Given finite i.i.d.~samples in a Hilbert space with zero mean and trace-class covariance operator $Σ$, the problem of recovering the spectral projectors of $Σ$ 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 $\hat Σ$, and offer a dimension-free framework where the complexity is characterized by the so-called relative rank of $Σ$. 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.