A Circle-Line Study of Mathematical Analysis

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