Convergence in Normed Dual Space implies Weak-* Convergence

Theorem

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

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

Let $\sequence {f_n}_{n \mathop \in \N}$ be a convergent sequence in $X^\ast$.

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

$\blacksquare$

Let $f \in X^\ast$ be the limit of $\sequence {f_n}_{n \mathop \in \N}$, i.e.:

$\norm {f_n - f}_{X^\ast} \stackrel{n \to \infty}{\longrightarrow} 0$

Thus, for each $x \in X$:

$\ds \size {\map {f_n} x - \map f x}$

$=$

$\ds \size {\map {\paren {f_n - f} } x}$

$\ds $

$\le$

$\ds \norm {f_n - f}_{X^\ast} \norm x_X$

$\ds $

$\ds \stackrel{n \to \infty}{\longrightarrow} 0$

$\blacksquare$