Lemma 12.12.2.label Let $E, F$ be Banach spaces, $E^{*}$ be the dual of $E$, equipped with the uniform topology, then the mapping
\[E^{*} \otimes F \to N(E; F) \quad \sum_{j = 1}^{n} \phi_{j} \otimes y_{j} \mapsto \sum_{j = 1}^{n} y_{j}\dpn{\cdot, \phi_j}{E}\]
extends continuously into a surjective linear map $E^{*} \tilde \otimes_{\pi} F \to N(E; F)$.
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