Concrete Hilbert Spaces

Abstract

In this chapter we consider an infinite-dimensional generalization of Euclidean space introduced by the mathematician David Hilbert. This generalization preserves two fundamental geometric notions of Euclidean space— namely, distance and perpendicularity. Both of these geometric properties depend on the existence of an inner product. In the infinite-dimensional case, however, we take the inner product of functions rather than of vectors. Our emphasis here will be on concrete examples of Hilbert spaces relevant to statistics. To keep our discussion within bounds, some theoretical facts are stated without proof. Relevant proofs can be found in almost any book on real or functional analysis [6, 12]. Applications of our examples to numerical integration, wavelets, and other topics appear in later chapters.