Convergent Sequence in Normed Vector Space is Weakly Convergent/Proof 2

Theorem

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

Let $x \in X$.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$ converging to $x$.

Then $\sequence {x_n}_{n \mathop \in \N}$ converges weakly to $x$.

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

Then, for each $f \in X^\ast$:

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

$\le$

$\ds \norm f_{X^\ast} \norm {x_n - x}_X$

$\ds $

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

$\blacksquare$