Lemma 13.12.1.label 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.
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 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 $1 \le j \le n$.$\square$
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