  Given finite i.i.d.~samples in a Hilbert space with zero mean and trace-class
covariance operator $\Sigma$, the problem of recovering the spectral projectors
of $\Sigma$ naturally arises in many applications. In this paper, we consider
the problem of finding distributional approximations of the spectral projectors
of the empirical covariance operator $\hat \Sigma$, and offer a dimension-free
framework where the complexity is characterized by the so-called relative rank
of $\Sigma$. In this setting, novel quantitative limit theorems and bootstrap
approximations are presented subject only to mild conditions in terms of
moments and spectral decay. In many cases, these even improve upon existing
results in a Gaussian setting.
