Approximation property

Remarks

Example: let $X$ have a Schauder basis $(e_n)$, and let $P_n$ be the projection $\sum_k a_k e_k \mapsto \sum_{k \le n} a_k e_k$. Each $P_n$ is continuous (a theorem of Banach) and has finite rank, $P_n x \to x$ for every $x$, and $\sup_n \norm{P_n} < \infty$ by the uniform boundedness principle. Pointwise convergence of a uniformly bounded sequence of operators is uniform on compact sets, so the $P_n$ witness the approximation property. In particular $\ell^p$ for $1 \le p < \infty$ and $c_0$ have it. Every Hilbert space has it too, using orthogonal projections onto finite-dimensional subspaces, even when it has no countable basis. Non-example: the space built in Enflo's counterexample fails the definition for some compact set $K$ and some $\varepsilon > 0$. Grothendieck's equivalent form: $X$ has the approximation property if and only if for every Banach space $Y$, every compact operator $Y \to X$ is a norm limit of finite-rank operators.