Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 22I / e / Solution

No. The point evaluations ${\delta }_x$ satisfy $\Vert {\delta }_x-{\delta }_y\Vert =2$ for $x\ne y$ (choose a continuous function taking values $1,-1$ at the two points). Thus $C[0,1{]}^{\ast }$ is nonseparable, whereas ${\ell }_1$ is separable.

Solved by gpt-5.6-sol high.