# Analyse réelle et complexe : Cours et exercices by Rudin

By Rudin

Cet ouvrage présente les thoughts de base et les théorèmes fondamentaux pour un cours de moment cycle. L'accent est mis sur les profondes connexions reliant les domaines traditionnellement disjoints de l'analyse: sont ainsi réunies l'analyse réelle et l'analyse complexe. Le livre aborde également quelques-unes des idées qui fondent l'analyse fonctionnelle. Cette troisième édition contient un nouveau chapitre consacré à los angeles différentiation, et il permet au lecteur de se familiariser avec les fonctions maximales. Les notions d'équicontinuité et de convergence sont présentées avec plus de précision, ainsi que le comportement à los angeles frontière des functions conformes étudiées par le moyen du théorème de Lindelof sur les valeurs asymptotiques des fonctions holomorphes bornées dans un disque.

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But since every non-empty subset of N has both a first and a last element, one sees that every subset of N + {m}which is not empty has this property as well. Therefore MeT, which means that the theorem is true for M. Theorem 45. If a set M is well-ordered and also inversely well-ordered, it is inductive finite. Proof. Let us assume the existence of elements y of M such that the set of all x = y was not inductive finite. Among these y there is then a least one, say m. There is a predecessor mi of m.

Proof. We obtain all sets M by taking in (1) all propositional functions which, by the operations of the predicate calculus, can be built from atomic statements yez and only contain x as a free variable. We may replace x by XQ, letting the bound variables be denoted by Xi, x2, .... Further, 0(x) may be written in prenex normal form, while its matrix is written in conjunctive normal form. Then we will get an enumeration of all 0(x) by enumerating all finite sequences consisting first of some pairs of integers corresponding to the quantifiers of the prefix, the first number being the index of the x which occurs as quantifier, the last number being 0 or 1 according as the quantifier is universal or existential.

We have seen that the ordinal numbers are well-ordered by the relation < . It is then natural to ask how the cardinal numbers behave. Because of the comparability of the ordinals it is immediately clear that the cardinal numbers are comparable; indeed, if M and N are any two sets and they are in some way well-ordered, then either M is similar to, and thus equivalent to, some initial part of N or inversely. Thus we have either M = N or N = M. Now let T be a set of sets. I assert that the cardinal numbers represented by the elements A,B,C,....