# 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 '10 at 19:46

...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