Corollary 13.8.8 (Existence of Continuous Trace).label Let $E$ a Banach space over $K \in \RC$ with the approximation property, then there exists a unique $\tr \in N(E; E)^{*}$ such that for each $\phi \in E^{*}$ and $x \in E$, $\tr(\phi \otimes y) = \dpn{y, \phi}{E}$.
Proof. By (U) of the projective tensor product and the isomorphism $E^{*} \wh \otimes_{\pi} E \iso N(E; E)$ from Corollary 13.8.7.$\square$
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