From d63c2e794f19180b8ca5adba0df28dcc8ba2f7fe Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Thu, 29 Jan 2026 12:28:15 -0500
Subject: [PATCH] Added some limits.

---
 src/fa/lc/index.tex       |  1 +
 src/fa/lc/inductive.tex   | 79 +++++++++++++++++++++++++++++++++++++++
 src/fa/tvs/inductive.tex  |  2 +-
 src/fa/tvs/projective.tex |  2 +-
 4 files changed, 82 insertions(+), 2 deletions(-)
 create mode 100644 src/fa/lc/inductive.tex

diff --git a/src/fa/lc/index.tex b/src/fa/lc/index.tex
index a5226e6..0f78bc7 100644
--- a/src/fa/lc/index.tex
+++ b/src/fa/lc/index.tex
@@ -6,4 +6,5 @@
 \input{./src/fa/lc/continuous.tex}
 \input{./src/fa/lc/quotient.tex}
 \input{./src/fa/lc/projective.tex}
+\input{./src/fa/lc/inductive.tex}
 \input{./src/fa/lc/hahn-banach.tex}
diff --git a/src/fa/lc/inductive.tex b/src/fa/lc/inductive.tex
new file mode 100644
index 0000000..34175f9
--- /dev/null
+++ b/src/fa/lc/inductive.tex
@@ -0,0 +1,79 @@
+\section{Inductive Limits}
+\label{section:lc-inductive}
+
+\begin{definition}[Inductive Locally Convex Topology]
+\label{definition:lc-inductive}
+    Let $\seqi{E}$ be locally convex spaces 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 locally convex space 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 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 $i \in I$.
+    \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}.
+    \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 locally convex space.
+
+    (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 convex, circled, and radial. By (2), $T_i^{-1}(U) \in \cn_{E_i}(0)$ for all $i \in I$. Thus the convex, circled, and radial neighbourhoods of $0$ in $(E, \mathcal{S})$ is a subset of $\mathcal{B}$.
+
+    (4): Let $i \in I$ and $U \in \cn_F(0)$ be convex, circled, and radial. 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:lc-inductive-limit}
+    Let $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system of locally convex spaces over $K \in \RC$, then there exists $(E, \bracsn{T^i_E}_{i \in I})$ such that:
+    \begin{enumerate}
+        \item $E$ is a locally convex space 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), (2), and (3), 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 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 $i \in I$.
+    \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:lc-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:lc-inductive}, $S \in L(E; F)$.
+
+    (5): By (5) of \ref{definition:lc-inductive}.
+\end{proof}
+
+\begin{remark}
+\label{remark:tvs-limits}
+    The projective topology behaves well across the constraints of topological vector spaces and locally convex spaces: the preimage of a vector space/locally convex topology is also a vector space/locally convex topology.
+
+    On the inductive side, the story is not as simple: In principle, the locally convex inductive topology is smaller than the vector space inductive topology, which is smaller than the inductive topology. As such, the same construction must be performed three separate times, each time restricting to a smaller collection of sets.
+
+    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}
diff --git a/src/fa/tvs/inductive.tex b/src/fa/tvs/inductive.tex
index 72b48b2..1749201 100644
--- a/src/fa/tvs/inductive.tex
+++ b/src/fa/tvs/inductive.tex
@@ -47,7 +47,7 @@
         }
         \]
 
-        \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
+        \item[(U)] For any pair $(F, \bracsn{S^i_F}_{i \in I})$ satisfying (1), (2), and (3), there exists a unique $S \in L({E, F})$ such that the following diagram commutes
 
         \[
         \xymatrix{
diff --git a/src/fa/tvs/projective.tex b/src/fa/tvs/projective.tex
index 5c7440b..8a429e8 100644
--- a/src/fa/tvs/projective.tex
+++ b/src/fa/tvs/projective.tex
@@ -57,7 +57,7 @@
         E \ar@{->}[u]^{T^E_i} \ar@{->}[ru]_{T^E_j} &
         }
         \]
-        \item[(U)] For any pair $(F, \bracsn{S^F_i}_{i \in I})$ satisfying (1) and (2), there exists a unique $S \in L(F; E)$ such that the following diagram commutes
+        \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
 
         \[
         \xymatrix{