Sketching Outside the Hilbert Space for the Infinite-Dimensional Randomised SVD

In randomised numerical linear algebra, common choices for distributions of sketch vectors $ω\in \mathbb R^n$ are isotropic, so that $\mathbb E[ωω^T]=I_n$. Unfortunately, the natural extension of an isotropic vector in $(\mathbb R^n, \|\cdot \|_{2})$ to an infinite-dimensional separable Hilbert spac