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Weak convergence question

  1. Nov 13, 2005 #1
    hello folks,

    I've got a question about weak convergence. I'm sure I'm missing something but can't see what it is (<--standard "I'm dumb apology")

    The problem concerns little l 1 and little l infinity (which is dual to little l 1) To make notation easier I'm going to denote these spaces by L1 and Linf.

    If a sequence in L1 weakly converges then it strongly converges.

    So we take a sequence in L1 {x_n} where each x_n is a inf-tuple (a_1,...) of real numbers such that Sum|a_k| is bounded. Take an element of Linf, call it g. We given that limg(x_n)=g(x) for an x in L1. Now we need to show that limx_n=x.

    So my dilemma is that I essentially keep proving that strong convergence implies weak convergence. I keep trying to work expressions like |g(x_m)-g(x_n)|<e and |g(x)-g(x_n)|<e into the analogs for the x_n (using of course the appropriate norm).

    As per usual i can't get my inequalities going in the right direction. I need only a tiny push, I'm sure. So advise sparingly.

    As always, your help is much appreciated,

    Kevin
     
  2. jcsd
  3. Nov 25, 2009 #2
    By Schur's lemma every weakly Cauchy sequence converges. So your answer lies in the proof of Schur's lemma.
     
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