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Theorems and problems in functional analysis

Author: A A Kirillov; A D Gvishiani
Publisher: New York : Springer-Verlag, ©1982.
Series: Problem books in mathematics.
Edition/Format:   Book : EnglishView all editions and formats
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Additional Physical Format: Online version:
Kirillov, A.A. (Aleksandr Aleksandrovich), 1936-
Theorems and problems in functional analysis.
New York : Springer-Verlag, ©1982
(OCoLC)624457501
Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: A A Kirillov; A D Gvishiani
ISBN: 038790638X 9780387906386 354090638X 9783540906384
OCLC Number: 8114713
Language Note: Translation of: Teoremy i zadachi funkt︠s︡ionalʹnogo analiza.
Notes: Translation of: Teoremy i zadachi funkt︠s︡ionalʹnogo analiza.
Description: ix, 347 pages : illustrations ; 24 cm.
Contents: 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.
Series Title: Problem books in mathematics.
Other Titles: Teoremy i zadachi funkt︠s︡ionalʹnogo analiza.
Responsibility: A.A. Kirillov, A.D. Gvishiani ; translated by Harold H. McFaden.
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