Added complexification.

src/fa/lc/inductive.tex

@@ -54,8 +54,8 @@
         \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:
         \[
         \xymatrix{
-        A \ar@{->}[r]^{T} & B \\
+        E \ar@{->}[r]^{T} & F \\
-        A_i \ar@{->}[u]^{\iota_i} \ar@{->}[ru]_{T_i} & 
+        E_i \ar@{->}[u]^{\iota_i} \ar@{->}[ru]_{T_i} & 
         }
         \]
         
@@ -71,13 +71,45 @@
     The space $E = \bigoplus_{i \in I}E_i$ is the \textbf{locally convex 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). 
+    Let $(E, \seqi{\iota})$ be the direct sum of $\seqi{E}$ as vector spaces, and equip it with the locally convex 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:lc-inductive}, $T \in L(E; F)$. 
 
     (4): By (6) of \autoref{definition:lc-inductive}.
 \end{proof}
 
+\begin{proposition}
+\label{proposition:finite-lc-product}
+    Let $\seqf{E_j}$ be TVSs over $K \in \RC$, then the following spaces coincide:
+    \begin{enumerate}
+        \item The product $\prod_{j = 1}^n E_j$.
+        \item The direct sum of $\seqf{E_j}$ as topological vector spaces.
+        \item The direct sum of $\seqf{E_j}$ as locally convex spaces. 
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    By \autoref{proposition:finite-tvs-product}, it is sufficient to show that (1) and (3) coincide. The proof is \textit{exactly} the same as \autoref{proposition:finite-tvs-product}, but included here for completeness.  
+
+    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:lc-direct-sum}. 
+
+    For any locally convex space $F$ 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:lc-direct-sum}, so the spaces coincide. 
+\end{proof}
+
+
+
 \begin{definition}[Inductive Limit]
 \label{definition:lc-inductive-limit}
     Let $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system of locally convex spaces over $K \in \RC$, then there exists $(E, \bracsn{T^i_E}_{i \in I})$ such that:

src/fa/tvs/complexify.tex

@@ -0,0 +1,119 @@
+\section{Complexification}
+\label{section:complexification}
+
+\begin{definition}[Complexification]
+\label{definition:complexification}
+    Let $E$ be a vector space over $\real$, then there exists a pair $(\complex(E), \iota)$ such that:
+    \begin{enumerate}
+        \item $\complex(E)$ is a vector space over $\complex$.
+        \item $\iota: E \to \complex(E)$ is a $\real$-linear map.
+        \item[(U)] For any pair $(F, T)$ satisfying (1) and (2), there exists a unique $\complex$-linear map $\complex(T): \complex(E) \to F$ such that the following diagram commutes:
+        \[
+        \xymatrix{
+        \mathbb{C}(E) \ar@{->}[r]^{\mathbb{C}(T)} & F \\
+        E \ar@{->}[u]^{\iota} \ar@{->}[ru]_{T} & 
+        }
+        \]
+        \item $\complex(E) = \iota(E) \oplus i\iota(E)$ as a vector space over $\real$. Under this decomposition, elements of $\complex(E)$ are written as $x + iy$, where $x, y \in E$. 
+    \end{enumerate}
+
+    The pair $(\complex(E), \iota)$ is the \textbf{complexification} of $E$, and
+    \begin{enumerate}
+        \item[(F)] For any vector space $F$ over $\real$ and $\real$-linear map $T: E \to F$, there exists a unique $\complex$-linear map $\complex(T): \complex(E) \to \complex(F)$ such that the following diagram commutes:
+        \[
+        \xymatrix{
+        \mathbb{C}(E) \ar@{->}[r]^{\mathbb{C}(T)} & \mathbb{C}(F) \\
+        E \ar@{->}[u]^{\iota} \ar@{->}[r]_{T} & F \ar@{->}[u]_{\iota}
+        }
+        \]
+
+        which is given by
+        \[
+        \complex(T)(x + iy) = Tx + iTy
+        \]
+    \end{enumerate}
+    
+\end{definition}
+\begin{proof}
+    (1): Let $\complex(E) = E \times E$ with coordinate-wise addition. For each $a, b \in \real$ and $x, y \in E$, let
+    \[
+    (a + bi)(x, y) = (ax - by, bx + ay)
+    \]
+
+    then $\complex(E)$ is a vector space over $\complex$. 
+
+    (2): Let $\iota: E \to \complex(E)$ be defined by $\iota(x) = (x, 0)$, then $\iota$ is $\real$-linear. 
+
+    (U): Let
+    \[
+    \complex(T): \complex(E) \to F \quad (x, y) \mapsto Tx + iTy
+    \]
+
+    then $\complex(T)$ is the unique $\complex$-linear map such that the given diagram commutes. 
+
+    (F): By (U) applied to $\iota \circ T$.
+\end{proof}
+
+\begin{definition}[Complexification of Topological Vector Space]
+\label{definition:complexification-tvs}
+    Let $E$ be a TVS over $\real$, then there exists a pair $(\complex(E), \iota)$ such that:
+    \begin{enumerate}
+        \item $\complex(E)$ is a TVS over $\complex$.
+        \item $\iota: E \to \complex(E)$ is a continuous $\real$-linear map. 
+        \item[(U)] For any pair $(F, T)$ satisfying (1) and (2), there exists a unique continuous $\complex$-linear map $\complex(T): \complex(E) \to F$ such that the following diagram commutes:
+        \[
+        \xymatrix{
+        \mathbb{C}(E) \ar@{->}[r]^{\mathbb{C}(T)} & F \\
+        E \ar@{->}[u]^{\iota} \ar@{->}[ru]_{T} & 
+        }
+        \]
+        \item $\complex(E) = \iota(E) \oplus i\iota(E)$ as a TVS over $\real$.
+    \end{enumerate}
+
+    The pair $(\complex(E), \iota)$ is the \textbf{complexification} of $E$ as a topological vector space, and
+    \begin{enumerate}[start=4]
+        \item If $E$ is locally convex, then so is $\complex(E)$. 
+        \item If $E$ is normed, then $\complex(E)$ is normable, and there exists a norm $\norm{\cdot}_{\complex(E)}: \complex(E) \to [0, \infty)$ such that $\iota: E \to \complex(E)$ is isometric. 
+        \item[(F)] For any vector space $F$ over $\real$ and continuous $\real$-linear map $T: E \to F$, there exists a unique continuous $\complex$-linear map $\complex(T): \complex(E) \to \complex(F)$ such that the following diagram commutes:
+        \[
+        \xymatrix{
+        \mathbb{C}(E) \ar@{->}[r]^{\mathbb{C}(T)} & \mathbb{C}(F) \\
+        E \ar@{->}[u]^{\iota} \ar@{->}[r]_{T} & F \ar@{->}[u]_{\iota}
+        }
+        \]
+
+        which is given by
+        \[
+        \complex(T)(x + iy) = Tx + iTy
+        \]
+    \end{enumerate}
+    
+\end{definition}
+\begin{proof}
+    (1), (2): Let $(\complex(E), \iota)$ be the complexification of $E$ as a vector space, and equip it with the \hyperref[direct sum]{definition:tvs-direct-sum} topology. 
+
+    (U): By (U) of the \hyperref[complexification]{definition:complexification}, there exists a $\complex$-linear map $\complex(T): \complex(E) \to F$ such that the given diagram commutes. Since $T \circ \iota$ and $iT \circ \iota$ are continuous, $T$ is continuous by (U) of the \hyperref[direct sum]{definition:tvs-direct-sum}.
+
+    (4): By \autoref{proposition:finite-lc-product}, the direct sum and product of finitely many locally convex spaces coincide. By \autoref{proposition:lc-projective-topology}, this topology is locally convex. 
+
+    (5): Let $\norm{\cdot}_E: E \to [0, \infty)$ be the norm of $E$, and define
+    \[
+    \norm{\cdot}_{\complex(E)}: \complex(E) \to [0, \infty) \quad (x, y) \mapsto \sup_{\theta \in [0, 2\pi]}\norm{\cos(\theta)x + \sin(\theta)y}_E
+    \]
+
+    then for any $\phi \in [0, 2\pi]$ and $x, y \in E$, 
+    \begin{align*}
+        \normn{e^{i \phi}(x, y)}_{\complex(E)} &= \normn{(\cos(\phi)x - \sin(\phi)y, \sin(\phi)x + \cos(\phi)y)}_{\complex(E)} \\
+        &= \sup_{\theta \in [0, 2\pi]}\norm{\cos(\theta - \phi)x + \sin(\theta - \phi)y}_E \\
+        &= \norm{(x, y)}_{\complex(E)}
+    \end{align*}
+
+    so $\norm{(x, y)}_{\complex(E)}$ is a norm. For any $x \in E$, 
+    \[
+    \norm{\iota x}_{\complex(E)} = \sup_{\theta \in [0, 2\pi]}\norm{\cos(\theta)x}_E = \norm{x}_E \\
+    \]
+
+    Therefore $\iota: E \to \complex(E)$ is isometric. 
+
+    (F): By (U) applied to $\iota \circ T$.
+\end{proof}

src/fa/tvs/index.tex

@@ -2,6 +2,7 @@
 \label{chap:tvs}
 
 \input{./definition.tex}
+\input{./complexify.tex}
 \input{./metric.tex}
 \input{./bounded.tex}
 \input{./dual.tex}

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: