1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Functional analysis, ortho basis, weakly convergent

  1. Dec 13, 2015 #1
    1. The problem statement, all variables and given/known data

    problem%205.26_zpsxlq56zgm.png

    This is a problem from Haim Brezis's functional analysis book.
    2. Relevant equations


    3. The attempt at a solution

    I'm assuming (e)n is the vectors like (e)1 = (1,0,0), (e)2=(0,1,0) and so on.

    We know every hilbert space has an orthonormal basis.

    I also need to know the difference between l^2 and L^2 spaces. I know that this might not be much of a solution, but I'm really stuck.
     
    Last edited by a moderator: Dec 13, 2015
  2. jcsd
  3. Dec 14, 2015 #2

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    You could start by stating clearly what weak convergence implies (here in the context of a Hilbert space).

    Once you have a clear understanding of weak convergence, 1. and 2. shouldn't be too difficult.
    As (##e_n)## is an orthornormal basis of the Hilbert space H, you can use ## \forall x \in H: x= \sum_{i=1}^\infty <x,e_i>e_i## (series converging in the norm of the Hilbert space).
    And also ##\forall x \in H:|x|²=\sum_{i=1}^\infty |<x,e_i>|²##.
     
    Last edited: Dec 14, 2015
  4. Dec 14, 2015 #3
    Thanks.

    I think I now have the answer to part 1 because you can use an idea from Parseval which automically shows that en weakly converges to zero. But I'm still not sure about what to do with an and un. I also think the boundedness of an plays a role.

    For part 2 I know that this is showing strongly convergence, but I don't really get how to show abs(un). Do I just pretend that the summation is really a sequence?
     
  5. Dec 14, 2015 #4

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    Yes, that is correct.
    You can compute an upper bound for ##|u_n|##, making use of the fact that ##(a_n)## is bounded, and ##(e_n)## orthonormal.
    You can use that, in general, you have:
    ##|\sum_{i=1}^n b_ie_i|²=\sum_{i=1}^n |b_i|²## (where ##(b_i)## are any numbers), because ##(e_n)## is orthonormal.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Functional analysis, ortho basis, weakly convergent
Loading...