[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?