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The continuous dual is Banach

  1. Jan 18, 2008 #1


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    [SOLVED] The continuous dual is Banach

    1. The problem statement, all variables and given/known data
    I'm trying to show that the continuous dual X' of a normed space X over K = R or C is complete.

    3. The attempt at a solution

    I have shown that if f_n is cauchy in X', then there is a functional f towards which f_n converge pointwise. Then I showed that the distance ||f_n - f|| can be made arbitrarily small. The only thing that remains then is to show that f is indeed linear and continuous. The linear part is easy but the continuous part eludes me.

    To say a linear functional f is continuous is equivalent to saying that ||f||<+oo. But I can't conclude that from the fact that f_n --> f (pointwise). I would need f_n --> f (uniformly). Or I would need a whole other approach. Ideas?
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  3. Jan 18, 2008 #2


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    Use the fact that (f_n) is cauchy to find an N such that if n,m>=N then ||f_n - f_m|| < 1. Then use the inequality |f(x)| <= |f(x) - f_N(x)| + |f_N(x)|.
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