CiNii 図書 - Real analysis and foundations

A Readable yet Rigorous Approach to an Essential Part of Mathematical Thinking Back by popular demand, Real Analysis and Foundations, Third Edition bridges the gap between classic theoretical texts and less rigorous ones, providing a smooth transition from logic and proofs to real analysis. Along with the basic material, the text covers Riemann-Stieltjes integrals, Fourier analysis, metric spaces and applications, and differential equations. New to the Third Edition Offering a more streamlined presentation, this edition moves elementary number systems and set theory and logic to appendices and removes the material on wavelet theory, measure theory, differential forms, and the method of characteristics. It also adds a chapter on normed linear spaces and includes more examples and varying levels of exercises. Extensive Examples and Thorough Explanations Cultivate an In-Depth Understanding This best-selling book continues to give students a solid foundation in mathematical analysis and its applications. It prepares them for further exploration of measure theory, functional analysis, harmonic analysis, and beyond.

Number Systems The Real Numbers The Complex Numbers Sequences Convergence of Sequences Subsequences Limsup and Liminf Some Special Sequences Series of Numbers Convergence of Series Elementary Convergence Tests Advanced Convergence Tests Some Special Series Operations on Series Basic Topology Open and Closed Sets Further Properties of Open and Closed Sets Compact Sets The Cantor Set Connected and Disconnected Sets Perfect Sets Limits and Continuity of Functions Basic Properties of the Limit of a Function Continuous Functions Topological Properties and Continuity Classifying Discontinuities and Monotonicity Differentiation of Functions The Concept of Derivative The Mean Value Theorem and Applications More on the Theory of Differentiation The Integral Partitions and the Concept of Integral Properties of the Riemann Integral Another Look at the Integral Advanced Results on Integration Theory Sequences and Series of Functions Partial Sums and Pointwise Convergence More on Uniform Convergence Series of Functions The Weierstrass Approximation Theorem Elementary Transcendental Functions Power Series More on Power Series: Convergence Issues The Exponential and Trigonometric Functions Logarithms and Powers of Real Numbers Differential Equations Picard's Existence and Uniqueness Theorem Power Series Methods Introduction to Harmonic Analysis The Idea of Harmonic Analysis The Elements of Fourier Series An Introduction to the Fourier Transform Fourier Methods and Differential Equations Functions of Several Variables A New Look at the Basic Concepts of Analysis Properties of the Derivative The Inverse and Implicit Function Theorems Advanced Topics Metric Spaces Topology in a Metric Space The Baire Category Theorem The Ascoli-Arzela Theorem Normed Linear Spaces What Is This Subject About? What Is a Normed Linear Space? Finite-Dimensional Spaces Linear Operators The Three Big Results Applications of the Big Three Appendix I: Elementary Number Systems Appendix II: Logic and Set Theory Appendix III: Review of Linear Algebra Table of Notation Glossary Bibliography Index Exercises are included at the end of each section.

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Real analysis and foundations

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