Theorems and Problems in Functional Analysis

Theory.- I Concepts from Set Theory and Topology.- x1. Relations. The Axiom of Choice and Zorn's Lemma.- x2. Completions.- x3. Categories and Functors.- II Theory of Measures and Integrals..- x1. Measure Theory.- 1. Algebras of Sets.- 2. Extension of a Measure.- 3. Constructions of Measures.- x2. Measurable Functions.- 1. Properties of Measurable Functions.- 2. Convergence of Measurable Functions.- x3. 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.- x1. General Theory.- 1. Topology, Convexity, and Seminorms.- 2. Dual Spaces.- 3. The Hahn-Banach Theorem.- 4. Banach Spaces.- x2. Linear Operators.- 1. The Space of Linear Operators.- 2. Compact Sets and Compact Operators.- 3. The Theory of Fredholm Operators.- x3. 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.- x4. Hilbert Spaces.- 1. The Geometry of Hilbert Spaces.- 2. Operators on a Hilbert Space.- IV The Fourier Transformation and Elements of Harmonic Analysis.- x1. Convolutions on an Abelian Group.- 1. Convolutions of Test Functions.- 2. Convolutions of Generalized Functions.- x2. 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.- x1. The Functional Calculus.- 1. Functions of Operators in a Finite-Dimensional Space..- 2. Functions of Bounded Selfadjoint Operators.- 3. Unbounded Selfadjoint Operators.- x2. 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.- x1. Relations. The Axiom of Choice and Zorn's Lemma.- x2. Completions.- x3. Categories and Functors.- II Theory of Measures and Integrals..- x1. Measure Theory.- 1. Algebras of Sets.- 2. Extension of a Measure.- 3. Constructions of Measures.- x2. Measurable Functions.- 1. Properties of Measurable Functions.- 2. Convergence of Measurable Functions.- x3. 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.- x1. General Theory.- 1. Topology, Convexity, and Seminorms.- 2. Dual Spaces.- 3. The Hahn-Banach Theorem.- 4. Banach Spaces.- x2. Linear Operators.- 1. The Space of Linear Operators.- 2. Compact Sets and Compact Operators.- 3. The Theory of Fredholm Operators.- x3. 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.- x4. Hilbert Spaces.- 1. The Geometry of Hilbert Spaces.- 2. Operators on a Hilbert Space.- IV The Fourier Transformation and Elements of Harmonic Analysis.- x1. Convolutions on an Abelian Group.- 1. Convolutions of Test Functions.- 2. Convolutions of Generalized Functions.- x2. 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.- x1. The Functional Calculus.- 1. Functions of Operators in a Finite-Dimensional Space..- 2. Functions of Bounded Selfadjoint Operators.- 3. Unbounded Selfadjoint Operators.- x2. 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.- x1. Relations. The Axiom of Choice and Zorn's Lemma.- x2. Completions.- x3. Categories and Functors.- II Theory of Measures and Integrals..- x1. Measure Theory.- 1. Algebras of Sets.- 2. Extension of a Measure.- 3. Constructions of Measures.- x2. Measurable Functions.- 1. Properties of Measurable Functions.- 2. Convergence of Measurable Functions.- x3. 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.- x1. General Theory.- 1. Topology, Convexity, and Seminorms.- 2. Dual Spaces.- 3. The Hahn-Banach Theorem.- 4. Banach Spaces.- x2. Linear Operators.- 1. The Space of Linear Operators.- 2. Compact Sets and Compact Operators.- 3. The Theory of Fredholm Operators.- x3. 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.- x4. Hilbert Spaces.- 1. The Geometry of Hilbert Spaces.- 2. Operators on a Hilbert Space.- IV The Fourier Transformation and Elements of Harmonic Analysis.- x1. Convolutions on an Abelian Group.- 1. Convolutions of Test Functions.- 2. Convolutions of Generalized Functions.- x2. 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.- x1. The Functional Calculus.- 1. Functions of Operators in a Finite-Dimensional Space..- 2. Functions of Bounded Selfadjoint Operators.- 3. Unbounded Selfadjoint Operators.- x2. Spectral Decomposition of Operators.- 1. Reduction of an Operator to the Form of Multiplication by a Function.- 2. The Spectral Theorem.- List of Notation.