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secchi s circle line study mathematical analysis 2023

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Textbook in PDF format 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. Preface Acknowledgements First Half of the Journey An Appetizer of Propositional Logic The Propositional Calculus Quantifiers Sets, Relations, Functions in a Naïve Way Comments Reference Numbers The Axioms of R Order Properties of R Natural Numbers Isomorphic Copies Complex Numbers Polar Representation of Complex Numbers A Construction of the Real Numbers Problems Comments References Elementary Cardinality Countable and Uncountable Sets The Schröder-Bernstein Theorem Problems Comments References Distance, Topology and Sequences on the Set of Real Numbers Sequences and Limits A Few Fundamental Limits Lower and Upper Limits Problems Comments Reference Series Convergence Tests for Positive Series Euler's Number as the Sum of a Series Alternating Series Product of Series Problems Comments Reference Limits: From Sequences to Functions of a Real Variable Properties of Limits Local Equivalence of Functions Comments Continuous Functions of a Real Variable Continuity and Compactness Intermediate Value Property Continuous Invertible Functions Problems Derivatives and Differentiability Rules of Differentiation, or the Algebra of Calculus Mean Value Theorems The Intermediate Property for Derivatives Derivatives at End-Points Derivatives of Derivatives Convexity Problems Comments References Riemann's Integral Partitions and the Riemann Integral Integrable Functions as Elements of a Vector Space Classes of Integrable Functions Antiderivatives and the Fundamental Theorem Problems Comments Elementary Functions Sequences and Series of Functions Uniform Convergence The Exponential Function Sine and Cosine Polynomial Approximation A Continuous Non-differentiable Function Asymptotic Estimates for the Factorial Function Problems Second Half of the Journey Return to Set Theory Kelley's System of Axioms From Sets to N A Summary of Kelley's Axioms Set Theory According to JD Monk ZF Axioms From N to Z From Z to Q From Q to R About the Uniqueness of R References Neighbors Again: Topological Spaces Topological Spaces The Special Case of RN Bases and Subbases Subspaces Connected Spaces Nets and Convergence Continuous Maps and Homeomorphisms Product Spaces, Quotient Spaces, and Inadequacy of Sequences Initial and Final Topologies Compact Spaces The Fundamental Theorem of Algebra Local Compactness Compactification of a Space Filters and Convergence Epilogue: The Limit of a Function Separation and Existence of Continuous Extensions Partitions of Unity and Paracompact Spaces Function Spaces Cubes and Metrizability Problems Comments References Differentiating Again: Linearization in Normed Spaces Normed Vector Spaces Bounded Linear Operators The Hahn-Banach Theorem Baire's Theorem and Uniform Boundedness The Open Mapping Theorem Weak and Weak* Topologies Isomorphisms Continuous Multilinear Applications Inner Product Spaces Linearization in Normed Vector Spaces Derivatives of Higher Order Partial Derivatives The Taylor Formula The Inverse and the Implicit Function Theorems Local Inversion A Global Inverse Function Theorem Critical and Almost Critical Points Problems Comments References A Functional Approach to Lebesgue Integration Theory The Riemann Integral in Higher Dimension Elementary Integrals Null and Full Sets The Class L+ The Class L of Integrable Functions Taking Limits Under the Integral Sign Measurable Functions and Measurable Sets Integration Over Measurable Sets The Concrete Lebesgue Integral Integration on Product Spaces Spaces of Integrable Functions The Space L∞ Changing Variables in Multiple Integrals Comments References Measures Before Integrals General Measure Theory Convergence Theorems Complete Measures Different Types of Convergence Measure Theory on Product Spaces Measure, Topology, and the Concrete Lebesgue Measure The Concrete Lebesgue Measure Mollifiers and Regularization Compactness in Lebesgue Spaces The Radon-Nykodim Theorem A Strong Form of the Fundamental Theorem of Calculus Problems Comments References

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secchi s circle line study mathematical analysis 2023