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From: Joubert on 31 May 2010 20:54
The space of smooth compactly supported functions is dense in the space
of tempered distributions.

Any proof of this (or reference)?
From: Chip Eastham on 1 Jun 2010 00:11
On May 31, 8:54 pm, Joubert <trappedinthecloset9...(a)yahoo.com> wrote:
> The space of smooth compactly supported functions is dense in the space
> of tempered distributions.
>
> Any proof of this (or reference)?

The late Walter Rudin develops the theory of tempered distributions
in Chapter 7 of _Functional Analysis_, having already the tool of
"test functions" D(R^n) (smooth compactly supported functions) and
distributions D'(R^n) defined in Chapter 6.

As a sketch tempered distributions S' are the bounded linear
functionals
on the space of "rapidly decreasing functions" S, i.e. those smooth
functions on R^n for which remain bounded upon multiplication by any
polynomial on R^n.

It is clear that D(R^n) is contained in S, and Rudin's Thm. 7.10(a)
states that D(R^n) is dense in S. However your question appears to
be a different one, about the density of D(R^n) in the space of
tempered distributions S'.

Have I at least got your question clear?

regards, chip





From: David C. Ullrich on 1 Jun 2010 05:43
On Tue, 01 Jun 2010 02:54:58 +0200, Joubert
<trappedinthecloset9985(a)yahoo.com> wrote:

>The space of smooth compactly supported functions is dense in the space
>of tempered distributions.
>
>Any proof of this (or reference)?

It must be clear from (an appropriate version of) the Hahn-Banach
theorem.

Say S is the space of Schwarz functions, D is the space of smooth
functions with compact support, and S' is the space of tempered
distributions. You want to show D is dense in S'. You must be
talking about the weak* topology on S'. The dual of
(S', weak*) is S.

So: Assume that f is in S and int fg = 0 for all g in D. If
this implies that f = 0 you're done by Hahn-Banach.

Hint: if int fg = 0 for all g in D then f*g = 0 for all g in D,
where * denotes convoolution...



From: Joubert on 1 Jun 2010 06:23
Yes, it is a different one.
From: Joubert on 1 Jun 2010 06:35
01/06/2010 11.43, David C. Ullrich wrote:

> Hint: if int fg = 0 for all g in D then f*g = 0 for all g in D,
> where * denotes convoolution...

Got it. Thanks.
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