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  "body": "\n## Summary\n\nA symmetric operator is a densely defined linear operator $T$ on a Hilbert space with $\\ip{Tx}{y} = \\ip{x}{Ty}$ for all $x, y$ in its domain.\nFor bounded operators defined on the whole space, symmetric is the same as self-adjoint.\nFor unbounded operators, such as differential operators, symmetry is weaker than self-adjointness.\nThe gap between the two is the subject of the theory of self-adjoint extensions.\n\n## Explanations\n\n### L1 human\n\nA symmetric operator can move from one side of the inner product to the other, as a symmetric matrix does.\nThink of $-i \\tfrac{d}{dx}$ acting on smooth functions that vanish near the ends of an interval: integration by parts moves the derivative across the inner product with no boundary terms.\nThe operator is symmetric, but it is only defined on a small set of functions.\nChoosing a larger domain on which it becomes self-adjoint is a real question, and the answer depends on boundary conditions.\n",
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