Definition 13.8.1 (Approximation Property).label Let $E$ be a separated locally convex space over $K \in \RC$, then the following are equivalent:
- (1)
The closure of $E^{*} \otimes E$ in $L_{c}(E; E)$ contains the identity map.
- (2)
$E^{*} \otimes E$ is dense in $L_{c}(E; E)$.
- (3)
For each locally convex space $F$ over $K$, $E^{*} \otimes F$ is dense in $L_{c}(E; F)$.
- (4)
For each locally convex space $F$ over $K$, $F^{*} \otimes E$ is dense in $L_{c}(F; E)$.
If the above holds, then $E$ has the approximation property.
Proof. (1) $\Rightarrow$ (2): Let $T \in L_{c}(E; E)$ and $A \subset E$ be precompact, then $T(A)$ is also precompact by Proposition 6.4.2. Let $U \in \cn_{E}(0)$, then there exists $S \in E^{*} \otimes E$ such that $Sx - x \in U$ for all $x \in T(A)$. In which case, $STx - Tx \in U$ for all $x \in A$.
(1) $\Rightarrow$ (3): Let $T \in L_{c}(E; F)$ and $A \subset E$ be precompact, and $U \in \cn_{F}(0)$, then there exists $S \in E^{*} \otimes E$ such that $Sx - x \in T^{-1}(U)$ for all $x \in A$. In which case, $TS \in E^{*} \otimes F$ and $TSx - Tx \in U$ for all $x \in A$.
(1) $\Rightarrow$ (4): Let $T \in L_{c}(F; E)$ and $A \subset F$ be precompact, then $T(A)$ is also precompact. Let $U \in \cn_{E}(0)$, then there exists $S \in E^{*} \otimes E$ such that $Sx - x \in U$ for all $x \in T(A)$. Thus $STx - Tx \in U$ for all $x \in A$.$\square$
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