Definition:Weak Convergence (Topological Vector Space)

Definition

Let $\GF \in \set {\R, \C}$.

Let $X$ be a topological vector space over $\GF$.

Let $X^\ast$ be the topological dual space of $X$.

Suppose that:

for each $x, y \in X$ with $x \ne y$, there exists $f \in X^\ast$ such that $\map f x \ne \map f y$.

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

Let $x \in X$.

Definition 1

Let $w$ be the weak topology on $X$.

We say that $\sequence {x_n}_{n \mathop \in \N}$ converges weakly if and only if it converges in $\struct {X, w}$.

We write:

$x_n \weakconv x$

Definition 2

We say that $\sequence {x_n}_{n \mathop \in \N}$ converges weakly to $x$ if and only if:

for each $f \in X^\ast$ we have $\map f {x_n} \to \map f x$ in $\GF$.

Also see

• Equivalence of Definitions of Weak Convergence on Topological Vector Space

• Results about weak convergence on topological vector spaces can be found here.