Mazur's Theorem

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:

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

where $\cl_w$ denotes the weak closure.

Corollary

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

Proof

From Topological Closure in Coarser Topology is Larger:

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

Now let $x \not \in \map \cl C$.

From Finite Topological Space is Compact, $\set x$ is compact.

Applying:

Hahn-Banach Separation Theorem: Compact Convex Set and Closed Convex Set (Real Case) if $\GF = \R$

Hahn-Banach Separation Theorem: Compact Convex Set and Closed Convex Set (Complex Case) if $\GF = \C$

for $A = \set x$ and $B = \map \cl C$, there exists a bounded linear functional $f : X \to \GF$ such that:

$\ds \map \Re {\map f x} < \inf_{y \in \map \cl C} \map \Re {\map f y}$

Now let $\alpha \in \R$ be such that:

$\ds \map \Re {\map f x} < \alpha < \inf_{y \in \map \cl C} \map \Re {\map f y}$

Consider:

$E = \set {x \in X : \map \Re {\map f x} \ge \alpha}$

We know that $C \subseteq E$.

From Characterization of Continuity of Linear Functional in Weak Topology, $f : \struct {X, w} \to \GF$ is continuous.

So, by Real and Imaginary Part Projections are Continuous and Composite of Continuous Mappings is Continuous, $\map \Re f : \struct {X, w} \to \R$ is continuous.

From Continuity Defined from Closed Sets, $E$ is then weakly closed.

From Topological Closure of Subset is Subset of Topological Closure, $\map {\cl_w} C \subseteq \map {\cl_w} E$.

From Set is Closed iff Equals Topological Closure, we have $\map {\cl_w} E = E$.

So:

$\map {\cl_w} C \subseteq E = \set {x \in X : \map \Re {\map f x} \ge \alpha}$

Since $\map \Re {\map f x} < \alpha$, we have $x \in X \setminus \map {\cl_w} C$.

So we obtain:

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

and so:

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

$\blacksquare$