Recent content by 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)?

Finite because it's bounded on a compact interval?

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)?

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?

Not following. Could you provide a little more detail?

No, b/c they are bounded.

They are uniformly continuous

Given any x_0 ∈ Rn, the delta function is the distribution, δ_{x_0} :D(Rn)→C given by the evaluation of a test function at x_0: ⟨δ_{x_0} , φ⟩ = φ(x_0)

I realize it's not a function in the classical sense, but how would one show that the delta dirac function is a distribution i.e. how do I show it's continuous and linear given that it's not truly a function?

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)