Came across this in a discussion of essential self-adjointedness:(adsbygoogle = window.adsbygoogle || []).push({});

LetPbe the densely defined operator with Dom(P) = [itex]C^{\infty}_c (\mathbb{R}) \subset L^2 ( \mathbb{R} )[/itex] and given by [itex]Pf = -i df/dx[/itex]. ThenPis essentially self-adjoint.

It is the [itex]C^{\infty}_c (\mathbb{R}) \subset L^2 ( \mathbb{R} )[/itex] part that bothered me. Doesn't this say the space of smooth complex functions on R is contained in the space of square-integrable functions on R? But isn't, say, f(x) = x an element of [itex]C^{\infty}_c (\mathbb{R}) [/itex] ? And isn't f(x) = x not square-integrable on R?

I am thinking maybe I don't know what the author means by the symbols [itex]C^{\infty}_c (\mathbb{R}) [/itex] or [itex]L^2 ( \mathbb{R} )[/itex]

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# Are all smooth functions square-integrable?

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