By Roger Godement

Les deux premiers volumes de cet ouvrage sont consacrés aux fonctions dans R ou C, y compris los angeles théorie élémentaire des séries et intégrales de Fourier et une partie de celle des fonctions holomorphes. L'exposé, non strictement linéaire, mix symptoms historiques et raisonnements rigoureux. Il montre los angeles diversité des voies d'accès aux principaux résultats afin de familiariser le lecteur avec les méthodes de raisonnement et idées fondamentales plutôt qu'avec les innovations de calcul, aspect de vue utile aussi aux personnes travaillant seules.

Les volumes three et four traiteront principalement des fonctions analytiques (théorie de Cauchy, théorie analytique des nombres et fonctions modulaires), ainsi que du calcul différentiel sur les variétés, avec un court docket exposé de l'intégrale de Lebesgue, en suivant d'assez près le célèbre cours donné longtemps par l'auteur à l'Université Paris 7.

On reconnaîtra dans ce nouvel ouvrage le kind inimitable de l'auteur, et pas seulement par son refus de l'écriture condensée en utilization dans de nombreux manuels.

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**Additional resources for Analyse mathématique I: Convergence, fonctions élémentaires**

**Example text**

If 1 ∫ 0 f ( x) dx ≠ 0, then there exists a point x in the interval [0, 1] such that f (x) ≠ 0. Symbolically, we have p ⇒ q, where p: 1 ∫ 0 f ( x) dx ≠ 0 q : ∃ x in [0, 1] f (x) ≠ 0. The contrapositive implication, ~ q ⇒ ~ p, can be written as If for every x in [0, 1], f ( x) = 0, then 1 ∫ 0 f ( x) = 0. This is much easier to prove. Instead of having to conclude the existence of an x in [0, 1] with a particular property, we are given that every x in [0, 1] has a different property. Indeed, the proof now follows directly from the definition of the integral, since each of the terms in any upper or lower Riemann sum will be zero.

Then {Ex : x ∈ S} is a partition of S. The relation “belongs to the same piece as” is the same as R. Conversely, if p is a partition of S, let P be defined by x P y iff x and y are in the same piece of the partition. Then P is an equivalence relation and the corresponding partition into equivalence classes is the same as p. Sets and Functions Proof: Let R be an equivalence relation on S. We have already shown that {Ex : x ∈ S} is a partition. ” Then xP y iff x, y ∈ Ez for some z ∈ S iff x R z and y R z for some z ∈ S iff x R y.

6 PRACTICE Let A = {1, 2, 3, 4, 5} B = {x : x = 2k for some k ∈ N} C = {x ∈ N : x < 6}. Which of the following statements are true? 2(d) we found that the collection D of all prime numbers between 8 and 10 is a legitimate set. This is so because the statement “ x ∈ D ” is always false, since there are no prime numbers between 8 and 10. Thus D is an example of the empty set, a set with no members. It is not difficult to show (Exercise 18) that there is only one empty set, and we denote it by ∅. For our first theorem we shall prove that the empty set is a subset of every set.