13.12 The Injective Tensor Product
Lemma 13.12.1.label Let $E, F$ be locally convex spaces over $K \in \RC$, then the canonical map
is injective.
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 $\psi \in F^{*}$,
By the Hahn-Banach Theorem, $\sum_{j = 1}^{n} x_{j}\dpn{y_j, \psi}{F}= 0$. 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 Hahn-Banach Theorem implies that $y_{j} = 0$ for each $1 \le j \le n$.$\square$
Definition 13.12.2 (Bi-Equicontinuous Convergence).label Let $E, F$ be locally convex spaces over $K \in \RC$ and
be the product of all equicontinuous subsets of $E^{*}$ and $F^{*}$, then the $\sigma$-topology on $L^{2}(E, F; K)$ is the topology of bi-equicontinuous convergence on $L^{2}(E, F; K)$. Under this topology, $L^{2}(E, F; K)$ is a locally convex space.
Proof. By Proposition 13.10.1, the $\sigma$-topology is a vector space topology.$\square$
Definition 13.12.3 (Injective Tensor Product).label 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 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 injective completion of $E$ and $F$.
Definition 13.12.4 (Injective Cross Seminorm).label 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
is the 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
is the injective cross norm on $E \otimes_{\eps} F$.
Definition 13.12.5 (Integral Bilinear Form).label Let $E, F$ be locally convex spaces over $K \in \RC$ and $\lambda \in L^{2}(E, F; K)$ be a bilinear form, then $\lambda$ is integral if there exists equicontinuous subsets $S \subset E^{*}$ and $T \subset F^{*}$, and a Radon measure $\mu \in M_{R}(S \times T; K)$ such that
for all $x, y \in E$.
The set $I(E, F)$ is the space of integral bilinear forms on $E$ and $F$.
Theorem 13.12.6.label Let $E, F$ be locally convex spaces over $K \in \RC$, then $(E \wh \otimes_{\eps} F)^{*} = I(E, F)$.
Proof. Let $\lambda \in (E \wh \otimes_{\eps} F)^{*}$, then there exists equicontinuous subsets $S \subset E^{*}$ and $T \subset F^{*}$ such that for each $x \in E$ and $y \in F$,
For any $(x, y) \in E \times F$ and $(\phi, \psi) \in S \times T$, let $f_{xy}(\phi, \psi) = \dpn{x, \phi}{E}\dpn{y, \psi}{F}$. By the Hahn-Banach Theorem, there exists $\Lambda \in C(S \times T; K)^{*}$ such that the following diagram commutes:
Now, since $S$ and $T$ are equicontinuous, using the Banach-Alaoglu Theorem, assume without loss of generality that $S$ and $T$ are weak*-compact. In which case, by the Riesz Representation Theorem, there exists $\mu \in M_{R}(S \times T; K)$ such that $\Lambda(f) = \int_{S \times T}f d\mu$ for all $f \in C(S \times T; K)$. Therefore
for all $(x, y) \in E \times F$, and $\lambda \in I(E; F)$.$\square$
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