By Roger Godement

ISBN-10: 3540634142

ISBN-13: 9783540634140

Les deux premiers volumes 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 l. a. 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 traitent principalement des fonctions analytiques (théorie de Cauchy, théorie analytique des nombres et fonctions modulaires), ainsi que du calcul différentiel sur les variétés, avec un 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 sort inimitable de l'auteur, et pas seulement par son refus de l'écriture condensée en utilization dans ce nombreux manuels.

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Extra resources for Analyse Mathématique II: Calculus différentiel et intégral, séries de Fourier, fonctions holomorphes

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Vii) :::} (i): Suppose C is a set . If UC admits a well ordering then, with respect to such an order, the map S H min S with domain C\ {0} is a choice function furC. 0 The Product Theorem has a claim to being intuitively obvious. It is easy to prove in Z F if the indexing set is finite or if revealing information is available about the co-ordinate sets ; if, for example, they are all equipped with a well ordering. With appropriate definitions and a little algebraic structure, it is easy to show that the number of elements in the product of a finite number of finite sets is got by multiplying the numbers of elements in each of them; it seems unreasonable to suggest that a larger product might be empty.

1 only if 151 < 00; 5 is said to be INFINITE otherwise. These terms are used also to describe families, products, unions and so on, where the meaning should be apparent from the context; so we might use the term finite product for a product in which the indexing set is finite, or finite union for the union of the members of a finite set . Suppose 5 is a set. 2 It is a consequence of the Axiom of Choice that every set admits a well ordering. But a set 5 with 151 :::; 00 can be characterized by being equinumerous with a counting numb er or with 00.

Even Larger Cardinals So far , 00 is the only cardinal we have encountered which is not a counting number. 19, the cardinals do not form a set. The theory of cardinalities thus has wider application than the primitive theory of counting, in that it enables us to distinguish certain classes of infinite sets from one another. 7 Suppose A is a set. For each subset B of A , we define the CHARACTERISTIC FUNCTION of B in A to be the function {(x , 1) I x E B} U {(x, 0) I x E A\B} and denote it by XB, the superset A being understood from the context .

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Analyse Mathématique II: Calculus différentiel et intégral, séries de Fourier, fonctions holomorphes by Roger Godement

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