Definition:Weak Convergence (Normed Vector Space)

Definition

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

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

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

Let $x \in X$.

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

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

We say that $x$ is a weak limit of $\sequence {x_n}_{n \mathop \in \N}$.

We denote this:

$x_n \weakconv x$

Also see

• Weak Limit in Normed Vector Space is Unique
• Weak Convergence in Hilbert Space
• Convergent Sequence in Normed Vector Space is Weakly Convergent
• Weakly Convergent Sequence in Hilbert Space with Convergent Norm is Convergent

• Results about weak convergence in a normed vector space can be found here.

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $27.1$: Weak Convergence