Hi,

The question: Let X, Y be Banach spaces, be a linear map. Prove that if is a continuous linear functional for every continuous linear functional , then is continuous.

The hypothesis of the problem (that X and Y both be Banach) suggest to me the Open Mapping Theorem, but we don't have T mapping onto Y, and T(X) may not be closed (which we need for it to be complete as a subspace of the complete space Y).

As always your help is much appreciated.