# Can we use sequences to test continuity of a weak$^*$-continuous operator?

Let $X,Y$ be Banach spaces. Now assume we have a map $T:X'\rightarrow Y'$ where $X'$ and $Y'$ are equipped with the weak$^*$ topology and not the norm topology. Can I infer from this that an operator satisfying $x_n'(x) \rightarrow x'(x) \Rightarrow (Tx_n')(x) \rightarrow (Tx')(x)$ for $x_n',x' \in X'$ is continuous in the weak $^*$ topology?

If anything is unclear, please let me know.

• If $X$ is not finite-dimensional then $X'$ is not metrizable. In this case, sequential continuity is weaker than continuity. I'm thus expecting the answer to be "no". – M.G May 27 '16 at 0:14
• Is $T$ supposed to be linear? – gerw May 27 '16 at 17:45
• @gerw yes, it is – Tzego May 27 '16 at 19:31