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\begin{corollary}[Existence of Continuous Trace]
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\label{corollary:trace-existence-approx}
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Let $E$ a Banach spaces 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 $x \in E$, $\tr(\phi \otimes y) = \dpn{y, \phi}{E}$.
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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 $x \in E$, $\tr(\phi \otimes y) = \dpn{y, \phi}{E}$.
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\end{corollary}
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\begin{proof}
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By (U) of the \hyperref[projective tensor product]{definition:projective-tensor-product} and the isomorphism $E^* \wh \otimes_\pi E \iso N(E; E)$ from \autoref{corollary:approximation-property-nuclear}.
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