Preliminaries for Semiparametric Inference

In this chapter, we present additional technical background, beyond that given in Chapter 6, needed for the development of semiparametric inference theory. We first discuss projections without direct reference to Hilbert spaces. Next, we present basic properties and results for Hilbert spaces and revisit projections as objects in Hilbert spaces. Finally, we build on the Banach space material presented in Chapter 6, and in other places such as Chapter 12. Concepts discussed include adjoints of operators and dual spaces. The background in this chapter provides the basic framework for a “semiparametric efficiency calculus” that is used for important computations in semiparametric inference research. The development of this calculus will be the main goal of the next several chapters.