Definition:Weakly Compact Set

Definition

Let $K$ be a topological field.

Let $X$ be a topological vector space with weak topology $w$.

Let $C \subseteq X$.

We say that $C$ is weakly compact (or $w$-compact) in $X$ if and only if $C$ is a compact in $\struct {X, w}$.

That is, if and only if for every collection of weakly open sets $\SS$ such that:

$\ds C \subseteq \bigcup_{S \in \SS} S$

there exists a finite subset $\set {S_1, \ldots, S_n} \subseteq \SS$ such that:

$\ds C \subseteq \bigcup_{k \mathop = 1}^n S_k$