A A Kirillov ; A D Gvishiani. creator A A Kirillov. text Mathematics.-Functional analysis.-Algebra.-A A Kirillov. xxu New York : Springer-Verlag , ©1982 monographic eng

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ix, 347 pages : illustrations ; 24 cm. Functional analysis takes up topological linear spaces, topological groups, normed rings, modules of representations of topological groups in topological linear spaces, and so on. Theory.- I Concepts from Set Theory and Topology.- 1. Relations. The Axiom of Choice and Zorn's Lemma.- 2. Completions.- 3. Categories and Functors.- II Theory of Measures and Integrals..- 1. Measure Theory.- 1. Algebras of Sets.- 2. Extension of a Measure.- 3. Constructions of Measures.- 2. Measurable Functions.- 1. Properties of Measurable Functions.- 2. Convergence of Measurable Functions.- 3. Integrals.- 1. The Lebesgue Integral.- 2. Functions of Bounded Variation and the Lebesque-Stieltjes Integral.- 3. Properties of the Lebesgue Integral.- III Linear Topological Spaces and Linear Operators.- 1. General Theory.- 1. Topology, Convexity, and Seminorms.- 2. Dual Spaces.- 3. The Hahn-Banach Theorem.- 4. Banach Spaces.- 2. Linear Operators.- 1. The Space of Linear Operators.- 2. Compact Sets and Compact Operators.- 3. The Theory of Fredholm Operators.- 3. Function Spaces and Generalized Functions.- 1. Spaces of Integrable Functions.- 2. Spaces of Continuous Functions.- 3. Spaces of Smooth Functions.- 4. Generalized Functions.- 5. Operations on Generalized Functions.- 4. Hilbert Spaces.- 1. The Geometry of Hilbert Spaces.- 2. Operators on a Hilbert Space.- IV The Fourier Transformation and Elements of Harmonic Analysis.- 1. Convolutions on an Abelian Group.- 1. Convolutions of Test Functions.- 2. Convolutions of Generalized Functions.- 2. The Fourier Transformation.- 1. Characters on an Abelian Group.- 2. Fourier Series.- 3. The Fourier Integral.- 4. Fourier Transformation of Generalized Functions.- V The Spectral Theory of Operators.- 1. The Functional Calculus.- 1. Functions of Operators in a Finite-Dimensional Space..- 2. Functions of Bounded Selfadjoint Operators.- 3. Unbounded Selfadjoint Operators.- 2. Spectral Decomposition of Operators.- 1. Reduction of an Operator to the Form of Multiplication by a Function.- 2. The Spectral Theorem.- Problems.- I Concepts from Set Theory and Topology.- 1. Relations. The Axiom of Choice and Zorn's Lemma.- 2. Completions.- 3. Categories and Functors.- II Theory of Measures and Integrals..- 1. Measure Theory.- 1. Algebras of Sets.- 2. Extension of a Measure.- 3. Constructions of Measures.- 2. Measurable Functions.- 1. Properties of Measurable Functions.- 2. Convergence of Measurable Functions.- 3. Integrals.- 1. The Lebesgue Integral.- 2. Functions of Bounded Variation and the Lebesque-Stieltjes Integral.- 3. Properties of the Lebesgue Integral.- III Linear Topological Spaces and Linear Operators.- 1. General Theory.- 1. Topology, Convexity, and Seminorms.- 2. Dual Spaces.- 3. The Hahn-Banach Theorem.- 4. Banach Spaces.- 2. Linear Operators.- 1. The Space of Linear Operators.- 2. Compact Sets and Compact Operators.- 3. The Theory of Fredholm Operators.- 3. Function Spaces and Generalized Functions.- 1. Spaces of Integrable Functions.- 2. Spaces of Continuous Functions.- 3. Spaces of Smooth Functions.- 4. Generalized Functions.- 5. Operations on Generalized Functions.- 4. Hilbert Spaces.- 1. The Geometry of Hilbert Spaces.- 2. Operators on a Hilbert Space.- IV The Fourier Transformation and Elements of Harmonic Analysis.- 1. Convolutions on an Abelian Group.- 1. Convolutions of Test Functions.- 2. Convolutions of Generalized Functions.- 2. The Fourier Transformation.- 1. Characters on an Abelian Group.- 2. Fourier Series.- 3. The Fourier Integral.- 4. Fourier Transformation of Generalized Functions.- V The Spectral Theory of Operators.- 1. The Functional Calculus.- 1. Functions of Operators in a Finite-Dimensional Space..- 2. Functions of Bounded Selfadjoint Operators.- 3. Unbounded Selfadjoint Operators.- 2. Spectral Decomposition of Operators.- 1. Reduction of an Operator to the Form of Multiplication by a Function.- 2. The Spectral Theorem.- Hints.- I Concepts from Set Theory and Topology.- 1. Relations. The Axiom of Choice and Zorn's Lemma.- 2. Completions.- 3. Categories and Functors.- II Theory of Measures and Integrals..- 1. Measure Theory.- 1. Algebras of Sets.- 2. Extension of a Measure.- 3. Constructions of Measures.- 2. Measurable Functions.- 1. Properties of Measurable Functions.- 2. Convergence of Measurable Functions.- 3. Integrals.- 1. The Lebesgue Integral.- 2. Functions of Bounded Variation and the Lebesque-Stieltjes Integral.- 3. Properties of the Lebesgue Integral.- III Linear Topological Spaces and Linear Operators.- 1. General Theory.- 1. Topology, Convexity, and Seminorms.- 2. Dual Spaces.- 3. The Hahn-Banach Theorem.- 4. Banach Spaces.- 2. Linear Operators.- 1. The Space of Linear Operators.- 2. Compact Sets and Compact Operators.- 3. The Theory of Fredholm Operators.- 3. Function Spaces and Generalized Functions.- 1. Spaces of Integrable Functions.- 2. Spaces of Continuous Functions.- 3. Spaces of Smooth Functions.- 4. Generalized Functions.- 5. Operations on Generalized Functions.- 4. Hilbert Spaces.- 1. The Geometry of Hilbert Spaces.- 2. Operators on a Hilbert Space.- IV The Fourier Transformation and Elements of Harmonic Analysis.- 1. Convolutions on an Abelian Group.- 1. Convolutions of Test Functions.- 2. Convolutions of Generalized Functions.- 2. The Fourier Transformation.- 1. Characters on an Abelian Group.- 2. Fourier Series.- 3. The Fourier Integral.- 4. Fourier Transformation of Generalized Functions.- V The Spectral Theory of Operators.- 1. The Functional Calculus.- 1. Functions of Operators in a Finite-Dimensional Space..- 2. Functions of Bounded Selfadjoint Operators.- 3. Unbounded Selfadjoint Operators.- 2. Spectral Decomposition of Operators.- 1. Reduction of an Operator to the Form of Multiplication by a Function.- 2. The Spectral Theorem.- List of Notation. / A A Kirillov. 515.7 KIR 9781461381556 International Book Distributors Flat No.-17 , Prakash Apartment , 5 Ansari Road, Daryaganj , New Delhi. 180404 20181112155949.0