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src/fa/tvs/continuous.tex

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+\section{Continuous Linear Maps}
+\label{section:tvs-linear-maps}
+
+
+\begin{definition}[Continuous Linear Map]
+\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 $T$ 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 $F$.
+\end{definition}
+\begin{proof}
+    $(1) \Rightarrow (2) \Rightarrow (3)$: By \ref{proposition:uniform-continuous} and \ref{definition:continuity}.
+
+    $(3) \Rightarrow (1)$: Let $U$ be an entourage of $F$, there exists an entourage $V$ of $E$ such that $T(V(0)) \subset U(0)$.  Using \ref{proposition:tvs-uniform} and \ref{lemma:translation-invariant-symmetric}, assume without loss of generality that $U$ and $V$ are symmetric and translation-invariant.
+
+    For any $x, y \in V$, $x - y \in V(0)$, so $Tx - Ty \in U(0)$, $Ty \in U(Tx)$ by symmetry, and $(Tx, Ty) \in U$. Therefore $T$ is uniformly continuous.
+\end{proof}
+
+\begin{definition}[Continuous Multilinear Map]
+\label{definition:continuous-multilinear}
+    Let $\seqf{E}$, $F$ be TVSs over $K \in \RC$, then the set $L^n(E_1, \cdots, E_n; F) = L^n(\seqf{E_j}; F)$ is the space of all continuous $n$-linear maps from $\prod_{j = 1}^n E_j$ to $F$.
+\end{definition}
+
+
+\begin{proposition}
+\label{proposition:continuous-bounded}
+    Let $E, F$ be TVSs over $K \in \RC$ and $T \in L(E; F)$, then for any $B \subset E$ bounded, $T(B)$ is also bounded.
+\end{proposition}
+\begin{proof}
+    Let $U \in \cn_F(0)$, then $T^{-1}(U) \in \cn_E(0)$, so there exists $\lambda \in K$ such that $\lambda T^{-1}(U) = T^{-1}(\lambda U) \supset B$ and $\lambda U \supset T(B)$.
+\end{proof}
+
+
+\begin{definition}[Initial Uniformity]
+\label{definition:tvs-initial}
+    Let $E$ be a vector space over $K \in \RC$, $\seqi{F}$ be TVSs, 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.
+    \end{enumerate}
+    The uniformity and its induced topology are the \textbf{initial 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 $\abs{\lambda - \lambda'} < \min_{j \in J}\eps$, $(x + y, x' + y'), (\lambda x, \lambda' x') \in V$.
+\end{proof}
+
+\begin{definition}[Product Topology]
+\label{definition:tvs-product}
+    Let $\seqi{E}$ be TVSs over $K \in \RC$ and $E = \prod_{i \in I}E_i$ be their product as a vector space, and $\fU$ be the initial uniformity generated by the projection maps, then
+    \begin{enumerate}
+        \item $E$ equipped with the topology induced by $\fU$ is a topological vector space.
+        \item[(U)] For any TVS $F$ over $K$ and $\seqi{T}$ where $T_i \in L(F; E_i)$ for each $i \in I$, there exists a unique $U \in L(F; E)$ such that the following diagram commutes
+
+        \[
+        \xymatrix{
+        F \ar@{->}[rd]^{T_i} \ar@{->}[d]_{T} &  \\
+        \prod_{i \in I}E_i \ar@{->}[r]_{\pi_i} & E_i
+        }
+        \]
+    \end{enumerate}
+    The uniformity $\fU$ and its induced topology are the \textbf{product uniformity/topology}, and $E$ equipped with $\fU$ is the \textbf{product TVS} of $\seqi{E}$.
+\end{definition}