By Roger Godement
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
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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