From 5c54ec088303a8ba301e42705b58c6242c806a9f Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Fri, 30 Jan 2026 11:54:08 -0500 Subject: [PATCH] Added properties of inductive limits. --- src/fa/lc/inductive.tex | 109 +++++++++++++++++++++++++++++++++++++-- src/fa/tvs/inductive.tex | 9 ++++ 2 files changed, 113 insertions(+), 5 deletions(-) diff --git a/src/fa/lc/inductive.tex b/src/fa/lc/inductive.tex index 34175f9..70b2c33 100644 --- a/src/fa/lc/inductive.tex +++ b/src/fa/lc/inductive.tex @@ -9,15 +9,16 @@ \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 locally convex space $F$ 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 $i \in I$. + \item The family + \[ + \mathcal{B} = \bracs{U \subset E|U \text{ convex, radial, circled}, T_i^{-1}(U) \in \cn_{E_i}(0) \forall i \in I} + \] + is a fundamental system of neighbourhoods for $E$ at $0$. \end{enumerate} The topology $\topo$ is the \textbf{inductive locally convex topology} on $E$ induced by $\seqi{T}$. \end{definition} \begin{proof} - (1): Let - \[ - \mathcal{B} = \bracs{U \subset E|U \text{ convex, radial, circled}, T_i^{-1}(U) \in \cn_{E_i}(0) \forall i \in I} - \] - 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}. + (1), (5): 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, $\frac{1}{2}U \in \mathcal{B}$. @@ -58,6 +59,11 @@ for all $i \in I$. \item For any locally convex space $F$ 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 $i \in I$. + \item The family + \[ + \mathcal{B} = \bracs{U \subset E|U \text{ convex, radial, circled}, (T^i_E)^{-1}(U) \in \cn_{E_i}(0) \forall i \in I} + \] + is a fundamental system of neighbourhoods for $E$ at $0$. \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} @@ -77,3 +83,96 @@ In addition to the neighbourhood construction given above, the inductive topology may also be constructed as the weak topology generated by all topologies satisfying certain properties. While this more non-constructive method is simpler, it does not directly provide an explicit fundamental system of neighbourhoods at $0$. \end{remark} + +\begin{lemma}[{{\cite[Lemma II.6.4]{SchaeferWolff}}}] +\label{lemma:lc-induct-separate} + Let $E$ be a locally convex space over $K$, $M \subset E$ be a subspace, $U \in \cn_M(0)$ be convex and circled, then + \begin{enumerate} + \item There exists $V \in \cn_E(0)$ circled and radial such that $U = M \cap V$. + \item For any $x \in E \setminus \ol M$, there exists $V \in \cn_E(0)$ circled and radial such that $U = M \cap V$ and $x \not\in U$. + \end{enumerate} +\end{lemma} +\begin{proof} + (1): Let $W \in \cn_E(0)$ be circled and radial with $W \cap M \subset U$, and $V = \text{Conv}(W \cup U)$. + \begin{itemize} + \item For any $u \in U$, $w \in W$, and $t \in [0, 1]$, there exists $\alpha \in (0, 1)$ such that $x = \alpha^{-1}w \in W$. In which case, + \[ + (1 - \alpha)U + w = (1 - \alpha) U + \alpha x \subset V + \] + so $V \in \cn(0)$. + \item For any $\lambda \in K$ with $\abs{\lambda} \le 1$, $u \in U$, $w \in W$, and $t \in [0, 1]$, + \[ + \lambda (1 - t)u + \lambda tw = (1 - t)\lambda u + t \lambda w \in V + \] + as $U$ and $W$ are both circled. + \end{itemize} + so $V \in \cn_E(0)$ is convex and circled. + + For any $u \in U$, $w \in W$, and $t \in [0, 1]$, if $(1 - t)u + tw \in M$, then $u \in M \cap U \subset M \cap W$, so $(1 - t)u + tw \in W$. + + (2): Since $x \not\in \ol M$, there exists $W \in \cn_E(0)$ circled and radial with $x + W \cap M = \emptyset$. Let $V$ as in (1), then $x \not\in W + M \supset W + U = V$. +\end{proof} + +\begin{proposition}[{{\cite[II.6.4, II.6.5, II.6.6]{SchaeferWolff}}}] +\label{proposition:lc-strict-inductive-countable} + Let $(\seq{E_n}, \seq{\iota_n^m|m, n \in \natp, m \le n})$ be a strict inductive system of locally convex spaces over $K \in \RC$ and $(E, \seq{\iota_n})$ be its direct limit, then: + \begin{enumerate} + \item For each $n \in \natp$, the topology of $E_n$ is induced by $\iota_n$, which allows the identification of $E_n$ as a subspace of $E$. + \item If $E_n$ is separated for all $n \in \natp$, then $E$ is also separated. + \item If $\iota^m_n(E_m) \subset E_n$ is closed for all $m, n \in \natp$ with $m \le n$, then the following are equivalent for any $B \subset E$: + \begin{enumerate} + \item[(a)] $B$ is bounded. + \item[(b)] There exists $n \in \natp$ such that $B \subset E_n$ is bounded. + \end{enumerate} + \item If $E_n$ is complete for each $n \in \natp$, then $E$ is also complete. + \end{enumerate} +\end{proposition} +\begin{proof} + (1): Let $U \in \cn_{E_n}(0)$. By \ref{lemma:lc-induct-separate}, there exists $\bracs{U_m| m \in \natp, m \ge n} \subset 2^E$ such that $U_n = U$, $U_m \in \cn_{E_m}(0)$ and $U_{m} = U_{m + 1} \cap E_m$ for all $m \in \natp$. Let $V = \bigcup_{m \ge n}U_m$, then $V \cap E_m = U_m$ for all $m \ge n$. In particular, $V \cap E_n = U_n = U$. + + (2): Let $x \in E \setminus \bracs{0}$, then there exists $n \in \natp$ such that $x \in E_n$. Since $E_n$ is separated, there exists $U \in \cn_{E_n}(0)$ with $x \not\in U$. By \ref{lemma:lc-induct-separate} and (1), there exists $V \in \cn_E(0)$ such that $V \cap E_n = U$, so $x \not\in V$. + + (3), $\neg (b) \Rightarrow \neg (a)$: If $B \not\subset E_n$ for all $n \in \natp$, then there exists a subsequence $\bracsn{n_k}_0^\infty \subset \natp$ and $\seq{x_k} \subset B$ such that $x_k \in E_{n_{k}} \setminus E_{n_{k - 1}}$ for all $k \in \natp$. + + Since $E_{n_k} \subset E_{n_{k+1}}$ is closed for all $k \in \natp$, there exists $\seq{U_k} \subset 2^E$ such that: + \begin{enumerate} + \item For each $k \in \natp$, $U_k \in \cn_{E_{n_k}}(0)$. + \item For each $k \in \natp$, $U_k = U_{k+1} \cap E_{n_k}$. + \item For each $k \in \natp$, $n^{-1}x_k \not\in U_k$. + \end{enumerate} + then $V = \bigcup_{k \in \natp}U_k \in \cn_E(0)$ with $V \cap E_{n_k} = U_k$ for all $k \in \natp$. For any $n \in \natp$, $x_k \not\in nU_k = nV \cap E_{n_k}$. Therefore $B$ is not bounded. + + (3), $(b) \Rightarrow (a)$: Let $U \in \cn_E(0)$, then $U \cap E_n \in \cn_{E_n}(0)$, so there exists $\lambda \in K$ with $\lambda (U \cap E_n) \supset B$. + + (4): Let $\fF \subset 2^E$ be a Cauchy filter and + \[ + \fU = \bracs{F + U|F \in \mathcal{F}, U \in \cn_E(0)} + \] + then $\fU$ is also a Cauchy filter, which converges if and only if $\fF$ does. + + Since each $E_n$ is complete, it is sufficient to show that there exists $n \in \natp$ such that $F + U \cap E_n \ne \emptyset$ for all $F \in \fF$ and $U \in \cn_E(0)$. + + Suppose for contradiction that for every $n \in \natp$, there exists $F_n \in \fF$ and $U_n \in \cn_E(0)$ such that $E_n \cap F_n + U_n = \emptyset$. Assume without loss of generality that for every $n \in \natp$, $U_n$ is convex and circled with $U_n \supset U_{n+1}$. Let + \[ + U = \text{Conv}\paren{\bigcup_{n \in \natp}(U_n \cap E_n)} + \] + then since each $U_n$ is circled, so is $U$. Thus $U \cap E_n \supset U_n \cap E_n \in \cn_{E_n}(0)$, and $U \in \cn_E(0)$. + + Now, suppose that $(F_n + U) \cap E_n \ne \emptyset$. Let $y \in (F_n + U) \cap E_n$, then there exists $N \in \natp$, $\bracs{x_k}_1^N \subset E$, $\bracs{\lambda_k}_1^N \subset [0, 1]$, and $z \in F_n$ such that + \begin{itemize} + \item For each $1 \le k \le N$, $x_k \in U_k \cap E_k$. + \item $\sum_{k = 1}^N \lambda_k = 1$. + \item $y = z + \sum_{k = 1}^N \lambda_k x_k$. + \end{itemize} + In which case, since $U_{k} \supset U_{k+1}$ for all $k \in \natp$, + \[ + \underbracs{y - \sum_{k = 1}^n \lambda_kx_k}_{\in E_n} = \underbrace{z + \sum_{k = n + 1}^N \lambda_kx_k}_{\in F_n + U_n} + \] + which is impossible. Therefore $(F_n + U) \cap E_n = \emptyset$ for all $n \in \natp$. + + Finally, since $\fF$ is a Cauchy filter, there exists $F \in \fF$ such that $F - F \subset U$. Let $z \in F$, then there exists $n \in \natp$ such that $z \in E_n$. In which case, for any $y \in F \cap F_n$, + \[ + z = y + (z - y) \in y + (F - F) \subset y + U \subset F_n + U + \] + which contradicts the fact that $(F_n + U) \cap E_n = \emptyset$. +\end{proof} diff --git a/src/fa/tvs/inductive.tex b/src/fa/tvs/inductive.tex index 1749201..6602bb4 100644 --- a/src/fa/tvs/inductive.tex +++ b/src/fa/tvs/inductive.tex @@ -68,3 +68,12 @@ (5): By (5) of \ref{definition:tvs-inductive}. \end{proof} + +\begin{definition}[Strict] +\label{definition:lc-inductive-strict} + Let $(\seqi{E}, \bracsn{\iota^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system of TVSs over $K \in \RC$, then the system is \textbf{strict} if: + \begin{enumerate} + \item For each $i, j \in I$ with $i \lesssim j$, $\iota^i_j: E_i \to E_j$ is injective. + \item For each $i, j \in I$ with $i \lesssim j$, the topology of $E_i$ is induced by $\iota^i_j$. + \end{enumerate} +\end{definition}