Weak Convergence in Normed Dual Space of Reflexive Normed Vector Space is Equivalent to Weak-* Convergence/Proof 2

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

In view of Definition of Reflexive Space, $J$ is especially a surjection.

Therefore:

$\ds f_n$

$\weakconv$

$\ds f$

$\ds \leadstoandfrom \ \ $

$\ds \forall x \in X^{**}: \, $

$\ds \map {x} {f_n}$

$\to$

$\ds \map {x} f$

$\ds \leadstoandfrom \ \ $

$\ds \forall x \in X: \, $

$\ds \map {\map J x} {f_n}$

$\to$

$\ds \map {\map J x} f$

as $J : X \to X^{**}$ is a surjection

$\ds \leadstoandfrom \ \ $

$\ds \forall x \in X: \, $

$\ds \map {f_n} x$

$\to$

$\ds \map f x$

Definition of $J$

$\ds \leadstoandfrom \ \ $

$\ds f_n$

$\weakstarconv$

$\ds f$

$\blacksquare$