Added another corollary.

src/fa/norm/ap.tex

@@ -203,9 +203,9 @@
     Let $E$ be a Banach space over $K \in \RC$. If $E^*$ has the approximation property, then so does $E$. 
 \end{corollary}
 \begin{proof}
-    By (3) of \autoref{theorem:approximation-property-dual}, for any Banach space $F$, the canonical map from $F^{*} \wh \otimes_\pi E^*$ to $L(F; E^*)$ is injective. Since $L(F; E^*)$ is canonically isomorphic to $L(E; F^*)$, the canonical map from $F^* \otimes_\pi E^*$ to $L(E; F^*)$ is then injective. 
+    By (3) of \autoref{theorem:approximation-property-dual}, for any Banach space $F$, the canonical map from $F^{*} \wh \otimes_\pi E^*$ to $L(F; E^*)$ is injective. Since $L(F; E^*)$ is canonically isomorphic to $L(E; F^*)$, the canonical map from $F^* \wh \otimes_\pi E^*$ to $L(E; F^*)$ is then injective. 
 
-    Now, let $F := E^*$, then the above yields an injection $E^{**} \wh \otimes_\pi E^*$ to $L(E; E^{**})$. Let $T \in E \wh \otimes_\pi E^*$. By \autoref{theorem:metrisable-tensor-product}, there exists $\seq{x_n} \subset E$ and $\seq{\phi_n} \subset E^*$ such that $\sum_{n \in \natp}\norm{x_n}_{E}\norm{\phi_n}_{E^*} < \infty$ and $T = \sum_{n =1}^\infty x_n \otimes \phi_n$. As an operator, for each $x \in E$, 
+    Now, let $F := E^*$, then the above yields an injection from $E^{**} \wh \otimes_\pi E^*$ to $L(E; E^{**})$. Let $T \in E \wh \otimes_\pi E^*$. By \autoref{theorem:metrisable-tensor-product}, there exists $\seq{x_n} \subset E$ and $\seq{\phi_n} \subset E^*$ such that $\sum_{n \in \natp}\norm{x_n}_{E}\norm{\phi_n}_{E^*} < \infty$ and $T = \sum_{n =1}^\infty x_n \otimes \phi_n$. As an operator, for each $x \in E$, 
     \[
     Tx = \sum_{n = 1}^\infty x_n \dpn{x, \phi_n}{E} \in E
     \]
@@ -225,12 +225,20 @@
 \begin{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 \autoref{theorem:approximation-property-dual}. 
 
-    If $E^*$ has the approximation property, then the canonical map
+    If $E^*$ has the approximation property, then by (3) of \autoref{theorem:approximation-property-dual}, the canonical map
     \[
-    F^{**} \wh \otimes E^{*} \to N(F^*; E^*) \iso N(E; F^{**})
+    F^{**} \wh \otimes_\pi E^{*} \to N(F^*; E^*) \iso N(E; F^{**})
     \]
     
-    is injective. Restricting to $F \wh \otimes E^*$ yields an injection into $N(E; F^{**})$. 
+    is injective. Restricting to $F \wh \otimes_\pi E^*$ yields an injection into $N(E; F)$. 
+\end{proof}
+
+\begin{corollary}[Existence of Continuous Trace]
+\label{corollary:trace-existence-approx}
+    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 $x \in E$, $\tr(\phi \otimes y) = \dpn{y, \phi}{E}$.
+\end{corollary}
+\begin{proof}
+    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}. 
 \end{proof}