This is a problem from Folland.
- Let $\mathcal{X}, \mathcal{Y}$ be Banach spaces. If $T : \mathcal{X} \rightarrow \mathcal{Y}$ is linear and $f \circ T \in \mathcal{X}^*$ for all $f \in \mathcal{Y}^*,$ then $T$ is bounded.
I am not sure how to go about proving it. It seems that I have to use the Baire category theorem in the form of the closed graph theorem; that is, it is sufficient to prove that $\Gamma(T) = \{(x, y) : x \in \mathcal{X}, y = Tx\}$ is closed in $\mathcal{X}\times\mathcal{Y}.$ But I am not sure how to do this.