Hi,

The question: Let X, Y be Banach spaces, $\displaystyle T:X\to Y$ be a linear map. Prove that if $\displaystyle L\circ T$ is a continuous linear functional $\displaystyle X\to \mathbb{R}$ for every continuous linear functional $\displaystyle L: Y\to \mathbb{R}$, then $\displaystyle T$ 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.