13.8 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 $x \in A$.$\square$

Proof. Let $V \in \fB$, $\pi_{V}: E \to \wh E_{V}$ be the canonical projection, and $A \subset E$ be precompact, then $\pi_{V}(A)$ is precompact as well. Since $\wh E_{V}$ has the approximation property, there exists $T \in E_{V}^{*} \otimes \wh E_{V}$ such that $Tx - x \in \pi_{V}(V)$ for all $x \in \pi_{V}(A)$. As $E_{V}$ is dense in $\wh E_{V}$, there exists $S \in E_{V}^{*} \otimes E_{V}$ such that $Sx - Tx \in \pi_{V}(V)$ for all $x \in \pi_{V}(A)$. In which case, $Sx - x \in 2\pi_{V}(V)$ for all $x \in \pi_{V}(A)$, and $S \circ \pi_{V}(x) - \pi_{V}(x) \in 2\pi_{V}(V)$.

Write $S = \sum_{j = 1}^{n} \phi_{j} \otimes y_{j}$. For each $1 \le j \le n$, choose any representative $x_{j} \in \pi_{V}^{-1}(y_{j})$, then for any $x \in A$,

Finally, since $\ker(\pi_{V}) = \bigcap_{\lambda > 0}\lambda V \subset V$, $x - \sum_{j = 1}^{n} x_{j}\dpn{x, \phi_j \circ \pi_V}{E}\in -3V = 3V$. Therefore if $R = \sum_{j = 1}^{n} (\phi_{j} \circ \pi_{V}) \otimes x_{j} \in E^{*} \otimes E$, then $Rx - x \in 3V$.$\square$