Weak convergence question

  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. By Schur's lemma every weakly Cauchy sequence converges. So your answer lies in the proof of Schur's lemma.
     
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