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 $\sequence {f_n}_{n \mathop \in \N}$ 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-$\ast$ (read as "weakly-star") if and only if:

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

We say that $f$ is a weak-$\ast$ limit of $\sequence {f_n}_{n \mathop \in \N}$.

We denote this:

$f_n \weakstarconv f$

Also see

• Weak-$\ast$ Limit in Normed Dual Space is Unique
• Convergence in Normed Dual Space implies Weak-$\ast$ Convergence

• Results about weak-$\ast$ convergence in a normed vector space can be found here.

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $27.4$: Weak-$\ast$ Convergence