Identifying L_p[-n,n] as a Subspace of L_p(R)

[Kreizhn]

I've been given an assignment question, where I've been asked to identify L_P[-n, n] as a subpsace of L_p(\mathbb R) in the obvious way. It seems to me though that this may be backwards, as if f \in L_p( \mathbb R) then its p-power should also be integrable on any subspace of \mathbb R. However, a function integrable on [-n,n] may not be p-power integrable on all of R. Do I have this backwards?

 

[Hurkyl]

a function integrable on [-n,n] may not be p-power integrable on all of R.

Wait a minute -- there isn't a restriction map from {functions on [-n,n]} to {functions on R}... What exactly do you mean here, and is it really what you want?



Incidentally, note that while you defined a map Lp(R) --> Lp[-n,n], it doesn't identify Lp(R) with a subspace of Lp[-n,n], because the map isn't injective.

(But even if you had an injective map, it's perfectly okay for there to exist maps in both directions that make Lp(R) a subspace of Lp[-n,n], and Lp[-n,n] a subspace of Lp(R))