An Introduction to Banach Space Theory
Springer Science & Business Media, 9 окт. 1998 г. - Всего страниц: 596
Many important reference works in Banach space theory have appeared since Banach's "Théorie des Opérations Linéaires", the impetus for the development of much of the modern theory in this field. While these works are classical starting points for the graduate student wishing to do research in Banach space theory, they can be formidable reading for the student who has just completed a course in measure theory and integration that introduces the L_p spaces and would like to know more about Banach spaces in general. The purpose of this book is to bridge this gap and provide an introduction to the basic theory of Banach spaces and functional analysis. It prepares students for further study of both the classical works and current research. It is accessible to students who understand the basic properties of L_p spaces but have not had a course in functional analysis. The book is sprinkled liberally with examples, historical notes, and references to original sources. Over 450 exercises provide supplementary examples and counterexamples and give students practice in the use of the results developed in the text.
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adjoint anxn argument assumed Banach algebra Banach space basic sequence bounded linear functional bounded linear operator bounded subset boundedness Cauchy sequence closed convex subset closed subspace closed unit ball compact subset complete continuous Corollary countable defined direct sum dual space easy to check element example Exercise F-space fact finishes the proof finite-dimensional following are equivalent following result formula Gateaux Hausdorff space implies internal direct sum isometric isomorphism Lemma locally convex metric space modulus natural map neighborhood nonnegative nonzero norm topology norm-attaining normed space notation Notice one-to-one open subset positive integer positive number Proposition Prove quotient real number respectively rotund normed space separable sequence xn sequentially standard unit vector strongly rotund subspace Suppose that xn theory topological space ultranet uniform uniformly smooth vector space operations vector topology weak topology weakly compact