Continuous dual of a weak-star topology ($(X^{\ast },\sigma (X^{\ast },X){)}^{\prime }=J_X(X)$)

A linear functional on $X^{\ast }$ is continuous for the weak-star topology exactly when it is evaluation at an element of $X$. Continuity makes it vanish on the common kernel of finitely many evaluations, so it is a linear combination of those evaluations. The canonical embedding into the bidual identifies this dual with $X$.