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  "body": "\n## Summary\n\nA densely defined operator is an unbounded operator whose domain is a dense subspace of the space.\nDensity of the domain is the condition under which the adjoint of an operator on a Hilbert space is well defined.\nDifferential operators on $L^2$ spaces, defined on smooth functions, are the standard examples.\n\n## Explanations\n\n### L1 human\n\nAn unbounded operator often cannot act on every vector, but it is good if it acts on vectors that come arbitrarily close to any vector.\nThat is what a dense domain gives.\nFor example, differentiation acts on polynomials, and every continuous function on $[0, 1]$ is a uniform limit of polynomials.\nA dense domain also makes the operator determined by enough data to define its adjoint.\n",
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