Continuously smooth functions and Lp space

[sdickey9480]

How might I prove the following?

1) If f ∈ C(Rn) and f has compact support, then f ∈ Lp(Rn) for every 1 ≤ p ≤ ∞.

2) If f ∈ C(Rn), then f ∈ Lp_{loc}(Rn) for every 1 ≤ p < ∞.

(Where C(Rn) is the space of continuous functions on Rn)

 

[theorem4.5.9]

What is special about continuous functions on compact domains? (apply this to both parts)

 

[sdickey9480]

They are uniformly continuous

 

[micromass]

They are uniformly continuous

Yes, but what else?? Can functions on compact domains grow arbitrarly large??

 

[sdickey9480]

No, b/c they are bounded.

 

[micromass]

Yes. So use that to find an estimate for the integral

[tex]\int_{\mathbb{R}^n} |f|^p[/tex]

 

[sdickey9480]

Not following. Could you provide a little more detail?

 

[sdickey9480]

So since we are dealing with bounded continuous functions, by finding an estimate to the aforementioned integral this will in turn justify that f must also belong to Lp?

 

[micromass]

If we can show that

[tex]\int_{\mathbb{R}^n} |f|^p [/tex]

is not infinite, then the function is in [itex]L^p[/itex]. So we must find some real number C such that

[tex]\int_{\mathbb{R}^n} |f|^p\leq C[/tex]

 

[Office_Shredder]

It might help to first answer the (hopefully easy) question

If f(x) is a continuous function on the reals, is [tex] \int_a^b |f(x)|^p dx[/tex] ever infinite?

If you can figure out the answer to this you can solve micromass's question

 

[sdickey9480]

Won't the integral always be finite? Hence do we even need to find a particular C, M, etc.? Can't we just assume there exists one, again b/c integral is finite? Am I using this same idea for 1) and 2)?

 

[micromass]

Why do you think the integral will always be finite?? Where did you use compactness??

 

[sdickey9480]

Finite because it's bounded on a compact interval?

 

[sdickey9480]

Can I just prove the space of smooth continuous functions is dense in Lp, hence if a function belongs to C(R) it belongs to Lp(R). If so, what's the difference in the proof of 1) & 2)?