Definition:Weak Closure

Definition
Let $K$ be a topological field.

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

Let $H \subseteq X$.

We define the weak closure $\map {\cl_w} H$ as the topological closure of $H$ in $\struct {X, w}$.

That is:


 * $\ds \map {\cl_w} H = \bigcap \leftset {C \supseteq H: C}$ is weakly closed in $\rightset X$

Also see

 * Mazur's Theorem shows that for convex sets, weak closures coincide with topological closures.