A characterization of L^p?

  • Thread starter Goklayeh
  • Start date
  • #1
17
0
Could someone confirm or refute the following statement?

[tex]f \in L^p\left(X, \mu\right) \: \Leftrightarrow \: \int_X{\lvert fg \rvert d\mu < \infty\: \forall g \in L^q\left(X, \mu\right)[/tex]

where [tex]1<p<\infty,\: \frac{1}{p}+\frac{1}{q}=1[/tex] and [tex](X, \mu)[/tex] is a measurable space (of course, the [tex](\Rightarrow)[/tex] is trivial by Holder inequality)

Thanks in advance!
 
Last edited:

Answers and Replies

  • #2
mathman
Science Advisor
7,942
496
It looks correct to me. From my recollection. Lp and Lq are adjoint, when p, q > 1 and 1/p + 1/q = 1.
 
  • #3
402
1
For those values of p, Lp is reflexive. What can you infer from this?
 

Related Threads on A characterization of L^p?

Replies
6
Views
3K
  • Last Post
Replies
17
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
4
Views
6K
  • Last Post
Replies
3
Views
2K
Replies
1
Views
9K
Replies
1
Views
2K
  • Last Post
Replies
3
Views
2K
Top