A practical guide to compact infinite dimensional parameter spaces

Compactness is a widely used assumption in econometrics. In this article, we gather and review general compactness results for many commonly used parameter spaces in nonparametric estimation, and we provide several new results. 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 illustrate how the choice of norms affects the parameter space, the strength of the conclusions, as well as other regularity conditions in two common settings: nonparametric mean regression and nonparametric instrumental variables estimation.