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support of a function

Post Icon Posted: Submitted by richardhp on 1 July 2008 - 8:28am.

Joined: 2007-10-01
Posts: 175

So I found this definition:

For $ f: \mathbb{R} \rightarrow \mathbb{C} \, $ define $ support(f) := \overline {\{ x \in \mathbb{R} | f(x) \ne 0 \} } $

Then we say $ C_0(\mathbb{R}) $ is the set of continuous functions with compact support. I was wondering what the significance of this set is since it is used to contruct the $ L^p $ spaces I don't see why you can't just use continuous functions instead.

‹ Maths/Physics Conjugate Numbers ›
Post Icon Posted: 1 July 2008 - 1:13pm

Joined: 2006-10-10
Posts: 519

Is that R/C with the Euclidean topology?
Post Icon Posted: 1 July 2008 - 3:38pm

Joined: 2007-10-03
Posts: 390

I think that as you're constructing all your $ L^p $ spaces, you need functions that are going to be p-integrable for all p, and using the result that a continuous on a compact set is bounded, and the fact that compact sets in R are also bounded, your integrals are going to be bounded. If you're just dealing with continuous functions, you could have them tend to a non-zero limit at $ +\infty $ for example, and then the integrals won't work most of the time (especially if the limit is greater than 1).
Post Icon Posted: 1 July 2008 - 8:34pm

Joined: 2006-10-10
Posts: 519

Well yeah, that's basically what I said
Post Icon Posted: 1 July 2008 - 8:42pm

Joined: 2007-10-03
Posts: 390

Oh yes sorry, now I see what you meant, I didn't quite understand at the time.
Post Icon Posted: 1 July 2008 - 9:51pm

Joined: 2007-10-01
Posts: 175

Thanks Sam that makes a lot of sense.

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