Closed Unit Ball in Normed Vector Space is Weakly Closed

Theorem

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

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

Let $B^-$ be the closed unit ball in $\struct {X, \norm {\, \cdot \,} }$.

Then $B^-$ is weakly closed.

Proof

From Closed Unit Ball is Convex Set, $B^-$ is convex.

From Closed Ball is Closed, $B^-$ is $\norm {\, \cdot \,}$-closed.

From Mazur's Theorem: Corollary, we can conclude that $B^-$ is weakly closed.

$\blacksquare$