# weak convergence question

by homology
Tags: convergence, weak
 P: 307 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)|
 P: 13 By Schur's lemma every weakly Cauchy sequence converges. So your answer lies in the proof of Schur's lemma.

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