Added complexification.

src/fa/tvs/inductive.tex

@@ -34,6 +34,58 @@
     (4): Let $U \in \cn_F(0)$ be circled and radial and $i \in I$. 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}[Direct Sum]
+\label{definition:tvs-direct-sum}
+    Let $\seqi{E}$ be TVSs over $K \in \RC$, then there exists $(E, \seqi{\iota})$ such that:
+    \begin{enumerate}
+        \item $E$ is a TVS over $K$. 
+        \item For each $i \in I$, $\iota_i \in L(E_i; E)$. 
+        \item[(U)] For each $(F, \seqi{T})$ satisfying (1) and (2), there exists a unique $T \in L(E; F)$ such that the following diagram commutes:
+        \[
+        \xymatrix{
+        E \ar@{->}[r]^{T} & F \\
+        E_i \ar@{->}[u]^{\iota_i} \ar@{->}[ru]_{T_i} & 
+        }
+        \]
+    \end{enumerate}
+
+    
+    The space $E = \bigoplus_{i \in I}E_i$ is the \textbf{direct sum} of $\seqi{E}$. 
+\end{definition}
+\begin{proof}
+    Let $(E, \seqi{\iota})$ be the direct sum of $\seqi{E}$ as vector spaces, and equip it with the inductive topology induced by $\seqi{\iota}$, then $(E, \seqi{\iota})$ satisfies (1) and (2). 
+
+    (U): By (U) of the \hyperref[direct sum]{definition:direct-sum}, there exists a unique $T \in \hom(E; F)$ such that the diagram commutes. In which case, by (4) of \autoref{definition:tvs-inductive}, $T \in L(E; F)$. 
+\end{proof}
+
+\begin{proposition}
+\label{proposition:finite-tvs-product}
+    Let $\seqf{E_j}$ be TVSs over $K \in \RC$, then
+    \[
+    \prod_{j = 1}^n E_j = \bigoplus_{j = 1}^n E_j
+    \]
+\end{proposition}
+\begin{proof}
+    Let $1 \le k \le n$, then for each $1 \le k, l \le n$, $\pi_l \circ \iota_k \in L(E_k, E_l)$, so by (U) of the \hyperref[product]{definition:tvs-product}, $\iota_k \in L(E_k; \prod_{j = 1}^n E_j)$. Thus $\prod_{j = 1}^n E_j$ satisfies (1) and (2) of the \hyperref[direct sum]{definition:tvs-direct-sum}. 
+
+    For any TVS $F$ over $K$ and $\seqf{T_j}$ with $T_j \in L(E_j; F)$ for each $1 \le j \le n$, let
+    \[
+    T: \prod_{j = 1}^n E_j \to F \quad (x_1, \cdots, x_n) \mapsto \sum_{j = 1}^n T_jx_j
+    \]
+
+    then $T \in L(\prod_{j = 1}^n E_j; F)$ is the unique continuous linear map such that the following diagram commutes:
+    \[
+    \xymatrix{
+    E \ar@{->}[r]^{T} & F \\
+    E_i \ar@{->}[u]^{\iota_i} \ar@{->}[ru]_{T_i} & 
+    }
+    \]
+
+    Hence $\prod_{j = 1}^n E_j$ satisfies (U) of the \hyperref[direct sum]{definition:tvs-direct-sum}, so the spaces coincide. 
+\end{proof}
+
+
+
 \begin{definition}[Inductive Limit]
 \label{definition:tvs-inductive-limit}
     Let $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system of TVSs over $K \in \RC$, then there exists $(E, \bracsn{T^i_E}_{i \in I})$ such that: