Added the injective tensor product.

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}
+
+
+

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}

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}

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
     \[
     L^1(X; K) \times E \to L^1(X; E) \quad f \otimes x \mapsto x \cdot f
     \]

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}
+