Compactness of infinite dimensional parameter spaces

We provide general compactness results for many commonly used parameter spaces in nonparametric estimation. We consider three kinds of functions: (1) functions with bounded domains which satisfy standard norm bounds, (2) functions with bounded domains which do not
satisfy standard norm bounds, and (3) functions with unbounded domains. In all three cases we provide two kinds of results, compact embedding and closedness, which together allow one to show that parameter spaces defined by a ||·||_s-norm bound are compact under a norm ||·||_c. We apply these results to nonparametric mean regression and nonparametric instrumental variables estimation.