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:
- $\map {\cl_w} C = \map \cl C$
$\blacksquare$
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