The following is a Theorem of Murphy's C*-algebra and operator theory:
To prove the theorem, the author claims compact linear map $u$ is weakly continuous. I know that every bounded linear map is weakly continuous, but I'm not sure each linear map is weakly continuous, too. Maybe compactness of $u$ implies weakly continuity of $u$ or he suppose that $u$ is bounded.
Please give me a hint to prove every linear map ( compact linear map ) is weakly continuous, or a counterexample for it. Thanks in advance.