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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 '10 at 19:46

2 Answers 2

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 '10 at 19:52
Thanks, Harry; my mistake. –  Bill Johnson Feb 2 '10 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 '10 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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