Let $X$ be a Banach space and suppose $T:X^*\to X^*$ is a linear mapping. If $T$ is norm-norm continuous, i.e. continuous from the normed space $X^*$ into the normed space $X^*$, is it also continuous when $X^*$ is equipped with its weak-star topology (in both the domain and codomain)?
I am a novice with respect to weak topologies and weak-star topologies, so any help is appreciated. Thank you.