Corollary 13.8.7.label Let $E$ and $F$ be Banach spaces over $K \in \RC$. If $E^{*}$ or $F$ has the approximation property, then the canonical map
\[E^{*} \otimes_{\pi} F \to N(E; F) \quad \braks{\sum_{j = 1}^n \phi_j \otimes y_j}(x) = \sum_{j = 1}^{n} y_{j} \dpn{x, \phi_j}{E}\]
extends to an isometric isomorphism.
Proof. If $F$ has the approximation property, then the canonical map $E^{*} \wh \otimes_{\pi} F \to N(E; F)$ is injective by (3) of Theorem 13.8.5.
If $E^{*}$ has the approximation property, then by (3) of Theorem 13.8.5, the canonical map
\[F^{**}\wh \otimes_{\pi} E^{*}\to N(F^{*}; E^{*}) \iso N(E; F^{**})\]
is injective. Restricting to $F \wh \otimes_{\pi} E^{*}$ yields an injection into $N(E; F)$.$\square$
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