Classical and Discrete Functional Analysis with Measure Theory 1st Edition by Martin Buntinas ISBN 9781107034143 1107034140

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Product details:

ISBN 10: 1107034140
ISBN 13: 9781107034143
Author: Martin Buntinas

Classical and Discrete Functional Analysis with Measure Theory 1st Edition Table of contents:

0 Preliminaries

0.1 Sets

0.2 Properties of R and C

0.3 Linear Spaces

0.4 Sequences and Series

0.5 Euclidean Space

0.6 Topology of Euclidean Space

0.7 Compact Sets in Euclidean Space

0.8 Continuity in Euclidean Space

0.9 Euclidean Measure

0.10 Sets of Measure Zero

*0.11 Overview of Integration

*0.12 Functions of Bounded Variation

*0.13 Inequalities

*0.14 The Axiom of Choice

Part I Measure and Integration

1 Lebesgue Measure

1.1 Outer and Inner Measure of Sets

1.2 Finite Union of Intervals

*1.3 Carathéodory Characterization

1.4 Countable Union of Intervals

1.5 Open and Closed Sets

1.6 The Approximation Theorem

1.7 General Measurable Sets

1.8 Borel Sets

*1.9 Nonmeasurable Sets

*1.10 A Measurable Set Which Is Not Borel

2 Lebesgue Integral

2.1 Measurable Functions

2.2 Egorov’s Theorem

2.3 The Lebesgue Integral

2.4 Limit Theorems for the Lebesgue Integral

3 Some Calculus

*3.1 Fubini’s Theorem

*3.2 The Riemann Integral

*3.3 Fundamental Theorem of Calculus

4 Abstract Measures

4.1 Abstract Measure Spaces

4.2 Lebesgue–Stieltjes Measure

4.3 Probability Measure

4.4 Signed Measure

*4.5 Absolute Continuity

*4.6 The Radon–Nikodym Theorem

Part II Elements of Classical Functional Analysis

5 Metric and Normed Spaces

5.1 Metric Spaces

5.2 Normed Spaces

5.3 Seminorms

5.4 Convergence in Metric Spaces

5.5 Topological Concepts

5.6 Continuity

5.7 Complete Metric Spaces

5.8 Banach Spaces

5.9 Compact Sets in Metric Spaces

*5.10 Finite-Dimensional Normed Spaces

*5.11 Total Boundedness

*5.12 Stone–Weierstrass Approximation

6 Linear Operators

6.1 Linear Operators

6.2 Operator Spaces

6.3 Linear Functionals

6.4 The Hahn–Banach Extension Principle

6.5 Second Dual Space

6.6 Category of a Set

6.7 Uniform Boundedness

6.8 Open Mapping Theorem

6.9 Closed Graph Theorem

6.10 The Lebesgue Spaces L[sup(p)]

*6.11 The Dual of C[a,b].

*6.12 Contraction Mappings

Part III Discrete Functional Analysis

7 Fourier Series

7.1 Inner Product Spaces

7.2 Fourier Series in Inner Product Spaces

7.3 Hilbert Space

*7.4 Adjoint Operators

7.5 Trigonometric Fourier Series

7.6 Examples of Trigonometric Series

7.7 Arbitrary Periods

7.8 Fourier Series and Summability

7.9 Convergence of Fourier Series

7.10 Square Wave Fourier Series

*7.11 Fourier Transform

8 Applications

8.1 Least Squares Methods

8.2 The Wave Equation

8.3 The Heat Equation

8.4 Harmonic Functions

8.5 Gambler’s Ruin and Random Walk

8.6 Atomic Theory of Matter

8.7 Sampling Theorem

8.8 Uncertainty Principle

8.9 Wavelets

8.10 A Continuous but Nowhere Differentiable Function

8.11 Group Structure and Fourier Series

9 Sequence Spaces

9.1 K-spaces

9.2 BK-spaces

9.3 Examples of BK-spaces

9.4 AD-spaces

9.5 Solid Spaces

9.6 AK-spaces

9.7 σK-spaces

10 Matrix Maps, Multipliers, and Duality

10.1 FK-spaces

10.2 Matrix Maps Between FK-spaces

10.3 Duality and Multipliers

10.4 Application: Matrix Mechanics

11 Summability

11.1 Introduction

11.2 Cesàro Method

11.3 Matrix Methods

11.4 Silverman–Toeplitz Theorem

11.5 Abel Summability

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Tags: Martin Buntinas, Functional Analysis, Measure Theory, Discrete