Definition:Weak-* Convergence (Normed Vector Space)

Definition
Let $\Bbb F \in \set {\R, \C}$.

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

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

Let $\sequence {f_n}_{n \mathop \in \N}$ be a sequence of bounded linear functionals on $X$.

Let $f : X \to \Bbb F$ be a bounded linear functional.

Then we say that $\sequence {f_n}_{n \mathop \in \N}$ converges weakly-* (read as "weakly-star") :


 * $\map {f_n} x \to \map f x$ for each $x \in X$.

We denote this:


 * $f_n \stackrel \ast \rightharpoonup f$