From b220d792e4659db1dfd75baa3f9a846db355ccd5 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Wed, 28 Jan 2026 13:39:00 -0500 Subject: [PATCH] Added projective limits. --- src/cat/cat/universal.tex | 49 +++++++++++++++++++++++++++++++++++---- src/fa/tvs/continuous.tex | 7 ++++-- src/fa/tvs/index.tex | 1 + src/fa/tvs/projective.tex | 37 +++++++++++++++++++++++++++++ 4 files changed, 88 insertions(+), 6 deletions(-) create mode 100644 src/fa/tvs/projective.tex diff --git a/src/cat/cat/universal.tex b/src/cat/cat/universal.tex index e7d2209..cb6f1e6 100644 --- a/src/cat/cat/universal.tex +++ b/src/cat/cat/universal.tex @@ -89,7 +89,7 @@ \begin{definition}[Direct Limit] \label{definition:direct-limit} - Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system, then a \textbf{direct limit} of $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ is a pair $(A, \bracsn{f^i_A}_{i \in I})$ such that: + Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system, then a \textbf{direct limit} of $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$, denoted $\lim_{\longrightarrow}A_i$, is a pair $(A, \bracsn{f^i_A}_{i \in I})$ such that: \begin{enumerate} \item For each $i \in I$, $f^i_A \in \mor{A_i, A}$. \item For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes: @@ -101,7 +101,7 @@ } \] - \item[(U)] For any pair $(B, \bracsn{g^i_A}_{i \in I})$ satsifying (1) and (2), there exists a unique $g \in \mor{A, B}$ such that the following diagram commutes + \item[(U)] For any pair $(B, \bracsn{g^i_A}_{i \in I})$ satisfying (1) and (2), there exists a unique $g \in \mor{A, B}$ such that the following diagram commutes \[ \xymatrix{ @@ -112,12 +112,11 @@ for all $i \in I$. \end{enumerate} - \end{definition} \begin{definition}[Inverse Limit] \label{definition:inverse-limit} -Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system, then an \textbf{inverse limit} of $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ is a pair $(A, \bracsn{f^A_i}_{i \in I})$ such that: +Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ be an downward-directed system, then an \textbf{inverse limit} of $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$, denoted $\lim_{\longleftarrow}A_i$, is a pair $(A, \bracsn{f^A_i}_{i \in I})$ such that: \begin{enumerate} \item For each $i \in I$, $f^A_i \in \mor{A, A_i}$. \item For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes: @@ -141,3 +140,45 @@ Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim for all $i \in I$. \end{enumerate} \end{definition} + +\begin{proposition} +\label{proposition:module-inverse-limit} + Let $R$ be a ring and $(\seqi{A}, \bracs{T^i_j|i, j \in I, i \lesssim j)}$ be a downward-directed system of $R$-modules, then there exists $(A, \bracsn{T^A_i}_{i \in I})$ such that: + \begin{enumerate} + \item For each $i \in I$, $T^A_i \in \hom(A; A_i)$. + \item For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes: + \[ + \xymatrix{ + A_i \ar@{->}[r]^{T^i_j} & A_j \\ + A \ar@{->}[u]^{T^A_i} \ar@{->}[ru]_{T^A_j} & + } + \] + \item[(U)] For any pair $(B, \bracsn{S^B_i}_{i \in I})$ satisfying (1) and (2), there exists a unique $S \in \hom(B; A)$ such that the following diagram commutes + + \[ + \xymatrix{ + & A_i \\ + B \ar@{->}[r]_{S} \ar@{->}[ru]^{S^B_i} & A \ar@{->}[u]_{T^A_i} + } + \] + + for all $i \in I$. + \end{enumerate} +\end{proposition} +\begin{proof} + Let + \[ + A = \bracs{x \in \prod_{i \in I}A_i \bigg | \pi_j(x) = T^i_j\pi_i(x) \forall i, j \in I, i \lesssim j} + \] + For each $i \in I$, let $T^A_i = \pi_i$, then $(A, (A, \bracsn{T^A_i}_{i \in I})$ satisfies (1) and (2) by definition of $A$. + + (U): Let $(B, \bracsn{S^B_i}_{i \in I})$ satisfying (1) and (2). Let + \[ + S: B \to \prod_{i \in I}A_i \quad \pi_i(Sx) = S^B_i + \] + then for any $x \in B$ and $i, j \in I$ with $i \lesssim j$, + \[ + \pi_j (Sx) = S^B_jx = T^i_j S^B_ix = T^i_j \pi_i(S x) + \] + so $S \in \hom(B; A)$, and the diagram commutes. Since any map $f: B \to A$ is uniquely determined by its composition with the projections, $S$ is unique. +\end{proof} diff --git a/src/fa/tvs/continuous.tex b/src/fa/tvs/continuous.tex index 60c223a..903b54c 100644 --- a/src/fa/tvs/continuous.tex +++ b/src/fa/tvs/continuous.tex @@ -6,8 +6,8 @@ \label{definition:continuous-linear} Let $E, F$ be TVSs over $K \in \RC$, and $T \in \hom({E, F})$ be a linear map, then the following are equivalent: \begin{enumerate} - \item $T$ is uniformly continuous. - \item $T$ is continuous. + \item $T \in UC(E; F)$. + \item $T \in C(E; F)$. \item $T$ is continuous at $0$. \end{enumerate} If the above holds, then $T$ is a \textbf{continuous linear map}. The set $L(E; F)$ denotes the vector space of all continuous linear maps from $E$ to $F$. @@ -46,6 +46,7 @@ \begin{enumerate} \item[(3)] $\fU$ is translation-invariant. \item[(4)] $E$ equipped with the topology induced by $\fU$ is a topological vector space. + \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 $i \in I$. \end{enumerate} The uniformity and its induced topology are the \textbf{initial uniformity/topology} induced by $\seqi{T}$. \end{definition} @@ -66,6 +67,8 @@ (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}. \end{proof} \begin{definition}[Product Topology] diff --git a/src/fa/tvs/index.tex b/src/fa/tvs/index.tex index 9540225..725e1e0 100644 --- a/src/fa/tvs/index.tex +++ b/src/fa/tvs/index.tex @@ -8,4 +8,5 @@ \input{./src/fa/tvs/continuous.tex} \input{./src/fa/tvs/quotient.tex} \input{./src/fa/tvs/completion.tex} +\input{./src/fa/tvs/projective.tex} \input{./src/fa/tvs/spaces-of-linear.tex} diff --git a/src/fa/tvs/projective.tex b/src/fa/tvs/projective.tex new file mode 100644 index 0000000..06a5127 --- /dev/null +++ b/src/fa/tvs/projective.tex @@ -0,0 +1,37 @@ +\section{Projective Limits} +\label{section:tvs-projective-limit} + +\begin{definition}[Projective Limit of Topological Vector Spaces] +\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 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: + \[ + \xymatrix{ + E_i \ar@{->}[r]^{T^i_j} & E_j \\ + E \ar@{->}[u]^{T^E_i} \ar@{->}[ru]_{T^E_j} & + } + \] + \item[(U)] For any pair $(F, \bracsn{S^F_i}_{i \in I})$ satisfying (1) and (2), there exists a unique $S \in L(F; E)$ such that the following diagram commutes + + \[ + \xymatrix{ + & E_i \\ + F \ar@{->}[r]_{S} \ar@{->}[ru]^{S^F_i} & A \ar@{->}[u]_{T^E_i} + } + \] + + 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 $T^E_i \circ F \in L(F; E_i)$ for all $i \in I$. + \end{enumerate} +\end{definition} +\begin{proof} + Let $(E, \bracsn{T^E_i}_{i \in I})$ be the inverse limit of $(\seqi{E}, \bracs{T^i_j|i, j \in I, i \lessim j})$ as $K$-vector spaces (\ref{proposition:module-inverse-limit}). + + Equip $E$ with the initial topology generated by $\bracsn{T^E_i}_{i \in I}$, then $(E, \bracsn{T^E_i}_{i \in I})$ satisfies (1) and (2). + + (4): By (5) of \ref{definition:initial-topology}. + + (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}