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  "body": "\n## Summary\n\nA Banach space has the approximation property when the identity operator can be approximated, uniformly on each compact set, by bounded operators of finite rank.\nGrothendieck showed that this is equivalent to a statement on operators.\nFor every Banach space $Y$, every compact operator from $Y$ into the space is a norm limit of finite-rank operators.\nSpaces with a Schauder basis, such as $\\ell^p$ for $1 \\le p \u003c \\infty$ and $c_0$, have the property.\n[[Enflo's counterexample|ZKP5KGFWCK]] shows that some separable Banach spaces do not.\n\n## Explanations\n\n### L1 human\n\nFinite-rank operators are the simplest operators: their values lie in a finite-dimensional space, so they are matrices in disguise.\nThe approximation property says that the identity map can be imitated by such operators on any compact set, up to any small error.\nIn $\\ell^2$, the map that keeps the first $n$ coordinates of a sequence and replaces the rest by $0$ does this for $n$ large.\n",
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