Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 106 / 5 / b / Solution

Identify $X$ with its canonical image in $X^{\ast \ast }$ and putFor $x\in S_X$ and $\lambda \in \mathbb{R}$, if $|\lambda |\le 1/(d+1)$ thenIf $|\lambda |\ge 1/(d+1)$, thenTherefore $d(x,\mathrm{span}\{\Phi \})\ge c$.

The restriction of $x\in X^{\ast \ast }$ to $\ker \Phi \subseteq X^{\ast }$ has normwhere the first equality is the Hahn-Banach distance formula and $(\ker \Phi {)}^{\perp }=\mathrm{span}\{\Phi \}$. Scaling from $S_X$ givesfor every $x\in X$. Hence $\ker \Phi$ is $c$-norming for $X$.