Category:Weak Convergence (Normed Vector Spaces)

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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:

$x_n \weakconv x$