Advanced Functional Analysis.

Cover -- Half Title -- Title Page -- Copyright Page -- Contents -- Preface -- Authors -- Symbol Description -- 1. Fundamentals of Linear and Topological Spaces -- 1.1 Introduction -- 1.2 Partially ordered sets and Zorn's lemma -- 1.3 Important notations -- 1.4 Inequalities -- 1.5 Linear spaces, algebraic bases and dimension -- 1.6 Linear maps -- 1.7 Semimetric and metric spaces -- 1.8 Seminormed and normed spaces -- 1.9 Topological spaces -- 1.10 Net convergence -- 1.11 Subnets -- 1.12 Compact sets -- 2. Linear Topological Spaces -- 2.1 Introduction -- 2.2 Set arithmetic and convexity -- 2.3 Convex and affine sets -- 2.4 Balloons and cones -- 2.5 Quotient spaces -- 2.6 Supremum and weak topologies -- 2.7 Product topology -- 2.8 Properties of linear topological spaces -- 2.9 Closed maps -- 2.10 Baire's category theorem -- 2.11 Locally convex spaces -- 2.12 Quotient topologies -- 3. Linear Metric Spaces -- 3.1 Introduction -- 3.2 Paranormed spaces -- 3.3 Properties of paranormed and seminormed spaces -- 3.4 Schauder basis -- 3.5 Open mapping theorem -- 3.6 The closed graph theorem -- 3.7 Uniform boundedness principle -- 3.8 Properties of seminorms -- 3.9 The Minkowski functional -- 3.10 Local convexity -- 3.11 Metrizability -- 4. Banach Spaces -- 4.1 Introduction -- 4.2 Some basic properties -- 4.3 Examples -- 4.4 Bounded linear operators -- 4.5 The Hahn-Banach theorem -- 4.6 Important theorems -- 4.7 Representation theorems -- 4.8 Reflexivity -- 4.9 Adjoint operators -- 4.10 Quotient spaces -- 4.11 Cauchy nets, summable families -- 4.12 Equivalent norms -- 4.13 Compactness and the Riesz lemma -- 4.14 Compact operators -- 4.15 Operators with closed range -- 5. Hilbert Spaces -- 5.1 Introduction -- 5.2 Inner product spaces -- 5.3 Elementary properties and Hilbert spaces -- 5.4 Examples -- 5.5 Theorem P. Jordan-J. von Neumann -- 5.6 Orthogonality.

5.7 Theorem of the element of minimal norm -- 5.8 Theorem on the orthogonal decomposition -- 5.9 Riesz representation theorem -- 5.10 Bessel's inequality, Fourier coefficients -- 5.11 Parseval's equality, Hilbert basis -- 5.12 Gram-Schmidt orthogonalization method -- 5.13 Hilbert adjoint operators -- 5.14 Hermitian, normal, positive and unitary operators -- 5.15 Projectors and orthogonal projectors -- 5.16 On the norm of idempotent operators -- 6. Banach Algebras -- 6.1 Introduction -- 6.2 The concept of a Banach algebra -- 6.3 Examples -- 6.4 Invertibility -- 6.5 Spectrum and resolvent -- 6.6 Spectral radius -- 6.7 Topological divisor of zero -- 6.8 Spectrum and subalgebras -- 6.9 One Hochwald-Morell and two Harte theorems -- 6.10 B(X) as a Banach algebra -- 6.11 The spectrum of self-adjoint and normal operators -- 6.12 Spectrum of compact operators -- 6.13 C*-algebras -- 6.14 Inverting the difference of projections -- 7. Measures of Noncompactness -- 7.1 Introduction -- 7.2 Preliminary results -- 7.3 Fixed point theorems -- 7.4 Hausdorff distance -- 7.5 The axioms of measures of noncompactness -- 7.6 The Kuratowski measure of noncompactness -- 7.7 The Hausdorff measure of noncompactness -- 7.8 The inner Hausdorff and the separation measures of noncompactness -- 7.9 The Golden stein-Goh'berg-Markus theorem -- 7.10 The Darbo-Sadovskiĭ theorem of operators -- 7.11 Measures of noncompactness of operators -- 7.12 Ascent and descent of operators -- 7.13 Fredholm theorems for operators with || . ||x &lt -- 1 -- 8. Fredholm Theory -- 8.1 Introduction -- 8.2 Preliminary results -- 8.3 Generalized kernel and generalized range of operators -- 8.4 Fredholm and semi-Fredholm operators -- 8.5 The index theorem -- 8.6 Atkinson's theorem -- 8.7 Yood results for Φ+- and Φ+- operators -- 8.8 Φ, Φi and Φr operators and Calkin algebras -- 8.9 Φ+ and Φ_ are open sets.

8.10 The punctured neighborhood theorem -- 8.11 The Kato decomposition theorem -- 8.12 Browder and semi-Browder operators -- 8.13 Essential spectra -- 8.14 Essential type subsets of the spectrum -- 9. Sequence Spaces -- 9.1 Introduction -- 9.2 FK and BK spaces -- 9.3 Matrix transformations into ℓ∞, c, c0, and ℓ1 -- 9.4 Dual spaces -- 9.5 Relations between di erent kinds of duals -- 9.6 Properties of the transposes of matrices -- 9.7 Matrix transformations between the classical sequence spaces -- 9.8 Compact matrix operators -- 9.9 The class K(c) -- 9.10 Compact matrix operators between the classical sequence spaces -- 10. Fixed Point Theory -- 10.1 Introduction -- 10.2 Banach contraction principle -- 10.3 Edelstein's results -- 10.4 Rakotch's results -- 10.5 Boyd and Wong's nonlinear contraction -- 10.6 Theorem of Meir-Keeler -- 10.7 Theorems by Kannan, Chatterjee, and Zamfirescu -- 10.8 Cirič's generalized contraction -- 10.9 The Reich and Hardy{Rogers theorems -- 10.10 Cirič's quasi-contraction -- 10.11 Caristi's theorem -- 10.12 A theorem by Bollenbacher and Hicks -- 10.13 Mann iteration -- 10.14 Continuous functions on compact real intervals -- Bibliography -- Russian Bibliography -- Index.

This book can be used as a textbook in advanced functional analysis, which is a modern and important field in mathematics, for graduate and postgraduate courses and seminars at universities. At the same time, it enables the interested readers to do their own research.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.