Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 201 / 6 / a / Solution

Every evaluation map ${\pi }_t$ is continuous in the uniform metric, so $\mathcal{F}\subseteq \mathcal{B}$. Conversely, for $g\in C[0,1]$,with the harmless restriction to $m$ for which $r-1/m>0$. Thus every open ball belongs to $\mathcal{F}$. Separability makes every open set a countable union of balls, so $\mathcal{B}\subseteq \mathcal{F}$ and equality follows.