I just did one exercise stating: Prove that the linear map $M: X \rightarrow C([0,1])$, is continuous iff for every $t\in[0,1]$, the rule $x\rightarrow (Mx)(t)$ defines a continuous linear functional on X. the next exercise stated: State, and prove a similar continuity criterion for linear maps $M:X\rightarrow Y$ where Y is an arbitrary Banach space.
Is there some theorem which states that $M$ is continuous iff $x\mapsto \ell(Mx)$ is continous for all linear functionals $\ell:Y\rightarrow \mathbb{K}$ in $Y'$? or what does it mean?
I posted a new try to a proof, can someone please confirm it or post another one?