Corollary 13.9.3.label Let $E$ be a separable Banach space over $K \in \RC$ with a Schauder basis, then $E$ enjoys the approximation property.
Proof. By Proposition 13.9.2, there exists $\seq{P_N}\subset L(E; E)$ such that $P_{N} \to \text{Id}$ uniformly on compact sets as $N \to \infty$.$\square$
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