Let $X$ be a separable space and let $x^{\*\*}\in X^{\*\*}$. If $x^{\*\*}(x^{\*}\_{n})\rightarrow 0$ for each weak\*-null sequence $(x^{\*}\_{n})\_{n}$ in $X^{\*}$, is $x^{\*\*}$ in $X$ ?


Thank you!



weak*-null sequences in the dual space of a separable space
Yes, because $(B\_{X^\*},w^\*)$ is compact metrizable. So $x^{\*\*}$ is $w^\*$-continuous at any $x^\*\in B\_{X^\*}$.


