• Support PF! Buy your school textbooks, materials and every day products Here!

Lp Subspaces

  • Thread starter Kreizhn
  • Start date
  • #1
743
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?
 

Answers and Replies

  • #2
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,916
19
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))
 
Last edited:

Related Threads on Lp Subspaces

  • Last Post
Replies
9
Views
696
  • Last Post
Replies
2
Views
683
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
10
Views
2K
  • Last Post
Replies
6
Views
8K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
0
Views
6K
Replies
2
Views
811
  • Last Post
Replies
4
Views
2K
Top