Closed Unit Ball in Normed Vector Space is Weakly Closed
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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.
$\blacksquare$