Banach-Alaoglu Theorem/Lemma 3
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Lemma for Banach-Alaoglu Theorem
Let $X$ be a normed vector space.
Denote by $B$ the closed unit ball in $X$.
Let $X^*$ be the dual of $X$.
Denote by $B^*$ the closed unit ball in $X^*$.
Let $\map \FF B = \closedint {-1} 1^B$ be the topological space of functions from $B$ to $\closedint {-1} 1$.
By Tychonoff's Theorem, $\map \FF B$ is compact with respect to the product topology.
We define the restriction map:
- $R: B^* \to \map \FF B$
by:
- $\map R \psi = \psi \restriction_B$
$\map R {B^*}$ is a closed subset of $\map \FF B$.
Proof
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