Understanding L1 vs. Lp Weak Convergence

In summary, the function u_n (x) = n if x belongs to (0, 1/n), 0 otherwise, is bounded in L1, but is not bounded in L2, and so cannot be used as a counterexample to show L1 is not weakly compact.
  • #1
muzialis
166
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

  • Weakconvergence L1.pdf
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  • #2
Sorry, not sure my response makes sense, edited it out. Will think about it some more.
 
Last edited:
  • #3
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.
 
  • #4
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.
 
  • #5
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.
 
  • #6
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
 

Related to Understanding L1 vs. Lp Weak Convergence

1. What is L1 and Lp weak convergence?

L1 and Lp weak convergence are mathematical concepts used to describe the convergence of a sequence of functions or random variables. L1 weak convergence refers to the convergence of a sequence of functions or random variables in the L1 norm, while Lp weak convergence refers to the convergence in the Lp norm. Both L1 and Lp weak convergence are important in the study of probability and measure theory.

2. What is the difference between L1 and Lp weak convergence?

The main difference between L1 and Lp weak convergence lies in the norm used for measuring convergence. L1 weak convergence uses the L1 norm, which is the sum of the absolute values of the function or random variable, while Lp weak convergence uses the Lp norm, which is the pth root of the sum of the pth powers of the function or random variable. Additionally, L1 weak convergence is stronger than Lp weak convergence, meaning that if a sequence converges weakly in L1, it also converges weakly in Lp but not vice versa.

3. How is L1 and Lp weak convergence related to strong convergence?

L1 and Lp weak convergence are weaker forms of convergence compared to strong convergence. Strong convergence requires the functions or random variables to converge in both the Lp norm and pointwise, while weak convergence only requires convergence in the Lp norm. However, if a sequence converges strongly, it also converges weakly in both L1 and Lp.

4. What are some applications of L1 and Lp weak convergence?

L1 and Lp weak convergence have various applications in statistics, probability, and measure theory. They are commonly used in the study of random processes, stochastic calculus, and functional analysis. L1 and Lp weak convergence also have applications in finance, where they are used to model and analyze financial data and time series.

5. How is L1 and Lp weak convergence tested or proven in practice?

In practice, L1 and Lp weak convergence can be tested or proven using various mathematical techniques such as the Vitali convergence theorem, the Helly-Bray theorem, and the Portmanteau theorem. These theorems provide necessary and sufficient conditions for L1 and Lp weak convergence. Additionally, simulations and numerical experiments can also be used to test for L1 and Lp weak convergence in specific cases.

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