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Let X be a normed linear space. Let X* be its dual space with the usual...

Let X be a normed linear space. Let X* be its dual space with the usual dual norm ||T|| = sup{ |T(x)| / ||x|| : x not equal to 0}. Show that X* is always complete. Hint: If Tn is a Cauchy sequence in X*, show that i) Tn(x) converges for each fixed x in X, ii) the resulting limits define a bounded linear functional T on X, and iii) the sequences Tn converges to T in the norm of X*.

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