By Roger Godement

ISBN-10: 3540438416

ISBN-13: 9783540438410

Ce 4?me quantity de l'ouvrage *Analyse math?matique* initiera le lecteur ? l'analyse fonctionnelle (int?gration, espaces de Hilbert, examine harmonique en th?orie des groupes) et aux m?thodes de l. a. th?orie des fonctions modulaires (s?ries L et theta, fonctions elliptiques, utilization de l'alg?bre de Lie de SL2). Tout comme pour les volumes 1 ? three, on reconna?tra ici encore, le variety inimitable de l'auteur et pas seulement par son refus de l'ecriture condens?e en utilization dans de nombreux manuels. Mariant judicieusement les math?matiques dites 'modernes' et' classiques', l. a. premi?re partie (Int?gration) est d'utilit? universelle tandis que los angeles seconde oriente le lecteur vers un domaine de recherche sp?cialis? et tr?s actif, avec de vastes g?n?ralisations possibles.

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**Additional info for Analyse mathématique IV: Intégration et théorie spectrale, analyse harmonique, le jardin des délices modulaires **

**Example text**

Anti-symmetry of follows from the fact that (bn − an ) cannot eventually be both strictly positive and strictly negative. This makes R a linearly ordered field. Given a real number x = [an ] = [bn ], let |x| := [|an |], which makes sense since |an | − |am | |an − am | ∞ 0 as n, m ∞ →, |an | − |bn | |an − bn | ∞ 0 as n ∞ →. In fact |x| = x when x > 0 and |x| = −x when x < 0, so it satisfies the properties |x| 0, |x| = 0 ⇐ x = 0, |−x| = |x|, and |x + y| |x| + |y|. 2(1). Q is dense in R: Note that a rational number a can be represented in R by the constant sequence [a, a, .

If F is a closed set in Y and f : X ∞ Y is a continuous function, then f −1 F is closed in X . ). 12. It is not enough that f (x, y) is continuous in x and y separately in order that f be continuous. For example, show that the function f (x, y) := x2 xy , + y2 f (0, 0) := 0, is discontinuous at (0, 0) even though f (xn , 0) ∞ 0, f (0, yn ) ∞ 0, when xn ∞ 0, yn ∞ 0. It needs to be “jointly continuous” in the sense that f (xn , yn ) ∞ f (x, y) for any (xn , yn ) ∞ (x, y). 36 3 Convergence and Continuity ⇐ 13.

The identity function I : X ∞ X, x ∩∞ x, is always continuous. 2. The functions that map the real number x to x + 1, 2x, x n (n ♦ N), a x (a > 0), and |x| are all continuous. 3. , if xn ∞ x and yn ∞ y then xn + yn ∞ x + y and xn yn ∞ x y. Deduce that if f, g : X ∞ R are continuous functions, then so are f + g and f g. For example, the polynomials on R are continuous. , max(xn , yn ) ∞ max(x, y). 4. The function f : ]0, √[ ∞ ]0, √[, defined by f (x) := 1/x is continuous. 5. Conjugation in C, z ∩∞ z¯ , is continuous.

### Analyse mathématique IV: Intégration et théorie spectrale, analyse harmonique, le jardin des délices modulaires by Roger Godement

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