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Functions in R and C, including the theory of Fourier series, Fourier integrals and part of that of holomorphic functions, form the focal topic of these two volumes. Based on a course given by the author to large audiences at Paris VII University for many years, the exposition proceeds somewhat nonlinearly, blending rigorous mathematics skilfully with didactical and historical considerations. It sets out to illustrate the variety of possible approaches to the main results, in order to initiate the reader to methods, the underlying reasoning, and fundamental ideas. It is suitable for both teaching and self-study. In his familiar, personal style, the author emphasizes ideas over calculations and, avoiding the condensed style frequently found in textbooks, explains these ideas without parsimony of words. The French edition in four volumes, published from 1998, has met with resounding success: the first two volumes are now available in English.

Mathematical analysis. --- Calculus. --- Fourier series. --- Holomorphic functions. --- Fourier integrals --- Series, Fourier --- Series, Trigonometric --- Trigonometric series --- Calculus --- Fourier analysis --- Harmonic analysis --- Harmonic functions --- Analysis (Mathematics) --- Fluxions (Mathematics) --- Infinitesimal calculus --- Limits (Mathematics) --- Mathematical analysis --- Functions --- Geometry, Infinitesimal --- 517.1 Mathematical analysis --- Functions, Holomorphic --- Functions of several complex variables --- Mathematics. --- Real Functions. --- Measure and Integration. --- Math --- Science --- Functions of real variables. --- Measure theory. --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Real variables --- Functions of complex variables

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Analysis Volume IV introduces the reader to functional analysis (integration, Hilbert spaces, harmonic analysis in group theory) and to the methods of the theory of modular functions (theta and L series, elliptic functions, use of the Lie algebra of SL2). As in volumes I to III, the inimitable style of the author is recognizable here too, not only because of his refusal to write in the compact style used nowadays in many textbooks. The first part (Integration), a wise combination of mathematics said to be modern and classical, is universally useful whereas the second part leads the reader towards a very active and specialized field of research, with possibly broad generalizations.

Mathematics. --- Real Functions. --- Mathématiques --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Functions of real variables. --- Math --- Science --- Functional analysis. --- Real variables --- Functions of complex variables

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This textbook covers the general theory of Lie groups. By first considering the case of linear groups (following von Neumann's method) before proceeding to the general case, the reader is naturally introduced to Lie theory. Written by a master of the subject and influential member of the Bourbaki group, the French edition of this textbook has been used by several generations of students. This translation preserves the distinctive style and lively exposition of the original. Requiring only basics of topology and algebra, this book offers an engaging introduction to Lie groups for graduate students and a valuable resource for researchers.

Mathematics. --- Topological groups. --- Lie groups. --- Topological Groups, Lie Groups. --- Topological Groups. --- Groups, Topological --- Continuous groups --- Groups & group theory. --- Mathematics --- Group Theory. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups

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Algebraic topology

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Mathématiques --- Mathematics. --- Analyse mathématique --- Mathematical analysis. --- Mathematical analysis --- Calcul différentiel --- Calcul intégral