Let $\mathcal{F}_1,\mathcal{F}_2$ be sets of continuous functions from $[0,1]$ to $[0,\infty)$ and suppose that, for every $\varepsilon>0$, then $\varepsilon$-covering numbers in the uniform norm are both finite and respectively denoted by $\mathcal{N}_1(\varepsilon)$ and by $\mathcal{N}_2(\varepsilon)$.
Now $\mathcal{F}$ be a set of continuous functions from $[0,1]^2$ to $[0,\infty)$ whose elements are characterized by: $f\in \mathcal{F}$ if and onlf if:
- For every $x\in [0,1]$, the map $f(x,\cdot)\in \mathcal{F}_2$,
- For every $y\in [0,1]$, the map $f(\cdot,y)\in \mathcal{F}_1$.
Can we bound the $\varepsilon$-covering number $\mathcal{N}(\varepsilon)$ of $\mathcal{F}$ in the uniform norm on $C([0,1]^2)$ in terms of that of $\mathcal{N}_1(\varepsilon)$ and by $\mathcal{N}_2(\varepsilon)$?
