Let $(H, \langle\cdot,\cdot\rangle)$ be a real separable Hilbert space with some fixed orthonormal basis $(e_i\mid i\in I)$, and for any given $v\equiv \sum_{i\in I} a_i e_i\in H$ and $J\subseteq I$ define $\pi_J(v):=\sum_{i\in J}a_i e_j$ (orthogonal projections).
Suppose that $\mathcal{S}\subset H$ is a compact subset which admits a bound $C\geq 1$ such that
$$\tag{1} \sup_{J\subseteq I \text{ finite}} \sup_{\lVert\ell\rVert=1} \frac{\sup_{q\in\mathcal{S}}\big|\langle\pi_J(\ell), q\rangle\big|}{\sup_{r\in\mathcal{S}}|\langle \ell, r\rangle|} ~ \leq ~ C$$
where the second supremum runs over all $\ell\in\mathrm{span}(e_i\mid i\in I)$ with $\langle \ell, \ell\rangle = 1$.
I am interested in whether condition (1) applies to interesting [i.e. 'large'] compact subsets $\mathcal{S}$ of $H$. (The sets $\mathcal{S} = \{0\}$ or $\mathcal{S}=\{e_j\}$ are trivial examples.)
Question: Suppose for interest that $I$ is infinite and $\mathcal{S}$ is not contained in a finite-dimensional subspace. Do you know of any interesting/nontrivial examples for $\mathcal{S}$, or references in which (sets defined by) conditions of type $(1)$ are studied?