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  "body": "\n## Summary\n\nA closed operator is an unbounded operator whose graph is a closed subset of the product space.\nClosedness is the weak substitute for continuity that most differential operators have.\nOn Banach spaces, the closed graph theorem says that a closed operator defined on the whole space is bounded.\nSo interesting closed operators are usually defined on a proper subspace.\n\n## Explanations\n\n### L1 human\n\nA continuous map sends convergent sequences to convergent sequences.\nA closed operator only promises something weaker: if $x_n \\to x$ and the outputs $T x_n$ also converge, then $x$ is in the domain and the limit of the outputs is $Tx$.\nDifferentiation is a typical example: if functions converge and their derivatives converge uniformly, then the limit is differentiable with the limit derivative.\nSo the operator never gives a wrong answer in the limit, even though it may blow up.\n",
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