# Lp Subspaces

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?

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Hurkyl
Staff Emeritus
Gold Member
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))

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