Understanding L1 vs. Lp Weak Convergence

  • Context: Graduate 
  • Thread starter Thread starter muzialis
  • Start date Start date
  • Tags Tags
    Convergence Weak
Click For Summary

Discussion Overview

The discussion revolves around the concepts of weak compactness in L1 and Lp spaces, particularly focusing on why L1 is not weakly compact while Lp for p > 1 is. Participants explore examples and counterexamples related to weak convergence and boundedness in these spaces.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents an example of the function u_n (x) = n if x belongs to (0, 1/n), 0 otherwise, to illustrate that L1 is not weakly compact, converging to the Dirac measure.
  • Another participant questions why a similar behavior would not occur in Lp spaces for p > 1, expressing confusion over the weak convergence of a related function.
  • It is noted that for weak compactness, any bounded sequence must have a convergent subsequence, and the sequence in question is not bounded in Lp for p > 1.
  • One participant clarifies that the sequence is bounded in L1 and presents a different function, u_n = x^(1/2) if x belongs to (0, 1/n), 0 otherwise, which is bounded in L2 and converges to an element of L2, indicating L2's weak compactness.
  • Another participant acknowledges a misunderstanding regarding the examples and the implications for weak compactness in L2.

Areas of Agreement / Disagreement

Participants express differing views on the boundedness of sequences in L1 and Lp spaces, leading to unresolved questions about the implications for weak compactness. There is no consensus on the relationship between the examples provided and the properties of weak compactness.

Contextual Notes

Participants reference specific examples and their boundedness in different Lp spaces, but there are unresolved assumptions regarding the definitions and implications of weak convergence and compactness.

muzialis
Messages
156
Reaction score
1
Hello,

I am struggling to understand why L1 is not weakly compact, while Lp, p>1, is.

The example I have seen put forward is the function u_n (x) = n if x belongs to (0, 1/n), 0 otherwise, the function being defined on (0,1).

It is shown this u_n converging to the Dirac measure, and this shows L1 not being weakly compact (as integrating u_n times the charactersitic function of the interval yields 1 as a result, while the result is zero for a function which is zero at the boundary.

I can not see why this should not happen for Lp, p > 1 too. In the attached short notes there is an example after which (pag. 6) it is stated that this function converges weakly to zero in Lp.

I can not understand this.

Many thanks

Regards
 

Attachments

Physics news on Phys.org
Sorry, not sure my response makes sense, edited it out. Will think about it some more.
 
Last edited:
Actually maybe what I said did make sense. So to be weakly compact any bounded sequence has to have a convergent sub-sequence from what I remember right? The sequence you defined is not bounded in Lp for p>1.
 
Thanks for replying, I am not sure I understand your reply. The sequence i mention is bounded in the L1 norm, and the example I mentioned, u_n = x ^(1/2) if x belongs to (0,1/n), 0 otherwise, is bounded in L2. But L2, on the contrary to L1, is weakly compact.
 
I thought your example was

"u_n (x) = n if x belongs to (0, 1/n), 0 otherwise"

So my point is that this is bounded in L1, and therefore it can be used as a counterexample to show L1 is not weakly compact.

However since it is not bounded in Lp for p>1, it can not be used as a counter example in those cases because the definition of weak compactness only puts restrictions on the behavior of bounded sequences.
 
Sorry, I was very unclear. I showed the example in L1, giving for granted that when considering the L2 case its analogue, mentioned in the attachment, would have considered.
The function u_n = x^(1/2) if x belongs to (0,1/n), 0 otherwise.
This is bounded in L2 and must converges to an element of L2, given that L2 is weakly compact.

But thank you anyhow, I think I discovered my silly mistake!

Bye
 

Similar threads

Undergrad Uniform convergence of family of functions with continuous index
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Convergence of sequences of functions with differing domains?
  • · Replies 11 ·
Replies
11
Views
2K
Graduate How Do Series Converge in Normed Spaces?
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Convergence of a Sequence: Point-wise, in Measure, Lp, and Uniformly?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Question about uniform convergence in a proof
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Understanding the nth-term test for Divergence
  • · Replies 17 ·
Replies
17
Views
6K
Graduate Why Don't We Learn This Necessary Criterion for Series Convergence in School?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Convergence of ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}##
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Pointwise convergence in Lp space
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Newton-Raphson Method With Complex Numbers
  • · Replies 7 ·
Replies
7
Views
3K