History of Functional Analysis presents functional analysis as a rather complex blend of algebra and topology, with its evolution influenced by the development of these two branches of mathematics. The book adopts a narrower definition―one that is assumed to satisfy various algebraic and topological conditions. A moment of reflections shows that this already covers a large part of modern analysis, in particular, the theory of partial differential equations. This volume comprises nine chapters, the first of which focuses on linear differential equations and the Sturm-Liouville problem. The succeeding chapters go on to discuss the ""crypto-integral"" equations, including the Dirichlet principle and the Beer-Neumann method; the equation of vibrating membranes, including the contributions of Poincare and H.A. Schwarz's 1885 paper; and the idea of infinite dimension. Other chapters cover the crucial years and the definition of Hilbert space, including Fredholm's discovery and the contributions of Hilbert; duality and the definition of normed spaces, including the Hahn-Banach theorem and the method of the gliding hump and Baire category; spectral theory after 1900, including the theories and works of F. Riesz, Hilbert, von Neumann, Weyl, and Carleman; locally convex spaces and the theory of distributions; and applications of functional analysis to differential and partial differential equations. This book will be of interest to practitioners in the fields of mathematics and statistics.

History of Functional Analysis (Volume 49) (North-Holland Mathematics Studies, Volume 49)

History

Dieudonné, J.

Europe

North-Holland

Foundations of modern analysis (Pure and applied mathematics; a series of monographs and textbooks)

Science & Math

Mathematics

Academic Press

Many of the earliest books, particularly those dating back to the 1900s and before, are now extremely scarce and increasingly expensive. We are republishing these classic works in affordable, high quality, modern editions, using the original text and artwork.

Foundations of Modern Analysis

Hesperides Press

FOUNDATIONS OF MODERN ANALYSIS. Enlarged and Corrected Printing J. DIEUDONNE This book is the first volume of a treatise which will eventually consist offour volumes. It is also an enlarged and corrected printing, essentially without changes, of my Foundations of Modern Analysis. Many readers, colleagues, and friends have urged me to write a sequelto that book, and in the end I became convinced that there was a place fora survey of modern analysis, somewhere between the minimum tool kitof an elementary nature which I had intended to write, and specialist monographs leading to the frontiers of research. My experience of teachinghas also persuaded me that the mathematical apprentice, after taking the firststep of Foundations, needs further guidance and a kind of general birdseye view of his subject before he is launched onto the ocean of mathematical literature or set on the narrow path of his own topic of research.Thus I have finally been led to attempt to write an equivalent, for the mathematicians of 1970, of what the Cours dAnalyse of Jordan, Picard,and Goursat were for mathematical students between 1880 and 1920.It is manifestly out of the question to attempt encyclopedic coverage, and certainly superfluous to rewrite the works of N. Bourbaki. I have therefore been obliged to cut ruthlessly in order to keep within limits comparable tothose of the classical treatises. I have opted for breadth rather than depth, inthe opinion that it is better to show the reader rudiments of many branchesof modern analysis rather than to provide him with a complete and detailedexposition of a small number of topics.Experience seems to show that the student usually finds a new theorydifficult to grasp at a first reading. He needs to return to it several times beforehe becomes really familiar with it and can distinguish for himself whichare the essential ideas and which results are of minor importance, and onlythen will he be able to apply it intelligently. The chapters of this treatise are therefore samples rather than complete theories: indeed, I have systematically tried not to be exhaustive. The works quoted in the bibliography willalways enable the reader to go deeper into any particular theory.However, I have refused to distort the main ideas of analysis by presentingthem in too specialized a form, and thereby obscuring their power andgenerality. It gives a false impression, for example, if differential geometryis restricted to two or three dimensions, or if integration is restricted to Lebesgue measure, on the pretext of making these subjects more accessible orintuitive.On the other hand I do not believe that the essential content of the ideasinvolved is lost, in a first study, by restricting attention to separable metrizabletopological spaces. The mathematicians of my own generation were certainlyright to banish, hypotheses of countability wherever they were not needed: thiswas the only way to get a clear understanding.

Foundations of Modern Analysis (Pure and Applied Mathematics, 10)

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Pour l'honneur de l'esprit humain

Hachette Littérature

It is a great satisfaction for a mathematician to witness the growth and expansion of a theory in which he has taken some part during its early years. When H. Weyl coined the words "classical groups", foremost in his mind were their connections with invariant theory, which his famous book helped to revive. Although his approach in that book was deliberately algebraic, his interest in these groups directly derived from his pioneering study of the special case in which the scalars are real or complex numbers, where for the first time he injected Topology into Lie theory. But ever since the definition of Lie groups, the analogy between simple classical groups over finite fields and simple classical groups over IR or C had been observed, even if the concept of "simplicity" was not quite the same in both cases. With the discovery of the exceptional simple complex Lie algebras by Killing and E. Cartan, it was natural to look for corresponding groups over finite fields, and already around 1900 this was done by Dickson for the exceptional Lie algebras G and E • However, a deep reason for this 2 6 parallelism was missing, and it is only Chevalley who, in 1955 and 1961, discovered that to each complex simple Lie algebra corresponds, by a uniform process, a group scheme (fj over the ring Z of integers, from which, for any field K, could be derived a group (fj(K).

The Classical Groups and K-Theory (Grundlehren der mathematischen Wissenschaften, 291)

Hahn, Alexander J.

O'Meara, O.Timothy

Springer

21. Kompakte Liesche Gruppen und halbeinfache Liesche Gruppen.- 22. Harmonische Analysis.- Anhang. Ergänzungen aus der Algebra (Fortsetzung des Anhangs zu Band 4).- A.22 Einfache Moduln.- A.23. Halbeinfache Moduln.- A.24. Beispiele.- A.25. Kanonische Zerlegung eines Endomorphismus.- Literatur.- Bezeichnungen.

Grundzüge der modernen Analysis: Band 5/6 (Logik und Grundlagen der Mathematik) (German Edition)

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