Weakly Convergent Sequence in Normed Dual Space is Weakly-* Convergent/Proof 2

Theorem

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

Let $\struct {X, \norm {\, \cdot \,}_X}$ be a 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 $f \in X^\ast$.

Let $\sequence {f_n}_{n \mathop \in \N}$ be a sequence in $X^\ast$ converging weakly to $f$.

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

Proof

Let $J$ be the evaluation linear transformation on $X$.

By Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual:

$J : X \to X^{\ast \ast}$

Thus, for each $x \in X$:

\(\ds \map {f_n} x\)

\(=\)

\(\ds \map {\map J x} {f_n}\)

\(\ds \)

\(\stackrel {n \to \infty} {\longrightarrow}\)

\(\ds \map {\map J x} f\)

\(\ds \)

\(\)

\(\, \ds = \, \)

\(\ds \map f x\)

$\blacksquare$