Let $(\mathcal{X},d)$ be a space of function $f: \mathbb{R}^d \to \mathbb{R}$ where $d=\| \cdot \|_\infty$ (i.e., $d(f)= \sup_{x\in \mathbb{R}^d} |f(x)|$ ).
Let $D_\alpha f= \frac{\partial^\alpha}{ \partial x_1^{p_1} .... \partial x_d^{p_d} }$ where $\alpha=p_1 +...+ p_d$ where $p_i$'s are non-negative integers and we let $D^0f =f$.
Suppose we consider the following set \begin{align} \mathcal{F}_\beta = \{ f: \| D_\alpha f \|_\infty<c_\alpha, \forall \alpha\le \beta \} . \end{align}
I am interested in an upper bound on the covering number of $\mathcal{F}_\beta$. Given the history of covering numbers and metric entropy, I would think that this result exists. However, I just cannot find anywhere a firm upper bound.
