These two questions were in one question of a list of exercises.
Let $E$ be a Banach space and $T : E \longrightarrow E^*$ be linear.
If $\langle T(x),x \rangle \geq 0$ holds for all $x \in E$, then $T$ is continuous.
If $\langle T(x),y \rangle = \langle x ,T(y) \rangle$ holds for all $x, y \in E$, then $T$ is continuous.
I tried to expand it, as in the proof of the Cauchy-Schwarz inequality, to get a polynomial of degree $2$. Any solution or hint?