Mazur's Theorem/Corollary

Corollary to Mazur's Theorem

Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \norm {\, \cdot \,} } $ be a normed vector space over $\GF$ with weak topology $w$.

Let $C \subseteq X$ be a convex subset of $X$.

Then:

$C$ is weakly closed if and only if it is $\norm {\, \cdot \,}$-closed.

Proof

From Mazur's Theorem, we have:

$\map {\cl_w} C = \map \cl C$

From Set is Closed iff Equals Topological Closure, we have that $C$ is weakly closed if and only if:

$\map {\cl_w} C = C$

That is, if and only if:

$\map \cl C = C$

From Set is Closed iff Equals Topological Closure, this is equivalent to $C$ being $\norm {\, \cdot \,}$-closed.

So $C$ is weakly closed if and only if it is $\norm {\, \cdot \,}$-closed.

$\blacksquare$