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Math Help - Weakly Cauchy

  1. #1
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    Weakly Cauchy

    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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  2. #2
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    Pick a weak Cauchy sequence (x_n) and call (\hat{x}_n) its corresponding sequence in the double dual then for any l\in X^* we have, since Cauchy sequences are bounded, |\hat{x}_n(l)|=|l(x_n)|\leq C_l and by the uniform boundednes principle we get that (x_n) is a bounded sequence, by reflexivity this gives a weakly convergent subsequence, call it (x_k) and a limit, call it x. Now just use the estimate |l(x-x_n)|-|l(x_k-x_n)|\leq |l(x-x_k)|<\varepsilon and if k,n are large enough then the result follows.
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