Chapter 1
Prerequisites

The purpose of this preliminary chapter is to clarify terminology and notation and to recall elementary properties of countability and of real numbers. The heart of the matter begins in Chapter 2, p. 11.

1.1. Sets, mappings, orders

Set theory notations. We utilize the usual notations of set theory1, namely = for "is equal to", ? for "is an element of", ? for "for all", ? for "there exists", ? for "implies", ? for "results from", ? for "is equivalent to". A barred symbol designates the negation of the property, for example ? means "is not equal to".

We denote {a, b, ., z} the set consisting of a, b, . and z, {a : P(a)} or {a}P(a) the set of the a having the property P(a), and Ø the empty set. We denote ? set inclusion, n their union and n their intersection. We denote ? for "is a superset of" and U \ V = {u ? U : u ? V}.

We denote (a, b, ., z) the ordered set composed of a, b, . and z. Therefore, (a, b) ? (b, a) if a? b. On the contrary, {a, b} = {b, a} always holds.

We denote U1× U2× ·· × Ud = { (u1, u2, ., ud) : ui ? Ui for all i} the product2 of the sets U1,. , Ud and denote Ud = { (u1, u2, ., ud) : ui ? U for every i}. In particular, U2 = U × U.

Given a set I and for each i ? I, a set Ui, we denote ?i ?I Ui the union of all the Ui, ?i ? I Ui their intersection and ?i ? I Ui = {(ui)i ? I:ui ? Ui, ? i ? I} their product. These quantities are related by De Morgan's laws3:

The axiomatic construction of the set theory is, for instance, achieved in [SCHWARTZ, 100, chap. I] or [BOURBAKI, 16].

Mappings. A mapping T from a set X into a set Y is the data, for each u ? X, of an element T(u) ? Y. It is said that T is defined on X and that T(u) is the image by T of the point u. The image4 by T of a subset U of X is the set

It is said that T is injective if T(u) = T(v) yields v = u, that it is surjective if T(X) = Y, and that it is bijective if it is injective and surjective. It is also said that T is an injection, a surjection or a bijection.

The preimage by T of a subset W of Y is the set

If T is bijective, there exists a unique mapping T-1 from Y into X, known as inverse of T, such that, for all u ? X,

Then, T is the inverse mapping of T-1, that is T(T-1(u)) = u.

The restriction of T to a subset U of X is the mapping T|U defined on U by T|U(u) = T(u) for all u ? U. An extension of T (to a set V ? X) is any mapping (defined on V) of which T is a restriction.

Ordering. An order on a set U is a relation ? among some pairs of its elements such that, for all u, v and w in U:u ? u;u ? v and v ? imply u = v;u ? v and v ? w imply u ? w. It is then said that U is ordered by ?. We denote u ? v if u ? v and u? v; and u ? v if v ? u; and u ? v if v ? u.

It is said that U is totally ordered by ? if, in addition, for all u and v in U, there is u ? v or v ? u.

If V is a subset of a set U ordered by ?, V ? U and m ? U, it is said that m is a majorant or an upper bound of V if every v ? V verifies v ? m. It is said that m is the supremum or the least upper bound of V if m is a majorant of V and if any other majorant m´ of V verifies m ? m´; if it exists, it is unique and denoted by sup V. It is said that m is the maximum of V if m is an element of V and is a majorant of V; if it exists, it is unique and denoted by max V.

We respectively call minorant, lower bound, infimum or greatest lower bound, and minimum the elements obtained by replacing ? by ? in these definitions. If they exist, the infimum is denoted inf V and the minimum is denoted min V and they are unique. A maximum is a supremum, and the latter is a majorant; a minimum is an infimum, and the latter is a minorant.

1.2. Countability

We denote N = {1,2, .} the set of "natural" numbers (or positive integers)5, N* = {0,1,2, .}6 the set of non-negative integers and Z = {., -2, -1, 0, 1, 2, .} the set of all integers7.

A sequence of a set U is the data, for every n ? N, of an element un ? U. It is an ordered set that is denoted (un)n ? N. This is equivalent to considering that it is a mapping from N into U, namely n ? un. A subsequence of (un)n ? N is any sequence of the form (us(n))n ? N, where s(n + 1) > s(n), that is where s is a strictly increasing mapping from N into itself.

A set U is said countable if there is a bijection from U onto a subset of N. It is said that U is finite if there is a bijection from U on an integer interval, that is a set of the form

with m and n in . It is said that U is infinite if it is not finite (an infinite countable set is sometimes said to be denumerable [LANG, 66, p. 6]).

Let us give countability properties, see, for instance, [SCHWARTZ, 100, p. 104-108] or [SIMON, 107].

THEOREM 1.1.- (a) Every finite or empty set is countable.

1. (b) Every subset of a countable set is countable.
2. (c) Every image of a countable set by a mapping is countable.
3. (d) Every finite product of countable sets is countable.
4. (e) Every countable union of countable sets is countable.
5. (f) The set of the finite subsets of a countable set is countable.
6. (g) The set of the finite sequences of a countable set is countable. ?

1.3. Construction of R

Motivation. The heart of the matter, which begins in Chapter 2, calling on R (for Definitions 2.1 of vector spaces and 2.2 of norms and semi-norms, etc.) it had seemed to us useful to recall a self-contained construction of the real numbers set and to establish its properties that we will use. Thus, here are the outlines of its construction by cuts (the construction by completion, frequently used, is based on more developed notions - Cauchy sequences, equivalent sequences - and is longer). For a detailed presentation, the reader is, for instance, referred to [ENDERTON, 39, p. 112-120] or [SIMON, 107].

The set of rational numbers. A rational number8 is any subset of Z× N of the form

where m ? N and n ? N. The set of all rational numbers is denoted by . Its addition, its multiplication and its order are defined by

The set R of real numbers. A real number9 is any semi-cut of , that is any set x such that:

The set of real numbers is denoted by R.

Given a ? , a real number is defined by . We identify a ? to which gives ? R. This is permissible because = gives a = ß. By identifying in addition any integer m to the rational number m/1, we get

Operations on R. A total order on R is defined by10

We denote x < y if x = y and x ?...