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!

Convergence in Hilbert space question

  1. Feb 1, 2008 #1

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    1. The problem statement, all variables and given/known data
    Is it true/possible to show that in a Hilbert space, if z_n is a sequence (not known to converge a priori) such that (z_n,y)-->0 for all y, then z_n-->0 ?


    3. The attempt at a solution

    I've shown that if z_n converges, then it must be to 0. But does it converge?
     
  2. jcsd
  3. Feb 1, 2008 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Consider an orthonormal basis of the Hilbert space, {e_n} and let the sequence be the basis vectors. Now (e_n,y) is the nth component of y, which must tend to zero as n goes to infinity for any y. Yet e_n clearly does not converge. So no, you can't show that.
     
  4. Feb 2, 2008 #3

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Good example!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Convergence in Hilbert space question
  1. Hilbert spaces (Replies: 5)

  2. Hilbert space (Replies: 2)

  3. Hilbert Spaces (Replies: 2)

  4. Hilbert Space (Replies: 3)

Loading...