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Homework Help: <uk,v> -> <u,k> for all v. prove uk goes to u.

  1. Oct 7, 2009 #1
    <uk,v> --> <u,k> for all v. prove uk goes to u.

    1. The problem statement, all variables and given/known data
    a)supposed for all v in Rn <uk,v>--> <u,v> prove {uk}-->u
    b)give an example of a sequence {uk} st. <uk,v> for some vector v. but uk does not converge to u.

    2. Relevant equations
    def of convergence.

    3. The attempt at a solution
    a)since this works for all v. we can let v1={1,0,0,...} then u1 --> u. now let v2={0,1,0,0,...} then u2 --> u.
    so we keep doing this and each term of uk converges to a term in u.
    so by point wise convergence we have convergence.

    b) I don't know how to do this part. for a) i think i got it right but i can't come up with an example.
  2. jcsd
  3. Oct 7, 2009 #2
    Re: <uk,v> --> <u,k> for all v. prove uk goes to u.

    your proof looks good. For the example, note that if uk converges to u then uk + w converges to u + w. Can you think of a way to choose w so that <uk + w, v> converges to <u, v>?
  4. Oct 7, 2009 #3


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    Re: <uk,v> --> <u,k> for all v. prove uk goes to u.

    You have the right idea for part 1 but you aren't writing it very well. Remember that each uk is a vector. So it doesn't make sense to say u1 --> u. u1 is a fixed vector. It might help you to adopt a notation where the components of, for example u1 are u1(1), u1(2),...u1(n), and try making the argument you are thinking of.

    Once you get the notation down you may see how to do the second part too.
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