Added projective limits for LC spaces.

src/fa/lc/index.tex

@@ -5,4 +5,5 @@
 \input{./src/fa/lc/convex.tex}
 \input{./src/fa/lc/continuous.tex}
 \input{./src/fa/lc/quotient.tex}
+\input{./src/fa/lc/projective.tex}
 \input{./src/fa/lc/hahn-banach.tex}

src/fa/lc/projective.tex

@@ -0,0 +1,23 @@
+\section{Projective Limits}
+\label{section:lc-projective}
+
+
+\begin{proposition}
+\label{proposition:lc-projective-topology}
+    Let $E$ be a vector space over $K \in \RC$, $\seqi{F}$ be locally convex spaces over $K$, and $\seqi{T}$ where $T_i \in \hom(E; F_i)$ for all $i \in I$, then the projective topology on $E$ is locally convex.
+\end{proposition}
+\begin{proof}
+    By \ref{definition:tvs-initial},
+    \[
+    \mathcal{B} = \bracs{\bigcap_{j \in J}T_j^{-1}(U_j) \bigg | J \subset I \text{ finite}, U_j \in \cn_{F_j}(0)}
+    \]
+    is a fundamental system of neighbourhoods at $0$. For each $i \in I$, $U_i \in \cn_{F_i}(0)$ convex, $T^{-1}(U_i)$ is also convex. Since each $F_i$ is locally convex, $\mathcal{B}$ contains a fundamental system of neighbourhoods at $0$ consisting of only convex sets.
+\end{proof}
+
+\begin{proposition}
+\label{proposition:lc-projective}
+    Let $(\seqi{E}, \bracsn{T^i_j|i, j \in I, i \lesssim j})$ be a downward-directed system of locally convex spaces over $K \in \RC$, then $E = \lim_{\longleftarrow}E_i$ is locally convex.
+\end{proposition}
+\begin{proof}
+    By (U) of \ref{definition:tvs-projective-limit} and \ref{definition:tvs-initial}, $E$ is equipped with the projective topology generated by the projection maps $E \to E_i$. By \ref{proposition:lc-projective-topology}, $E$ is locally convex.
+\end{proof}

src/fa/tvs/projective.tex

@@ -1,6 +1,49 @@
 \section{Projective Limits}
 \label{section:tvs-projective-limit}
 
+\begin{definition}[Projective Uniformity]
+\label{definition:tvs-initial}
+    Let $E$ be a vector space over $K \in \RC$, $\seqi{F}$ be TVSs over $K$, and $\seqi{T}$ where $T_i \in \hom(E; F_i)$ for all $i \in I$, then there exists a uniformity $\fU$ on $E$ such that:
+    \begin{enumerate}
+        \item For each $i \in I$, $T_i \in L(E; F_i)$.
+        \item[(U)] If $\mathfrak{V}$ is a uniformity on $E$ satisfying $(1)$, then $\mathfrak{V} \supset \fU$.
+    \end{enumerate}
+    Moreover,
+    \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 $i \in I$.
+        \item[(6)] The collection
+        \[
+        \bracs{\bigcap_{j \in J}T_j^{-1}(U_j) \bigg | J \subset I \text{ finite}, U_j \in \cn_{F_j}(0)}
+        \]
+        is a fundamental system of neighbourhoods for $E$ at $0$.
+    \end{enumerate}
+    The uniformity and its induced topology are the \textbf{projective uniformity/topology} induced by $\seqi{T}$.
+\end{definition}
+\begin{proof}
+    (1), (U): By \ref{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}.
+
+    (6): By \ref{definition:initial-uniformity}.
+\end{proof}
+
 \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:
@@ -29,9 +72,9 @@
 \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).
+    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).
 
-    (4): By (5) of \ref{definition:initial-topology}.
+    (4): 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}