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.

Let X and Y be normed affine spaces and let A be a subset of X and B a subset of Y. A map f: A → B is said to be a smooth homeomorphism if it is a homeomorphism and if each of the maps X ↣ Y; x ⇝ f(x) and is smooth (C1). A map f : X ↣ Y is said to be locally a smooth homeomorphism at a point a ∈ X if there are open neighbourhoods A of a in X and fi of f(a) in Y such that f⊢(A) = B; x ⇝ f(x) and the map is a smooth homeomorphism.

The main theorem of this chapter, the ‘inverse function theorem’, is a criterion for a map f: X ↣ Y to be locally a smooth homeomorphism, when X and Y are complete normed affine spaces. Important corollaries include the ‘implicit function theorem’ and various propositions preliminary to the study of smooth submanifolds. Another corollary is the ‘fundamental theorem of algebra’.

Higher differentials are considered briefly at the end of the chapter.

The increment formula

One of the main tools used in the proof of the inverse function theorem is the ‘increment formula’ (‘la formule des accroissements finis’). This inequality replaces the ‘mean value theorem’, which occurs at this stage in many treatments of the calculus of real-valued functions of one real variable.

In this chapter we discuss briefly the definitions of group, ring and field and the additive, multiplicative and order properties of the ring of integersZ and the rational, real and complex fieldsQ, R and C. Since there are many detailed accounts of these topics available, most of the proofs are omitted. For more details of the construction of R see, for example,.

Groups

Products on a set were defined in Chapter 1, and we have already encountered several specimens, for example, composition either on XX or on X!, where X is any set. Further examples are union and intersection on Sub X, and addition, multiplication and exponentiation on the set of natural numbers ω.

A group consists of a set, G, say, and a product with unity on G that is associative and admits inverses, such a product being said to be a group structure for the set G.

For example, composition is a group structure for X!. This group is called the group of permutations of X. On the other hand composition is not a group structure for Xx, unless # X = 0 or 1, for in all other cases there exist non-invertible transformations of X.

A set may have many group structures. When we are only interested in one such structure it is common practice to use the same letter, G, say, to denote both the group and the underlying set, and to use the notational conventions of ordinary multiplication or of composition.