Definition:Weak Convergence (Normed Vector Space)
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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