 Solution

Put $d=\mathrm{dist}(x,Z)$. If $d=0$, the zero functional works. If $d>0$, then $x\notin \overline{Z}$, and on $M=Z+\mathrm{span}\{x\}$ defineThe representation is unique, so this is linear. For $a\ne 0$,and the bound also holds for $a=0$. Thus $\Vert f_0\Vert \le 1$. The Hahn-Banach theorem extends it to $f\in X^{\prime }$ with the same bound, provingNo closedness assumption on $Z$ was required; distance is unchanged on taking its closure.