Results 1 to 2 of 2

Math Help - Weak Convergence

  1. #1
    Newbie
    Joined
    May 2010
    Posts
    8

    Weak Convergence

    Hi

    The question is:
    Find an example of 1\leq p,q < \infty and a sequence (x_n) which converges strongly to 0 in l^p, and weakly but not strongly in l^q.

    Any help would be greatly appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by markwolfson16900 View Post
    Hi

    The question is:
    Find an example of 1\leq p,q < \infty and a sequence (x_n) which converges strongly to 0 in l^p, and weakly but not strongly in l^q.
    If p < q and x_n\to0 in l^p then x_n\to0 in l^q. So for your example you need to take q < p.

    Suppose for example that q = 2 and p = 4. Define the k'th coordinate of x_n by x_n(k) = \begin{cases}n^{-1/2}&\text{if $n+1\leqslant k\leqslant 2n$,}\\ 0&\text{otherwise.}\end{cases}

    Then \|x_n\|_4 = n^{-1/4}, so x_n\to0 strongly in l^4. But \|x_n\|_2 = 1, so x_n\not\to0 strongly in l^2. I'll leave it to you to figure out why x_n\to0 weakly in l^2.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Weak convergence, characteristic functions.
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: December 28th 2011, 02:11 PM
  2. Weak convergence of the sum of dependent variables, question
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: December 13th 2011, 04:46 PM
  3. Weak Convergence
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: August 15th 2010, 11:55 PM
  4. Weak convergence of measures
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: December 13th 2009, 02:12 PM
  5. weak convergence
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: April 24th 2009, 08:02 PM

Search Tags


/mathhelpforum @mathhelpforum