Approximation properties of certain operator-induced norms on Hilbert spaces

We consider a class of operator-induced norms, acting as finite-dimensional surrogates to the $L^2$ norm, and study their approximation properties over Hilbert subspaces of $L^2$. The class includes, as a special case, the usual empirical norm encountered, for example, in the context of nonparametric regression in a reproducing kernel Hilbert space (RKHS). Our results have implications to the analysis of $M$-estimators in models based on finite-dimensional linear approximation of functions, and also to some related packing problems.