Linear Differential Equations and the Sturm-Liouville Problem. The ``Crypto-Integral'' Equations. The Equation of Vibrating Membranes. The Idea of Infinite Dimension. The Crucial Years and the Definition of Hilbert Space. Duality and the Definition of Normed Spaces. Spectral Theory after 1900. Locally Convex Spaces and the Theory of Distributions. Applications of Functional Analysis to Differential and Partial Differential Equations. References. Author and Subject Index.
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.
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- © North Holland 1981
- 1st January 1983
- North Holland
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@qu:An excellent supplement to any standard course in functional analysis. @source:American Mathematical Monthly @qu:The admirable book under review, written by an eminently qualified mathematician, makes a notable contribution to the understanding of the historical process that has shaped what is known today as functional analysis...Dieudonné has given us a very readable and exciting account of how functional analysis has evolved...This is essential reading for functional analysts who wish to know how their subject came into existence. @source:Bulletin of the American Mathematical Society