Theorem 5.21.4.label Let $X$ be a LCH space and $E$ be a complete locally convex space over $K \in \RC$, then the canonical map
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.
Proof. To see that the canonical map is continuous, let $\rho: E \to [0, \infty)$ be a continuous seminorm on $E$. By the Hahn-Banach Theorem, 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
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 $\phi \in T$,
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 Proposition 5.21.2, $C_{0}(X; K)$ is complete. By Proposition 5.21.3, $C_{0}(X; K) \otimes_{\eps} E$ is dense in $C_{0}(X; E)$. Therefore the canonical map extends to an isomorphism through the Linear Extension Theorem.$\square$
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