= Norming subspace of a dual space
{title2=$c\|x\|\leq\sup_{z\in Z,\ \|z\|\leq1}|z(x)|$}
= Norming subspace
{synonym}
A <vector subspace> $Z\subseteq X^*$ is norming if some $c>0$ satisfies the displayed inequality. It retains enough continuous <linear functionals> to control the <norm> of every vector. Every such <vector subspace> is dense for the <weak-star topology>: otherwise a nonzero evaluation <linear functional> would annihilate its weak-star closure, contradicting the inequality. In the real case its symmetric <unit ball> makes the absolute-value and signed supremum versions equivalent.
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