From 8faaa8dab40f72e3cbd3fbf73dcff0b0c100a7d5 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Fri, 7 Aug 2026 15:50:54 -0400 Subject: [PATCH] Added the injective tensor product. --- src/fa/lc/beqtensor.tex | 62 ++++++++++++++++++++++++++++++++++++++++ src/fa/lc/index.tex | 1 + src/fa/lc/tensor.tex | 2 +- src/fa/lp/definition.tex | 2 +- src/topology/main/c0.tex | 39 +++++++++++++++++++++++++ 5 files changed, 104 insertions(+), 2 deletions(-) create mode 100644 src/fa/lc/beqtensor.tex diff --git a/src/fa/lc/beqtensor.tex b/src/fa/lc/beqtensor.tex new file mode 100644 index 0000000..c516ead --- /dev/null +++ b/src/fa/lc/beqtensor.tex @@ -0,0 +1,62 @@ +\section{The Injective Tensor Product} +\label{section:beq-tensor-product} + +\begin{lemma} +\label{lemma:tensor-product-dual-injection} + Let $E, F$ be locally convex spaces over $K \in \RC$, then the canonical map + \[ + E \otimes F \to L^2(E^*, F^*; K) \quad (x \otimes y)(\phi, \psi) = \dpn{x, \phi}{E}\dpn{y, \psi}{F} + \] + + is injective. +\end{lemma} +\begin{proof} + Let $\lambda = \sum_{j = 1}^n x_j \otimes y_j \in E \otimes F$ such that $\lambda(\phi, \psi) = 0$ for all $\phi \in E^*$ and $\psi \in F^*$. Assume without loss of generality that $\bracsn{x_j}_1^n \subset E$ is a linearly independent set. Fix $\phi \in E^*$, then for every $\psi \in F^*$, + \[ + 0 = \lambda(\phi, \psi) = \sum_{j = 1}^n \dpn{x_j, \phi}{E} \dpn{y_j, \psi}{F} = \angles{\sum_{j = 1}^n x_j\dpn{y_j, \psi}{F}, \phi}_E + \] + + By the \hyperref[Hahn-Banach Theorem]{proposition:hahn-banach-utility}, $\sum_{j = 1}^n x_j\dpn{y_j, \psi}{F} = 0$. Since $\bracs{x_j}_1^n \subset E$ is linearly independent, $\dpn{y_j, \psi}{F} = 0$ for each $1 \le j \le n$. + + As the above holds for all $\psi \in F^*$, the \hyperref[Hahn-Banach Theorem]{proposition:hahn-banach-utility} implies that $y_j = 0$ for each $1 \le j \le n$. +\end{proof} + +\begin{definition}[Bi-Equicontinuous Convergence] +\label{definition:beq} + Let $E, F$ be locally convex spaces over $K \in \RC$ and + \[ + \sigma = \bracsn{S \times T| S \subset E^* \text{ equipcontinuous}, T \subset F^* \text{ equicontinuous}} + \] + + be the product of all equicontinuous subsets of $E^*$ and $F^*$, then the $\sigma$-topology on $L^2(E, F; K)$ is the \textbf{topology of bi-equicontinuous convergence} on $L^2(E, F; K)$. Under this topology, $L^2(E, F; K)$ is a locally convex space. +\end{definition} +\begin{proof} + By \autoref{proposition:lc-spaces-linear-map}, the $\sigma$-topology is a vector space topology. +\end{proof} + + + +\begin{definition}[Injective Tensor Product] +\label{definition:beq-tensor-product} + Let $E, F$ be locally convex spaces over $K \in \RC$, and identify $E \otimes F$ as a subspace of $L^2(E, F; K)$, then $E \otimes F$ equipped with the topology of bi-equicontinuous convergence is the \textbf{injective tensor product} of $E$ and $F$, denoted $E \otimes_\eps F$. + + The Hausdorff completion $E \wh{\otimes}_\eps F$ of $E \otimes_\eps F$ is the \textbf{injective completion} of $E$ and $F$. +\end{definition} + +\begin{definition}[Injective Cross Seminorm] +\label{definition:beq-cross-norm} + Let $E, F$ be locally convex spaces over $K \in \RC$, $S \subset E^*$ and $T \subset F^*$ be equicontinuous, and $\lambda = \sum_{j = 1}^n x_j \otimes y_j \in E \otimes F$, then + \[ + [\lambda]_{S, T} = \braks{\sum_{j = 1}^n x_j \otimes y_j}_{S, T} = \sup_{\phi \in S, \psi \in T} \abs{\sum_{j = 1}^n \dpn{x_j, \phi}{E}\dpn{y_j, \psi}{F}} + \] + + is the \textbf{injective cross seminorm} of $\lambda$ with respect to $S$ and $T$. The family of all such norms induces the topology on $E \otimes_\eps F$. In particular, if $E, F$ are normed vector spaces, then + \[ + \norm{\lambda}_{E \otimes_\eps F} = \norm{\sum_{j = 1}^n x_j \otimes y_j} = \sup_{\substack{\phi \in E^* \\ \norm{\phi}_{E^*} \le 1}}\sup_{\substack{\psi \in E \\ \norm{\psi}_{F^*} \le 1}}\abs{\sum_{j = 1}^n \dpn{x_j, \phi}{E}\dpn{y_j, \psi}{F}} + \] + + is \textit{the} \textbf{injective cross norm} on $E \otimes_\eps F$. +\end{definition} + + + diff --git a/src/fa/lc/index.tex b/src/fa/lc/index.tex index 8d5eed1..de9d98f 100644 --- a/src/fa/lc/index.tex +++ b/src/fa/lc/index.tex @@ -13,5 +13,6 @@ \input{./hahn-banach.tex} \input{./spaces-of-linear.tex} \input{./tensor.tex} +\input{./beqtensor.tex} \input{./nuclear.tex} \input{./nuclear-space.tex} diff --git a/src/fa/lc/tensor.tex b/src/fa/lc/tensor.tex index f24dd03..dfee3ce 100644 --- a/src/fa/lc/tensor.tex +++ b/src/fa/lc/tensor.tex @@ -26,7 +26,7 @@ \end{enumerate} - The space $E \otimes_\pi F$ is the \textbf{projective tensor product} of $E$ and $F$, and the mapping $\iota \in L^2(E, F; E \otimes_\pi F)$ is the \textbf{canonical embedding}. + The space $E \otimes_\pi F$ is the \textbf{projective tensor product} of $E$ and $F$, and the mapping $\iota \in L^2(E, F; E \otimes_\pi F)$ is the canonical embedding. The space $E \wh{\otimes}_\pi F$ denotes the Hausdorff completion of $E \otimes_\pi F$. \end{definition} diff --git a/src/fa/lp/definition.tex b/src/fa/lp/definition.tex index c0e5922..caea3f3 100644 --- a/src/fa/lp/definition.tex +++ b/src/fa/lp/definition.tex @@ -132,7 +132,7 @@ \begin{theorem}[{{\cite[III.6.5]{SchaeferWolff}}}] \label{theorem:l1-tensor} - Let $(X, \cm, \mu)$ be a measure space and $E$ be a Banach space over $K \in \RC$, then the map $L^1(X; K) \td{\otimes}_\mu E \to L^1(X; E)$ defined by extending + Let $(X, \cm, \mu)$ be a measure space and $E$ be a Banach space over $K \in \RC$, then the map $L^1(X; K) \td{\otimes}_\pi E \to L^1(X; E)$ defined by extending \[ L^1(X; K) \times E \to L^1(X; E) \quad f \otimes x \mapsto x \cdot f \] diff --git a/src/topology/main/c0.tex b/src/topology/main/c0.tex index 2d412f4..045c45f 100644 --- a/src/topology/main/c0.tex +++ b/src/topology/main/c0.tex @@ -64,4 +64,43 @@ For details regarding the complex-valued cased, in particular its properties as \end{proof} +\begin{theorem} +\label{theorem:c0-beq-tensor} + Let $X$ be a LCH space and $E$ be a complete locally convex space over $K \in \RC$, then the canonical map + \[ + C_0(X; K) \otimes_\eps E \to C_0(X; E) \quad \sum_{j = 1}^n f_j \otimes y_j \mapsto \sum_{j = 1}^n y_j \cdot f_j + \] + + extends into an isomorphism between $C_0(X; K) \wh{\otimes}_\eps E$ and $C_0(X; E)$. Moreover, if $E$ is a Banach space, then the isomorphism is an isometry. +\end{theorem} +\begin{proof} + To see that the canonical map is continuous, let $\rho: E \to [0, \infty)$ be a continuous seminorm on $E$. By the \hyperref[Hahn-Banach Theorem]{proposition:hahn-banach-utility}, there exists an equicontinuous family $T \subset E^*$ such that for each $y \in E$, $\rho(y) = \sup_{\phi \in T}|\dpn{y, \phi}{E}|$. + + Let $\lambda = \sum_{j = 1}^n f_j \otimes y_j \in C_0(X; K) \otimes_\eps E$, then + \[ + \sup_{x \in X}\rho(\lambda(x)) = \sup_{x \in X}\sup_{\phi \in T}|\dpn{\lambda(x), \phi}{E}| = \sup_{x \in X}\sup_{\phi \in T}\abs{\sum_{j = 1}^n f_j(x) \dpn{y_j, \phi}{E}} + \] + + Since the evaluation maps $\bracsn{\pi_x: C_0(X; K) \to K|x \in X}$ are equicontinuous, the uniform seminorm on $C_0(X; E)$ with respect to $\rho$ is bounded above by a cross seminorm of the injective tensor product. Thus the inclusion is continuous. + + On the other hand, let $T \subset E^*$ be equicontinuous, then there exists a continuous seminorm $\rho: E \to [0, \infty)$ such that $|\phi| \le \rho$ for all $\phi \in T$. In which case, for any $\lambda = \sum_{j = 1}^n f_j \otimes y_j \in C_0(X; K) \otimes_\eps E$, $I \in C_0(X; K)^*$, and $\phi \in T$, + \begin{align*} + \abs{\sum_{j = 1}^n \dpn{f_j, I}{C_0(X; K)}\dpn{y_j, \phi}{E}} &= \abs{\angles{\sum_{j = 1}^nf_j\dpn{y_j, \phi}{E}, I}_{C_0(X; K)}} \\ + &\le \norm{I}_{C_0(X; K)} \cdot \sup_{x \in X} \abs{\angles{\sum_{j = 1}^n f_j(x) y_j, \phi}_{E}} \\ + &\le \norm{I}_{C_0(X; K)} \cdot \sup_{x \in X} \rho(\lambda(x)) + \end{align*} + + Hence the cross seminorm corresponding to $B_{C_0(X; K)^*}(0, 1)$ and $T$ is bounded above by the uniform seminorm on $C_0(X; E)$ with respect to $\rho$, so the inclusion is an embedding. + + Finally, by \autoref{proposition:c0-properties}, $C_0(X; K)$ is complete. By \autoref{proposition:c0-tensor}, $C_0(X; K) \otimes_\eps E$ is dense in $C_0(X; E)$. Therefore the canonical map extends to an isomorphism through the \hyperref[Linear Extension Theorem]{theorem:linear-extension-theorem-tvs}. +\end{proof} + +\begin{corollary} +\label{corollary:c0-seq-beq-tensor} + Let $I$ be a set and $E$ be a complete locally convex space over $K \in \RC$, then $c_0(I; K) \wh \otimes_\eps E \iso c_0(I; E)$. +\end{corollary} +\begin{proof} + By \autoref{theorem:c0-beq-tensor}. +\end{proof} +