In 1902, modern function theory began when Henri Lebesgue described a new integral calculus. His Lebesgue integral handles more functions than the traditional integral so many more that mathematicians can study collections (spaces) of functions. For example, it defines a distance between any two functions in a space. This book describes these ideas in an elementary, accessible way. Anyone who has mastered calculus concepts of limits, derivatives, and series can enjoy the material. Unlike any other text, this book brings analysis research topics within reach of readers even just beginning to think about functions from a theoretical point of view.
Author(s): William Johnston
Series: MAA Textbooks
Publisher: American Mathematical Society
Year: 2015
Language: English
Pages: 297
Tags: Lebesgue Integration
Cover
Half title
Copyright
Title
Series
Contents
Preface
Introduction
1 Lebesgue Integrable Functions
1.1 Two Infinities: Countable and Uncountable
1.2 A Taste of Measure Theory
1.3 Lebesgue's Integral for Step Functions
1.4 Limits
1.5 The Lebesgue Integral and L1
Notes for Chapter 1
2 Lebesgue's Integral Compared to Riemann's
2.1 The Riemann Integral
2.2 Properties of the Lebesgue Integral
2.3 Dominated Convergence and Further Properties of the Integral
2.4 Application: Fourier Series
Notes for Chapter 2
3 Function Spaces
3.1 The Spaces Lp
3.2 The Hilbert Space Properties of L2 and l2
3.3 Orthonormal Basis for a Hilbert Space
3.4 Application: Quantum Mechanics
Notes for Chapter 3
4 Measure Theory
4.1 Lebesgue Measure
4.2 Lebesgue Integrals with Respect to Other Measures
4.3 The Hilbert Space L2(μ)
4.4 Application: Probability
Notes for Chapter 4
5 Hilbert Space Operators
5.1 Bounded Linear Operators on L2
5.2 Bounded Linear Operators on General Hilbert Spaces
5.3 The Unilateral Shift Operator
5.4 Application: A Spectral Theorem Example
Notes for Chapter 5
Solutions to Selected Problems
Bibliography
Index
