Let X be a reflexive normed linear space.Then show that X is weakly complete..i'e;Every Cauchy sequence is weakly convergent..

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- January 18th 2011, 07:17 AMrishirichWeakly Cauchy
Let X be a reflexive normed linear space.Then show that X is weakly complete..i'e;Every Cauchy sequence is weakly convergent..

- January 18th 2011, 08:30 AMJose27
Pick a weak Cauchy sequence and call its corresponding sequence in the double dual then for any we have, since Cauchy sequences are bounded, and by the uniform boundednes principle we get that is a bounded sequence, by reflexivity this gives a weakly convergent subsequence, call it and a limit, call it . Now just use the estimate and if are large enough then the result follows.