Added projective limits.

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}