Weakly Convergent Sequence in Normed Dual Space is Weakly-* Convergent/Proof 1

Proof
Let $x \in X$.

We aim to show that:


 * $\map {f_n} x \to \map f x$

Then, since $x \in X$ was arbitrary, we will obtain that $\sequence {f_n}_{n \mathop \in \N}$ converges weakly-$\ast$ to $f$.

Since $\sequence {f_n}_{n \mathop \in \N}$ converges weakly to $f$, we have:


 * $\map F {f_n} \to \map F f$ for each $F \in \paren {X^\ast}^\ast$.

Define $x^\wedge : X^\ast \to \mathbb F$ by:


 * $\map {x^\wedge} f = \map f x$

for each $f \in X^\ast$.

From Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual, we have:


 * $x^\wedge \in \paren {X^\ast}^\ast$

So, setting $F = x^\wedge$, we obtain:


 * $\map {x^\wedge} {f_n} \to \map {x^\wedge} f$

That is:


 * $\map {f_n} x \to \map f x$