The book addresses the rigorous foundations of mathematical analysis. The first part presents a complete discussion of the fundamental topics: a review of naive set theory, the structure of real numbers, the topology of R, sequences, series, limits, differentiation and integration according to Riemann.
The second part provides a more mature return to these topics: a possible axiomatization of set theory, an introduction to general topology with a particular attention to convergence in abstract spaces, a construction of the abstract Lebesgue integral in the spirit of Daniell, and the discussion of differentiation in normed linear spaces.
The book can be used for graduate courses in real and abstract analysis and can also be useful as a self-study for students who begin a Ph.D. program in Analysis. The first part of the book may also be suggested as a second reading for undergraduate students with a strong interest in mathematical analysis.
Author(s): Simone Secchi
Series: UNITEXT, 141
Edition: 1
Publisher: Springer
Year: 2023
Language: English
Pages: 488
City: Cham
Tags: Calculus; Real Analysis; Abstract Analysis; Topological and Normed Vector Spaces; Differentiation in Banach Spaces; Integration Theory; Axiomatic Set Theory
Preface
Acknowledgements
Contents
Part I First Half of the Journey
1 An Appetizer of Propositional Logic
1.1 The Propositional Calculus
1.2 Quantifiers
2 Sets, Relations, Functions in a Naïve Way
2.1 Comments
Reference
3 Numbers
3.1 The Axioms of R
3.2 Order Properties of R
3.3 Natural Numbers
3.4 Isomorphic Copies
3.5 Complex Numbers
3.6 Polar Representation of Complex Numbers
3.7 A Construction of the Real Numbers
3.8 Problems
3.9 Comments
References
4 Elementary Cardinality
4.1 Countable and Uncountable Sets
4.2 The Schröder-Bernstein Theorem
4.3 Problems
4.4 Comments
References
5 Distance, Topology and Sequences on the Set of Real Numbers
5.1 Sequences and Limits
5.2 A Few Fundamental Limits
5.3 Lower and Upper Limits
5.4 Problems
5.5 Comments
Reference
6 Series
6.1 Convergence Tests for Positive Series
6.2 Euler's Number as the Sum of a Series
6.3 Alternating Series
6.3.1 Product of Series
6.4 Problems
6.5 Comments
Reference
7 Limits: From Sequences to Functions of a Real Variable
7.1 Properties of Limits
7.2 Local Equivalence of Functions
7.3 Comments
8 Continuous Functions of a Real Variable
8.1 Continuity and Compactness
8.2 Intermediate Value Property
8.3 Continuous Invertible Functions
8.4 Problems
9 Derivatives and Differentiability
9.1 Rules of Differentiation, or the Algebra of Calculus
9.2 Mean Value Theorems
9.3 The Intermediate Property for Derivatives
9.4 Derivatives at End-Points
9.5 Derivatives of Derivatives
9.6 Convexity
9.7 Problems
9.8 Comments
References
10 Riemann's Integral
10.1 Partitions and the Riemann Integral
10.2 Integrable Functions as Elements of a Vector Space
10.3 Classes of Integrable Functions
10.4 Antiderivatives and the Fundamental Theorem
10.5 Problems
10.6 Comments
11 Elementary Functions
11.1 Sequences and Series of Functions
11.2 Uniform Convergence
11.3 The Exponential Function
11.4 Sine and Cosine
11.5 Polynomial Approximation
11.6 A Continuous Non-differentiable Function
11.7 Asymptotic Estimates for the Factorial Function
11.8 Problems
Part II Second Half of the Journey
12 Return to Set Theory
12.1 Kelley's System of Axioms
12.2 From Sets to N
12.3 A Summary of Kelley's Axioms
12.4 Set Theory According to J.D. Monk
12.5 ZF Axioms
12.6 From N to Z
12.7 From Z to Q
12.8 From Q to R
12.9 About the Uniqueness of R
References
13 Neighbors Again: Topological Spaces
13.1 Topological Spaces
13.2 The Special Case of RN
13.3 Bases and Subbases
13.4 Subspaces
13.5 Connected Spaces
13.6 Nets and Convergence
13.7 Continuous Maps and Homeomorphisms
13.8 Product Spaces, Quotient Spaces, and Inadequacy of Sequences
13.9 Initial and Final Topologies
13.10 Compact Spaces
13.10.1 The Fundamental Theorem of Algebra
13.10.2 Local Compactness
13.11 Compactification of a Space
13.12 Filters and Convergence
13.13 Epilogue: The Limit of a Function
13.14 Separation and Existence of Continuous Extensions
13.15 Partitions of Unity and Paracompact Spaces
13.16 Function Spaces
13.17 Cubes and Metrizability
13.18 Problems
13.19 Comments
References
14 Differentiating Again: Linearization in Normed Spaces
14.1 Normed Vector Spaces
14.2 Bounded Linear Operators
14.3 The Hahn-Banach Theorem
14.4 Baire's Theorem and Uniform Boundedness
14.5 The Open Mapping Theorem
14.6 Weak and Weak* Topologies
14.7 Isomorphisms
14.8 Continuous Multilinear Applications
14.9 Inner Product Spaces
14.10 Linearization in Normed Vector Spaces
14.11 Derivatives of Higher Order
14.12 Partial Derivatives
14.13 The Taylor Formula
14.14 The Inverse and the Implicit Function Theorems
14.14.1 Local Inversion
14.15 A Global Inverse Function Theorem
14.16 Critical and Almost Critical Points
14.17 Problems
14.18 Comments
References
15 A Functional Approach to Lebesgue Integration Theory
15.1 The Riemann Integral in Higher Dimension
15.2 Elementary Integrals
15.3 Null and Full Sets
15.4 The Class L+
15.5 The Class L of Integrable Functions
15.6 Taking Limits Under the Integral Sign
15.7 Measurable Functions and Measurable Sets
15.8 Integration Over Measurable Sets
15.9 The Concrete Lebesgue Integral
15.10 Integration on Product Spaces
15.11 Spaces of Integrable Functions
15.12 The Space L∞
15.13 Changing Variables in Multiple Integrals
15.14 Comments
References
16 Measures Before Integrals
16.1 General Measure Theory
16.2 Convergence Theorems
16.3 Complete Measures
16.4 Different Types of Convergence
16.5 Measure Theory on Product Spaces
16.6 Measure, Topology, and the Concrete Lebesgue Measure
16.6.1 The Concrete Lebesgue Measure
16.7 Mollifiers and Regularization
16.8 Compactness in Lebesgue Spaces
16.9 The Radon-Nykodim Theorem
16.10 A Strong Form of the Fundamental Theorem of Calculus
16.11 Problems
16.12 Comments
References
