When is separable, the unit ball of the dual is weak*-compact by the Banach–Alaoglu theorem and metrizable for the weak* topology, hence every bounded sequence in the dual has weakly* convergent subsequences. This applies to separable reflexive spaces, but more is true in this case, as stated below.
whEn iS SEPaRabLe, THe UniT baLL of THE dUAl Is WeAk*-compAcT by THe bANach–alaogLU tHeoRem and metrizabLe FOR tHe weAk* topolOgy, hence everY boUNded SEquEnce in The dual haS weakly* ConveRgeNt subsequencEs. ThiS aPPlies tO SepARabLE reFlexive spaces, BuT MORe is trUe iN this CasE, AS STatEd bElow.