We review here some of the analytic concepts and facts that will be used in later chapters. Most of this material forms part of the standard textbook literature on real analysis or functional analysis and it is not necessary to repeat here the pertinent proofs. However, a few facts of a more special character and not generally known will be formulated as lemmas and proved.

Throughout this book we let Ed denote the Euclidean d-dimensional space. If x is a point of Ed the coordinates of x will be denoted by xi; hence, x = (x1, …, xd). The letter o denotes the origin (0, …, 0) of Ed. If u, v ∈ Ed we let u · v denote the inner product, and |u| the Euclidean norm. Of course, for points in E1, that is, for real numbers, | · | is the ordinary absolute value. The Lebesgue measure of a subset S of Ed will usually be called the volume of S and denoted by v(S). We write Bd(p, r) for the closed ball in Ed of radius r centered at p, and Bd = Bd(o, 1) for the closed unit ball in Ed centered at o. Furthermore, we let Sd–1 denote the boundary of Bd, that is, the unit sphere in Ed. The spherical Lebesgue measure on Sd–1 will be denoted by σ, the volume of Bd by κd, and the surface area of Bd by σd.