From 719e9a1f7aed587d033c78dd510a01e873a456e1 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Sat, 23 May 2026 21:19:04 -0400 Subject: [PATCH] Added complexification. --- src/fa/lc/inductive.tex | 38 +++++++++++- src/fa/tvs/complexify.tex | 119 ++++++++++++++++++++++++++++++++++++++ src/fa/tvs/index.tex | 1 + src/fa/tvs/inductive.tex | 52 +++++++++++++++++ 4 files changed, 207 insertions(+), 3 deletions(-) create mode 100644 src/fa/tvs/complexify.tex diff --git a/src/fa/lc/inductive.tex b/src/fa/lc/inductive.tex index e498f2d..825bc83 100644 --- a/src/fa/lc/inductive.tex +++ b/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: \[ \xymatrix{ - A \ar@{->}[r]^{T} & B \\ - A_i \ar@{->}[u]^{\iota_i} \ar@{->}[ru]_{T_i} & + E \ar@{->}[r]^{T} & F \\ + E_i \ar@{->}[u]^{\iota_i} \ar@{->}[ru]_{T_i} & } \] @@ -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: diff --git a/src/fa/tvs/complexify.tex b/src/fa/tvs/complexify.tex new file mode 100644 index 0000000..b0c455a --- /dev/null +++ b/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} diff --git a/src/fa/tvs/index.tex b/src/fa/tvs/index.tex index 1a4b5dc..22335c1 100644 --- a/src/fa/tvs/index.tex +++ b/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} diff --git a/src/fa/tvs/inductive.tex b/src/fa/tvs/inductive.tex index c28365a..412afce 100644 --- a/src/fa/tvs/inductive.tex +++ b/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: