How can I show that the continuous dual X' of a normed space X is complete?

[quasar987]

[SOLVED] The continuous dual is Banach

Homework Statement

I'm trying to show that the continuous dual X' of a normed space X over K = R or C is complete.

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?

 

[morphism]

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)|.