Finite-codimensional weak-star dense dual subspace is norming ($c=\mathrm{dist}(S_X,Y^{\perp })>0$)

Let $X$ be a Banach space, and let $Y\subseteq X^{\ast }$ be a norm-closed linear subspace with finite codimension, dense for the weak-star topology. Then $Y$ is a norming subspace: some $c>0$ satisfiesPut $F=Y^{\perp }\subseteq X^{\ast \ast }$. Finite codimension makes $F$ finite-dimensional, while density for the weak-star topology gives $F\cap JX=\{0\}$. The canonical embedding into the bidual has closed image $JX$, whose unit sphere has positive distance from $F$: otherwise bounded approximants in $F$ have a convergent subsequence with a nonzero limit in both spaces. The Hahn-Banach theorem identifies the norm of evaluation restricted to $Y$ with $\mathrm{dist}(Jx,F)$, proving the bound. Completeness of $X$ is needed for this closedness argument.