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Lp Subspaces

  1. Mar 2, 2009 #1
    I've been given an assignment question, where I've been asked to identify [itex] L_P[-n, n] [/itex] as a subpsace of [itex] L_p(\mathbb R) [/itex] in the obvious way. It seems to me though that this may be backwards, as if [itex] f \in L_p( \mathbb R) [/itex] then its p-power should also be integrable on any subspace of [itex] \mathbb R [/itex]. However, a function integrable on [-n,n] may not be p-power integrable on all of R. Do I have this backwards?
     
  2. jcsd
  3. Mar 2, 2009 #2

    Hurkyl

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    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))
     
    Last edited: Mar 2, 2009
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