Welcome to MathOverflow

Question

-2

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}) $.

flag	asked Feb 2 2010 at 19:42 Jimmy 1●1

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

Bill Johnson · Answer 1 · 2010-02-02 19:49:29Z UTC

2

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

answered Feb 2 2010 at 19:49

Bill Johnson
14.4k●1●33●60

3

@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

3

...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

Harry Gindi · Answer 2 · 2010-02-02 19:47:48Z UTC

0

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).

answered Feb 2 2010 at 19:47

Harry Gindi
8,868●2●33●96

Schwartz space dense in L2

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

2 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.