Understanding Weak Convergence in Little l1 and Little l∞ Spaces

[homology]

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

 

[NSAC]

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