= Finite-codimensional weak-star dense dual subspace is norming
{title2=$c=\operatorname{dist}(S_X,Y^\perp)>0$}
A norm-closed finite-codimensional <vector subspace> $Y\subseteq X^*$ that is weak-star dense is a <norming subspace>. Its annihilator $F=Y^\perp\subseteq X^{**}$ is finite-dimensional and disjoint from $JX$. The positive distance $d$ of the unit sphere from $F$ gives a triple-dual <linear functional> vanishing on $F$ and taking a value at least $d$ at a unit vector. Apply the <finite-dimensional interpolation form of Goldstine's theorem> to the <Banach space> $X^*$, then normalize and take a supremum to obtain norming constant $d$.
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