Mathematicians frequently use geometrical examples as aids to the study of more abstract concepts and these examples can be of great interest in their own right. Yet at the present time little of this is to be found in undergraduate textbooks on mathematics. The main reason seems to be the standard division of the subject into several watertight compartments, for teaching purposes. The examples get excluded since their construction is normally algebraic while their greatest illustrative value is in analytic subjects such as advanced calculus or, at a slightly more sophisticated level, topology and differential topology.

Experience gained at Liverpool University over the last few years, in teaching the theory of linear (or, more strictly, affine) approximation along the lines indicated by Prof. J. Dieudonné in his pioneering book Foundations of Modern Analysis, has shown that an effective course can be constructed which contains equal parts of linear algebra and analysis, with some of the more interesting geometrical examples included as illustrations. The way is then open to a more detailed treatment of the geometry as a Final Honours option in the following year.

This book is the result. It aims to present a careful account, from first principles, of the main theorems on affine approximation and to treat at the same time, and from several points of view, the geometrical examples that so often get forgotten.

The theory of affine approximation is presented as far as possible in a basis-free form to emphasize its geometrical flavour and its linear algebra content and, from a purely practical point of view, to keep notations and proofs simple.

A finite-dimensional linear space over a commutative field K is a linear space over K that is isomorphic to Kn, for some finite number n. For example, any linear space isomorphic to R2 is finite-dimensional. Among the many corollaries of the main theorem of the chapter, Theorem 6.12, is Corollary 6.13 which states that, for any finite m, n the K-linear spaces Km and Kn are isomorphic if, and only if, m = n. It is therefore meaningful to say that a K-linear space isomorphic to Kn has dimension n or is n-dimensionaly, and, in particular, to say that a real linear space isomorphic to R2 is two-dimensional.

The theory of finite-dimensional linear spaces is in some ways simpler than the theory of linear spaces in general. For example, if X and Y are linear spaces each isomorphic to Kn, then a linear map t : X → Y is injective if, and only if, it is surjective (Cor. 6.33). Therefore, to prove that t is an isomorphism it is necessary only to verify that X and Y have the same dimension and that t is injective. Any finite-dimensional linear space has the same dimension as its dual space and may be identified with the dual of its dual (Exercise 6.44), this being the origin of the term ‘dual’ space.

We begin with a discussion of linear dependence and independence.

This short guide is intended to help the reader to find his way about the book.

As we indicated in the Foreword, the book consists of a basic course on affine approximation, which we refer to colloquially as ‘the Dieudonné course’, linked at various stages with various geometrical examples whose construction is algebraic and to which the topological and differential theorems are applied.

Chapters 1 and 2, on sets, maps and the various number systems serve to fix basic concepts and notations. The Dieudonné course proper starts at Chapter 3. Since the intention is to apply linear algebra in analysis, one has to start by studying linear spaces and linear maps, and this is done here in Chapters 3 to 7 and in the first part of Chapter 8. Next one has to set up the theory of topological spaces and continuous maps. This is done in Chapter 16, this being prefaced, for motivational and for technical reasons, by a short account of normed linear spaces in Chapter 15. The main theorems of linear approximation are then stated and proved in Chapters 18 and 19, paralleling Chapters 8 and 10, respectively, of Prof. Dieudonné's book.

The remainder of the book is concerned with the geometry. We risk a brief consideration of the simplest geometrical examples here, leaving the reader to come back and fill in the details when he feels able to do so.

The language of sets and maps is basic to any presentation of mathematics. Unfortunately, in many elementary school books sets are discussed at length while maps are introduced clumsily, if at all, at a rather late stage in the story. In this chapter, by contrast, maps are introduced as early as possible. Also, by way of a change, more prominence than is usual is given to the von Neumann construction of the set of natural numbers.

Most of the material is standard. Non-standard notations include f⊢ and f⊢, to denote the forward and backward maps of subsets induced by a map f, and X! to denote the set (and in Chapter 2 the group) of permutations of a set X. The notation ω for the set of natural numbers is that used in and in. An alternative notation in common use is N.

Membership

Membership of a set is denoted by the symbol ∈, to be read as an abbreviation for ‘belongs to’ or ‘belonging to’ according to its grammatical context. The phrase ‘x is a member of X’ is denoted by x ∈ X. The phrase ‘x is not a member of X’ is denoted by x ∉ X. A member of a set is also said to be an element or a point of the set. Sets X and Y are equal, X = Y, if, and only if, each element of X is an element of Y and each element of Y is an element of X.