Appendix A. Background Topics

Abstract

This appendix treats some topics that are likely to be well known by some readers and less well known by others. Section A1 deals with set theory and with functions: it discusses the role of formal set theory, it works in a simplified framework that avoids too much formalism and the standard pitfalls, it establishes notation, and it mentions some formulas. Some emphasis is put on distinguishing the image and the range of a function, as this distinction is important in algebra and algebraic topology and therefore plays a role when real analysis begins to interact seriously with algebra.

Sections A2 and A3 assume knowledge of Section I.1 and discuss topics that occur logically between the end of Section I.1 and the beginning of Section I.2. The first of these establishes the Mean Value Theorem and its standard corollaries and then goes on to define the notion of a continuous derivative for a function on a closed interval. The other section gives a careful treatment of the differentiability of an inverse function in one-variable calculus.

Section A4 is a quick review of complex numbers, real and imaginary parts, complex conjugation, and absolute value. Complex-valued functions appear in the book beginning in Section I.5. Section A5 states and proves the classical Schwarz inequality, which is used in Chapter II to establish the triangle inequality for certain metrics but is needed before that in Chapter I in the context of Fourier series.

Sections A6 and A7 are not needed until Chapter II. The first of these defines equivalence relations and establishes the basic fact that they lead to a partitioning of the underlying set into equivalence classes. The other section discusses the connection between linear functions and matrices in the subject of linear algebra and summarizes the basic properties of determinants.

Section A8, which is not needed until Chapter IV, establishes unique factorization for polynomials with real or complex coefficients and defines “multiplicity” for roots of complex polynomials.

Sections A9 and A10 return to set theory. Section A9 defines partial orderings and includes Zorn's Lemma, which is a powerful version of the Axiom of Choice, while Section A10 concerns cardinality. The material in these sections first appears in problems in Chapter V; it does not appear in the text until Chapter X in the case of Section A9 and until Chapter XII in the case of Section A10.