tex @@ -9,4 +9,5 @@ \input{./src/fa/tvs/quotient.tex} \input{./src/fa/tvs/completion.tex} \input{./src/fa/tvs/projective.tex} +\input{./src/fa/tvs/inductive.tex} \input{./src/fa/tvs/spaces-of-linear.tex} diff --git a/src/fa/tvs/inductive.tex b/src/fa/tvs/inductive.tex new file mode 100644 index 0000000..72b48b2 --- /dev/null +++ b/src/fa/tvs/inductive.tex @@ -0,0 +1,70 @@ +\section{Inductive Limits} +\label{section:tvs-inductive} + +\begin{definition}[Inductive Topology] +\label{definition:tvs-inductive} + Let $\seqi{E}$ be TVSs over $K \in \RC$, $\seqi{T}$ such that $T_i \in \hom(E_i; E)$ for all $i \in I$, and $E$ be a vector space over $K$, then there exists a topology $\topo$ on $E$ such that: + \begin{enumerate} + \item $(E, \topo)$ is a TVS over $K$. + \item For each $i \in I$, $T_i \in L(E_i; E)$. + \item[(U)] For any topology $\mathcal{S}$ on $E$ satisfying (1) and (2), $\mathcal{S} \subset T$. + \item For any TVS $F$ and $T \in \hom(E; F)$, $T \in L(E; F)$ if and only if $T \circ T_i \in L(E_i; F)$ for all $i \in I$. + \end{enumerate} + The topology $\topo$ is the \textbf{inductive topology} on $E$ induced by $\seqi{T}$. +\end{definition} +\begin{proof} + (1): Let + \[ + \mathcal{B} = \bracs{U \subset E|U \text{ radial, circled}, T_i^{-1}(U) \in \cn_{E_i}(0) \forall i \in I} + \] + To see that $\mathcal{B}$ is a fundamental system of neighbourhoods at $0$ for a vector space topology on $E$, it is sufficient to verify the following and apply \ref{proposition:tvs-0-neighbourhood-base}. + \begin{enumerate} + \item[(TVB1)] Every set in $\mathcal{B}$ is radial and circled by definition. + \item[(TVB2)] For any $U \in \mathcal{B}$, $U$ is circled, so $\frac{1}{2}U + \frac{1}{2}U \subset U$. Since $\frac{1}{2}U$ is also circled and radial, $\frac{1}{2}U \in \mathcal{B}$. + \end{enumerate} + + Let $\topo$ be the vector space topology such that $\mathcal{B}$ is a fundamental system of neighbourhoods at $0$, then $(E, \topo)$ is a TVS. + + (2): For each $i \in I$ and $U \in \mathcal{B}$, $T_i^{-1}(U) \in \cn_{E_i}(0)$, so $T_i \in L(E_i; E)$. + + (U): Let $U \in \cn_{(E, \mathcal{S})}(0)$ be circled and radial, then by (2), $T_i^{-1}(U) \in \cn_{E_i}(0)$ for all $i \in I$. Thus the circled and radial neighbourhoods of $0$ in $(E, \mathcal{S})$ is a subset of $\mathcal{B}$. + + (4): Let $U \in \cn_F(0)$ be circled and radial and $i \in I$. Since $T \circ E_i \in L(E_i; F)$, $T_i^{-1}(T^{-1}(U)) \in \cn_{E_i}(0)$, so $T^{-1}(U) \in \mathcal{B} \subset \cn_E(0)$. +\end{proof} + +\begin{definition}[Inductive Limit] +\label{definition:tvs-inductive-limit} + Let $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system of TVSs over $K \in \RC$, then there exists $(E, \bracsn{T^i_E}_{i \in I})$ such that: + \begin{enumerate} + \item $E$ is a TVS over $K$. + \item For each $i \in I$, $T^i_E \in L({E_i, E})$. + \item For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes: + + \[ + \xymatrix{ + E_i \ar@{->}[rd]_{T^i_E} \ar@{->}[r]^{T^i_j} & E_j \ar@{->}[d]^{T^j_E} \\ + & E + } + \] + + \item[(U)] For any pair $(F, \bracsn{S^i_F}_{i \in I})$ satisfying (1) and (2), there exists a unique $S \in L({E, F})$ such that the following diagram commutes + + \[ + \xymatrix{ + E_i \ar@{->}[d]_{T^i_E} \ar@{->}[rd]^{S^i_F} & \\ + E \ar@{->}[r]_{S} & F + } + \] + + for all $i \in I$. + \item For any TVS $F$ and $T \in \hom(E; F)$, $T \in L(E; F)$ if and only if $T \circ T^i_E \in L(E_i; F)$ for all $i \in I$. + \end{enumerate} + The pair $(E, \bracsn{T^i_E}_{i \in I})$ is the \textbf{inductive limit} of $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$. +\end{definition} +\begin{proof} + Let $(E, \bracsn{T^i_E}_{i \in I})$ be the direct limit of $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$ as vector spaces over $K$ (\ref{proposition:module-direct-limit}). Equip $E$ with the inductive topology (\ref{definition:tvs-inductive}) induced by $\bracsn{T^i_E}_{i \in I}$, then $(E, \bracsn{T^i_E}_{i \in I})$ satisfies (1), (2), and (3). + + (U): By (U) of \ref{proposition:module-direct-limit}, there exists a unique $S \in \hom(E; F)$ such that the given diagram commutes. By (4) of \ref{definition:tvs-inductive}, $S \in L(E; F)$. + + (5): By (5) of \ref{definition:tvs-inductive}. +\end{proof} diff --git a/src/fa/tvs/projective.tex b/src/fa/tvs/projective.tex index 9dce9ee..5c7440b 100644 --- a/src/fa/tvs/projective.tex +++ b/src/fa/tvs/projective.tex @@ -19,7 +19,7 @@ \] is a fundamental system of neighbourhoods for $E$ at $0$. \end{enumerate} - The uniformity and its induced topology are the \textbf{projective uniformity/topology} induced by $\seqi{T}$. + 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}. @@ -48,6 +48,7 @@ \label{definition:tvs-projective-limit} 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: \begin{enumerate} + \item $E$ is a TVS over $K$. \item For each $i \in I$, $T^E_i \in L(E; E_i)$. \item For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes: \[ @@ -66,58 +67,16 @@ \] 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 $T^E_i \circ F \in L(F; E_i)$ 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}). - 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) and (2). + 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). - (4): By (5) of \ref{definition:tvs-initial}. + (5): By (5) of \ref{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), $S \in L(F; E)$. \end{proof} - -\begin{proposition}[{{\cite[II.5.4]{SchaeferWolff}}}] -\label{proposition:complete-lc-projective-limit} - Let $E$ be a Hausdorff complete locally convex space over $K \in \RC$, $\mathcal{B} \subset \cn_E(0)$ be a fundamental system of neighbourhoods consisting of convex, circled, and radial sets, directed under inclusion. - - For each $U \in \mathcal{B}$, let $[\cdot]_U$ be its gauge, $M_U = \bracs{x \in E|[\cdot]_U = 0}$, $E_U = E/M_U$, and $\norm{\cdot}_U: E_U \to [0, \infty)$ be the quotient of $[\cdot]_U$ by $M_U$, then - \begin{enumerate} - \item For each $U, V \in \mathcal{B}$ with $U \subset V$, let - \[ - \pi^U_V: E_U \to E_V \quad x + M_U \mapsto x + M_V - \] - then $\pi^U_V \in L(E_U; E_V)$. - \item $(\bracsn{E_U}_{U \in \mathcal{B}}, \bracs{\pi^U_V|U, V \in \mathcal{B}, U \subset V})$ is a downward-directed system of topological vector spaces. - \item The map $\pi \in L(E, \lim_{\longleftarrow}E_U)$ induced by $\bracs{\pi_U}_{U \in \mathcal{B}}$ is a bijection. - \item For each $U, V \in \mathcal{B}$, let $\ol E_U$ be the completion of $E_U$, $\ol{\pi_U} \in L(E; \ol E_U)$, and $\ol{\pi^U_V} \in L(\ol E_U; \ol E_V)$ be the unique extensions of $\pi_U$ and $\pi^U_V$, respectively. Then, - \[ - E = \lim_{\longleftarrow}E_U = \lim_{\longleftarrow} \ol E_U - \] - \end{enumerate} -\end{proposition} -\begin{proof} - (1): Since $V \supset U$, $[\cdot]_V \ge [\cdot]_U$, so $M_V \supset M_U$. Thus $\ker(\pi_V) \supset M_U$. By (U) of the quotient (\ref{definition:tvs-quotient}), $\pi_V$ factors through $E_U$ as $\pi^U_V$, so $\pi^U_V \in L(E_U; E_V)$. - - (2): Since $\mathcal{B}$ is a fundamental system of neighbourhoods, it is downward-directed under inclusion. For any $U, V, W \in \mathcal{B}$ with $U \subset V \subset W$, $M_U \supset M_V \supset M_W$. Thus $\pi^U_W = \pi^V_W \circ \pi^U_V$. - - (3): Let $\lim E_U$ be the projective limit. For each $U \in \mathcal{B}$, let $p_U: \lim E_U \to E_U$ be the canonical map. - - Let $x \in E$. Since $E$ is Hausdorff and $\mathcal{B}$ is a fundamental system of neighbourhoods at $0$, there exists $U \in \mathcal{B}$ such that $\pi_U(x) \ne 0$. In which case, $p_U \circ \pi(x) = \pi_U(x) \ne 0$, so $\pi$ is injective. - - Let $x \in \lim E_U$. For each $U \in \mathcal{B}$, let $x_U \in E$ such that $\pi_U(x_U) = p_U(x)$. For any $V \in \cn_E(0)$, there exists $W \in \mathcal{B}$ with $W \subset V$. In which case, for any $U \in \mathcal{B}$ with $U \subset W$, - \[ - \pi_W(x_U) = \pi_W^U \circ \pi_U(x_U) = \pi_W^U p_U(x) - \] - Thus for any $U' \in \mathcal{B}$ with $U \subset W$, $[x_U - x_{U'}]_W = 0$, and $x_U - x_{U'} \in W$. Therefore $\bracs{x_U}_{U \in \mathcal{B}}$ is a Cauchy net, and converges to $x_0 \in E$ by completeness of $E$. - - For any $U \in \mathcal{B}$, $\pi_U(x_0) = \lim_{V \in \mathcal{B}}\pi_U(x_V) = p_U(x)$, so $\pi(x_0) = x$, and $\pi$ is surjective. - - (4): Since $\mathcal{B} \subset \cn_E(0)$ is a fundamental system of neighbourhoods, the topology on $E$ is the projective topology generated by $\bracs{\pi_U|U \in \mathcal{B}}$. As $\pi_U \circ \pi^{-1} = p_U \in L(\lim E_U; E_U)$ for all $U \in \mathcal{B}$, $\pi^{-1} \in L(\lim E_U; E)$ by (U) of the projective topology (\ref{definition:tvs-initial}). - - Let $x \in \lim\ol{E}_U$ and $V \in \cn(x)$. Since $\mathcal{B}$ is downward-directed and $\lim\ol{E}_U$ is equipped with the projective topology induced by $\bracs{p_U|U \in \mathcal{B}}$, there exists $U \in \mathcal{B}$ and $W \in \cn_{\ol E_U}(x)$ such that $p_U^{-1}(W) \subset V$. As $E_U$ is dense in $\ol E_U$, there exists $y_U \in W$, and $y \in E$ such that $y_U = \pi_U(y)$. Therefore $\pi(y_U) \in p_U^{-1}(W) \subset V$, and $\lim E_U$ is dense in $\lim \ol{E}_U$. - - Since $E$ is complete and isomorphic to $\lim E_U$, $\lim E_U$ is a complete, and thus closed subset of $\lim \ol{E}_U$. Therefore $E = \lim E_U = \lim \ol{E}_U$. -\end{proof}