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 $\struct {X, w}$.

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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.

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{V}$ Weak Topologies: $\S 1.$ Duality