This book discusses major topics in measure theory, Fourier transforms, complex analysis and algebraic topology. It presents material from a mature mathematical perspective. The text is suitable for a two-semester graduate course in analysis and will help students prepare for a research career in mathematics. After a short survey of undergraduate analysis and measure theory, the book highlights the essential theorems that have now become ubiquitous in mathematics. It studies Fourier transforms, derives the inversion theorem and gives diverse applications ranging from probability theory to mathematical physics. It reviews topics in complex analysis and gives a synthetic, rigorous development of the calculus of residues as well as applications to a wide array of problems. It also introduces algebraic topology and shows the symbiosis between algebra and analysis. Indeed, algebraic archetypes were providing foundational support from the start. Multivariable calculus is comprehended in a single glance through the algebra of differential forms. Advanced complex analysis inevitably leads one to the study of Riemann surfaces, and so the final chapter gives the student a hint of these motifs and underlying algebraic patterns.
Author(s): M. Ram Murty
Series: HBA Lecture Notes in Mathematics
Edition: 1
Publisher: Springer
Year: 2022
Language: English
Pages: 346
Tags: Analysis; Complex Analysis; Measure Theory; Fourier Transforms; Algebraic Topology
Preface
Acknowledgements
Contents
1 Background
1.1 Axioms of Set Theory
1.2 Constructing Numbers from Sets
1.3 Set-Theoretic Construction of the Real Numbers
1.4 Sequences of Real Numbers
1.5 Infinite Series
1.6 Sequences of Functions
1.7 Power Series
1.8 Metric Spaces and Euclidean Spaces
1.9 The Heine–Borel Theorem
1.10 Vector-Valued Functions
1.11 Derivatives of Multivariable Functions
1.12 The Inverse Function Theorem
1.13 The Implicit Function Theorem
1.14 The Lagrange Multiplier Method
1.15 Level Sets and Tangent Spaces
1.16 Changing Variables in Integrals
1.17 Volume and Surface Area of the Hypersphere
1.18 Green's Theorem
1.19 Theorems of Gauss and Stokes
1.20 Differential Forms
References
2 Measure Theory
2.1 Topological Spaces and Measure Spaces
2.2 The Lebesgue Integral
2.3 Inner Product Spaces
2.4 Orthonormal Sets
2.5 Trigonometric Series
2.6 Banach Spaces
2.7 Baire's Theorem
2.8 Hahn–Banach Theorem
2.9 Examples of Dual Spaces
References
3 Fourier Transforms
3.1 Fubini's Theorem and Convolutions
3.2 The Fourier Transform
3.3 Differentiation Under the Integral Sign
3.4 Further Examples of Fourier Transforms
3.5 A Convolution Theorem
3.6 The Inversion Theorem
3.7 Further Properties of the Fourier Transform
3.8 The Plancherel Theorem
3.9 The Uncertainty Principle
3.10 Trigonometric Polynomials
3.11 The Isoperimetric Inequality
3.12 Weyl's Criterion and Uniform Distribution
3.13 Fourier Series
3.14 The Poisson Summation Formula
3.15 A Fourier Analytic Proof of the Central Limit Theorem
References
4 Complex Analysis
4.1 Basic Definitions
4.2 Integration over Paths
4.3 The Local Cauchy Theorem
4.4 Zeros and Singularities
4.5 The Maximum Modulus Principle
4.6 The Global Cauchy Theorem
4.7 The Calculus of Residues
4.8 Further Examples
4.9 Rouché's Theorem
4.10 Infinite Products and Weierstrass Factorization
4.11 The Logarithm
4.12 The Phragmén–Lindelöf Theorem and Jensen's Theorem
4.13 Entire Functions of Order 1
4.14 The Gamma Function
4.15 Stirling's Formula
4.16 The Wiener–Ikehara Tauberian Theorem
4.17 The Analytic Theorem
4.18 The Proof of the Tauberian Theorem
4.19 The Prime Number Theorem
4.20 Further Applications
4.21 The Paley–Wiener Theorems
References
5 Introduction to Algebraic Topology
5.1 A Very Brief Historical Introduction
5.2 Homotopic Paths
5.3 The Fundamental Group
5.4 Examples of Some Fundamental Groups
5.5 Covering Spaces
5.6 Applications
5.7 Group Actions and Orbit Spaces
5.8 Automorphisms of Covering Spaces
5.9 The Universal Covering Space
5.10 Suggestions for Further Reading
Index
