Weak Convergence in Normed Dual Space of Reflexive Normed Vector Space is Equivalent to Weak-* Convergence

Theorem
Let $\mathbb F$ be a subfield of $\C$.

Let $\struct {X, \norm {\, \cdot \,}_X}$ be a reflexive normed vector space over $\mathbb F$.

Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.

Let $f \in X^\ast$.

Then:


 * $\sequence {f_n}_{n \mathop \in \N}$ converges weakly to $f$ $\sequence {f_n}_{n \mathop \in \N}$ converges weakly-$\ast$ to $f$.