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  "body": "\n## Definition\n\nLet $X$ be a [[Banach space|06GZ956657]] over $\\R$ or $\\C$.\nA [[continuous linear map|03TM135G0N]] $T \\colon X \\to X$ has *finite rank* if its image $T(X)$ is finite-dimensional.\nThe space $X$ has the *approximation property* if the following holds.\nFor every [[compact|0R6EWFKRCF]] set $K \\subseteq X$ and every $\\varepsilon \u003e 0$ there is a continuous linear map $T \\colon X \\to X$ of finite rank such that $\\norm{Tx - x} \u003c \\varepsilon$ for all $x \\in K$.\n\n## Remarks\n\nExample: let $X$ have a Schauder basis $(e_n)$, and let $P_n$ be the projection $\\sum_k a_k e_k \\mapsto \\sum_{k \\le n} a_k e_k$.\nEach $P_n$ is continuous (a theorem of Banach) and has finite rank, $P_n x \\to x$ for every $x$, and $\\sup_n \\norm{P_n} \u003c \\infty$ by the [[uniform boundedness principle|YG0042V8CJ]].\nPointwise convergence of a uniformly bounded sequence of operators is uniform on compact sets, so the $P_n$ witness the approximation property.\nIn particular $\\ell^p$ for $1 \\le p \u003c \\infty$ and $c_0$ have it.\nEvery [[Hilbert space|HM2HHE3J51]] has it too, using orthogonal projections onto finite-dimensional subspaces, even when it has no countable basis.\nNon-example: the space built in [[Enflo's counterexample|ZKP5KGFWCK]] fails the definition for some compact set $K$ and some $\\varepsilon \u003e 0$.\nGrothendieck'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.\n",
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