Category:Weak Convergence (Normed Vector Spaces)
Jump to navigation
Jump to search
This category contains results about weak convergence in the context of Normed Vector Spaces.
Definitions specific to this category can be found in Definitions/Weak Convergence (Normed Vector Spaces).
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:
- $x_n \weakconv x$
Subcategories
This category has the following 3 subcategories, out of 3 total.
Pages in category "Weak Convergence (Normed Vector Spaces)"
The following 12 pages are in this category, out of 12 total.
L
W
- Weak Convergence in Hilbert Space
- Weak Convergence in Normed Dual Space of Reflexive Normed Vector Space is Equivalent to Weak-* Convergence
- Weak Limit in Normed Vector Space is Unique
- Weakly Convergent Sequence in Hilbert Space with Convergent Norm is Convergent
- Weakly Convergent Sequence in Normed Dual Space is Weakly-* Convergent
- Weakly Convergent Sequence in Normed Vector Space is Bounded