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AuthorZaanen, Adriaan C. author
TitleIntroduction to Operator Theory in Riesz Spaces [electronic resource] / by Adriaan C. Zaanen
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1997
Connect tohttp://dx.doi.org/10.1007/978-3-642-60637-3
Descript XI, 312p. online resource

SUMMARY

The book deals with the structure of vector lattices, i.e. Riesz spaces, and Banach lattices, as well as with operators in these spaces. The methods used are kept as simple as possible. Almost no prior knowledge of functional analysis is required. For most applications some familiarity with the oridinary Lebesgue integral is already sufficient. In this respect the book differs from other books on the subject. In most books on functional analysis (even excellent ones) Riesz spaces, Banach lattices and positive operators are mentioned only briefly, or even not at all. The present book shows how these subjects can be treated without undue extra effort. Many of the results in the book were not yet known thirty years ago; some even were not known ten years ago


CONTENT

1 Lattices and Boolean Algebras -- 1 Partially Ordered Sets -- 2 Lattices -- 3 Boolean Algebras -- 2 Riesz Spaces -- 4 Riesz Spaces -- 5 Equalities and Inequalities -- 6 Distributive Laws, the Birkhoff Inequalities and the Riesz Decomposition Property -- 3 Ideals, Bands and Disjointness -- 7 Ideals and Bands -- 8 Disjointness -- 4 Archimedean Spaces and Convergence -- 9 Archimedean Riesz spaces -- 10 Order Convergence and Uniform Convergence -- 5 Projections and Dedekind Completeness -- 11 Projection Bands -- 12 Dedekind Completeness -- 6 Complex Riesz Spaces -- 13 Complex Riesz spaces -- 7 Normed Riesz Spaces and Banach Lattices -- 14 Normed Spaces and Banach Spaces -- 15 Normed Riesz Spaces and Banach Lattices -- 8 The Riesz-Fischer Property and Order Continuous Norms -- 16 The Riesz-Fischer Property -- 17 Order Continuous Norms -- 9 Linear Operators -- 18 Linear Operators in Normed Spaces and in Riesz Spaces -- 19 Riesz Homomorphisms and Quotient Spaces -- 10 Order Bounded Operators -- 20 Order Bounded Operators -- 11 Order Continuous Operators -- 21 Order Continuous Operators -- 22 The Band of Order Continuous Operators -- 12 Carriers of Operators -- 23 Order Denseness -- 24 The Carrier of an Operator -- 13 Order Duals and Adjoint Operators -- 25 The Order Dual of a Riesz Space -- 26 Adjoint Operators -- 14 Signed Measures and the Radon-Nikodym Theorem -- 27 The Space of Signed Measures -- 28 The Radon-Nikodym Theorem -- 15 Linear Functionals on Spaces of Measurable Functions -- 29 Linear Functionals on Spaces of Measurable Functions -- 16 Embedding into the Bidual -- 30 Annihilators and Inverse Annihilators -- 31 Embedding into the Order Bidual -- 17 Freudenthalโ{128}{153}s Spectral Theorem -- 32 Projection Bands and Components -- 33 Freudenthalโ{128}{153}s Spectral Theorem -- 18 Functional Calculas and Multiplication -- 34 Functional Calculus -- 35 Multiplication -- 19 Complex Operators -- 36 Complex Operators -- 37 Synnatschkeโ{128}{153}s Theorem -- 20 Results with the Hahn-Banach Theorem -- 38 The Hahn-Banach Theorem in Normed Vector Spaces -- 39 The Hahn-Banach Theorem in Normed Riesz Spaces -- 21 Spectrum, Resolvent Set and the Krein-Rutman Theorem -- 40 Spectrum and Resolvent Set -- 41 The Krein-Rutman Theorem -- 22 Spectral Theory of Positive Operators -- 42 Irreducible Operators -- 43 The Spectrum of a Compact Irreducible Operator -- 44 The Peripheral Spectrum of a Positive Operator


Mathematics Functional analysis Operator theory Mathematics Functional Analysis Operator Theory



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