An Introduction to Banach Space Theory
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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algebra argument assumed Banach space basic sequence basis bounded linear operator called Cauchy characterization closed subspace collection compact complete condition consequence contains continuous converges convex subset Corollary countable defined Definition denoted dense dual space easy element equivalent example Exercise extension fact finite finite-dimensional follows formula Give given Hausdorff holds identity immediately implies includes induced isometric isomorphism Lemma limit linear functional locally measure metric space natural needed neighborhood nonempty nonzero normed space Notice obtained origin particular positive integer preceding proof Proposition Prove reflexive relatively respectively result rotund satisfies scalars sense separable sequence sequentially shown smooth space X standard statement subsequence Suppose Theorem theory topological space topology uniform uniformly rotund usual vector space weak weakly weakly compact