Added projective limits.

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]