## Schwartz space dense in L2

How can one prove that $S( \mathbb{R}^{d})$, the Schwartz space, is dense in $L^{2}(\mathbb{R}^{d})$?

A slightly stronger result is that $C_{0}^{\infty}( \mathbb{R}^{d})$, the set of infinitely differentiable functions with compact support, is dense in $L^{2}(\mathbb{R}^{d})$, but I think this is probably easy to show once it has been proved that $S( \mathbb{R}^{d})$ is dense in $L^{2}(\mathbb{R}^{d})$.

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 Bump functions approximate piecewise linear functions, which are automatically dense in L^2. – Qiaochu Yuan Feb 2 2010 at 19:46

This is an easy homework problem and not appropriate for MO.

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@Bill, you should probably have posted this as a comment rather than a response. – Harry Gindi Feb 2 2010 at 19:52
Thanks, Harry; my mistake. – Bill Johnson Feb 2 2010 at 21:09
...but to make it an answer, $C_0^\infty$ is uniformly dense in the the continuous functions that vanish at infinity by Stone-Weierstrauss, whence also dense in $L_2$. – Bill Johnson Feb 2 2010 at 23:27

Typically, one proves first that simple functions are dense, then extends this characterization to step functions, then smooth functions with compact support (this involves the classic "bump" construction).

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