Introduction to Operator Theory I: Elements of Functional AnalysisThis book was written expressly to serve as a textbook for a one- or two-semester introductory graduate course in functional analysis. Its (soon to be published) companion volume, Operators on Hilbert Space, is in tended to be used as a textbook for a subsequent course in operator theory. In writing these books we have naturally been concerned with the level of preparation of the potential reader, and, roughly speaking, we suppose him to be familiar with the approximate equivalent of a one-semester course in each of the following areas: linear algebra, general topology, complex analysis, and measure theory. Experience has taught us, however, that such a sequence of courses inevitably fails to treat certain topics that are important in the study of functional analysis and operator theory. For example, tensor products are frequently not discussed in a first course in linear algebra. Likewise for the topics of convergence of nets and the Baire category theorem in a course in topology, and the connections between measure and topology in a course in measure theory. For this reason we have chosen to devote the first ten chapters of this volume (entitled Part I) to topics of a preliminary nature. In other words, Part I summarizes in considerable detail what a student should (and eventually must) know in order to study functional analysis and operator theory successfully. |
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Introduction to Operator Theory I: Elements of Functional Analysis A. Brown,C. Pearcy No preview available - 2013 |
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analytic function arbitrary Banach algebra Banach space belongs Borel measure Borel sets Cauchy closed coincides collection compact Hausdorff space compact subset complex Borel measure complex numbers complex-valued function contains converges definition direct sum disjoint dual pair element Example finite measure function defined function f Hausdorff space Hence Hint indexed family infinite interval invertible Lebesgue integral Lemma let f let ƒ linear manifold linear space linear submanifold linear topology linear transformation mapping measurable function measurable set measurable subset measure space metric space Moreover neighborhood nonempty normed space o-finite o-ring open set open subset ordinal number partition positive integer positive number Prob Problem PROOF Proposition pseudonorm quasinorm real number real-valued function respect to µ scalars sequence Show subspace summable theorem topological linear space topological space topology induced vector space Verify weak topology weakly zero