A Guide to Real Variables.

Intro -- Contents -- Preface -- 1 Basics -- 1.1 Sets -- 1.2 Operations on Sets -- 1.3 Functions -- 1.4 Operations on Functions -- 1.5 Number Systems -- 1.5.1 The Real Numbers -- 1.6 Countable and Uncountable Sets -- 2 Sequences -- 2.1 Introduction to Sequences -- 2.1.1 The Definition and Convergence -- 2.1.2 The Cauchy Criterion -- 2.1.3 Monotonicity -- 2.1.4 The Pinching Principle -- 2.1.5 Subsequences -- 2.1.6 The Bolzano-Weierstrass Theorem -- 2.2 Limsup and Liminf -- 2.3 Some Special Sequences -- 3 Series -- 3.1 Introduction to Series -- 3.1.1 The Definition and Convergence -- 3.1.2 Partial Sums -- 3.2 Elementary Convergence Tests -- 3.2.1 The Comparison Test -- 3.2.2 The Cauchy Condensation Test -- 3.2.3 Geometric Series -- 3.2.4 The Root Test -- 3.2.5 The Ratio Test -- 3.2.6 Root and Ratio Tests for Divergence -- 3.3 Advanced Convergence Tests -- 3.3.1 Summation by Parts -- 3.3.2 Abel's Test -- 3.3.3 Absolute and Conditional Convergence -- 3.3.4 Rearrangements of Series -- 3.4 Some Particular Series -- 3.4.1 The Series for e -- 3.4.2 Other Representations for e -- 3.4.3 Sums of Powers -- 3.5 Operations on Series -- 3.5.1 Sums and Scalar Products of Series -- 3.5.2 Products of Series -- 3.5.3 The Cauchy Product -- 4 The Topology of the Real Line -- 4.1 Open and Closed Sets -- 4.1.1 Open Sets -- 4.1.2 Closed Sets -- 4.1.3 Characterization of Open and Closed Sets in Terms of Sequences -- 4.1.4 Further Properties of Open and Closed Sets -- 4.2 Other Distinguished Points -- 4.2.1 Interior Points and Isolated Points -- 4.2.2 Accumulation Points -- 4.3 Bounded Sets -- 4.4 Compact Sets -- 4.4.1 Introduction -- 4.4.2 The Heine-Borel Theorem -- 4.4.3 The Topological Characterization of Compactness -- 4.5 The Cantor Set -- 4.6 Connected and Disconnected Sets -- 4.6.1 Connectivity -- 4.7 Perfect Sets -- 5 Limits and the Continuity of Functions.

5.1 Definitions and Basic Properties -- 5.1.1 Limits -- 5.1.2 A Limit that Does Not Exist -- 5.1.3 Uniqueness of Limits -- 5.1.4 Properties of Limits -- 5.1.5 Characterization of Limits Using Sequences -- 5.2 Continuous Functions -- 5.2.1 Continuity at a Point -- 5.2.2 The Topological Approach to Continuity -- 5.3 Topological Properties and Continuity -- 5.3.1 The Image of a Function -- 5.3.2 Uniform Continuity -- 5.3.3 Continuity and Connectedness -- 5.3.4 The Intermediate Value Property -- 5.4 Monotonicity and Classifying Discontinuities -- 5.4.1 Left and Right Limits -- 5.4.2 Types of Discontinuities -- 5.4.3 Monotonic Functions -- 6 The Derivative -- 6.1 The Concept of Derivative -- 6.1.1 The Definition -- 6.1.2 Properties of the Derivative -- 6.1.3 The Weierstrass Nowhere Differentiable Function -- 6.1.4 The Chain Rule -- 6.2 The Mean Value Theorem and Applications -- 6.2.1 Local Maxima and Minima -- 6.2.2 Fermat's Test -- 6.2.3 Darboux's Theorem -- 6.2.4 The Mean Value Theorem -- 6.2.5 Examples of the Mean Value Theorem -- 6.3 Further Results on the Theory of Differentiation -- 6.3.1 l'Hopital's Rule -- 6.3.2 Derivative of an Inverse Function -- 6.3.3 Higher Derivatives -- 6.3.4 Continuous Differentiability -- 7 The Integral -- 7.1 The Concept of Integral -- 7.1.1 Partitions -- 7.1.2 Refinements of Partitions -- 7.1.3 Existence of the Riemann Integral -- 7.1.4 Integrability of Continuous Functions -- 7.2 Properties of the Riemann Integral -- 7.2.1 Existence Theorems -- 7.2.2 Inequalities for Integrals -- 7.2.3 Preservation of Integrable Functions Under Composition -- 7.2.4 The Fundamental Theorem of Calculus -- 7.2.5 Mean Value Theorems -- 7.3 Further Results on the Riemann Integral -- 7.3.1 The Riemann-Stieltjes Integral -- 7.3.2 Riemann's Lemma -- 7.4 Advanced Results on Integration Theory.

7.4.1 Existence for the Riemann-Stieltjes Integral -- 7.4.2 Integration by Parts -- 7.4.3 Linearity Properties -- 7.4.4 Bounded Variation -- 8 Sequences and Series of Functions -- 8.1 Partial Sums and Pointwise Convergence -- 8.1.1 Sequences of Functions -- 8.1.2 Uniform Convergence -- 8.2 More on Uniform Convergence -- 8.2.1 Commutation of Limits -- 8.2.2 The Uniform Cauchy Condition -- 8.2.3 Limits of Derivatives -- 8.3 Series of Functions -- 8.3.1 Series and Partial Sums -- 8.3.2 Uniform Convergence of a Series -- 8.3.3 The Weierstrass M-Test -- 8.4 The Weierstrass Approximation Theorem -- 8.4.1 Weierstrass's Main Result -- 9 Advanced Topics -- 9.1 Metric Spaces -- 9.1.1 The Concept of a Metric -- 9.1.2 Examples of Metric Spaces -- 9.1.3 Convergence in a Metric Space -- 9.1.4 The Cauchy Criterion -- 9.1.5 Completeness -- 9.1.6 Isolated Points -- 9.2 Topology in a Metric Space -- 9.2.1 Balls in a Metric Space -- 9.2.2 Accumulation Points -- 9.2.3 Compactness -- 9.3 The Baire Category Theorem -- 9.3.1 Density -- 9.3.2 Closure -- 9.3.3 Baire's Theorem -- 9.4 The Ascoli-Arzela Theorem -- 9.4.1 Equicontinuity -- 9.4.2 Equiboundedness -- 9.4.3 The Ascoli-Arzela Theorem -- Glossary of Terms from Real Variable Theory -- Bibliography -- Index -- About the Author.

The purpose of A Guide to Real Variables is to provide an aid and conceptual support for the student studying for the qualifying exam in real variables. Beginning with the foundations of the subject, the text moves rapidly but thoroughly through basic topics like completeness, convergence, sequences, series, compactness, topology and the like. All the basic examples like the Cantor set, the Weierstrass nowhere differentiable function, the Weierstrass approximation theory, the Baire category theorem, and the Ascoli-Arzela theorem are treated.The book contains over 100 examples, and most of the basic proofs. It illustrates both the theory and the practice of this sophisticated subject. Graduate students studying for the qualifying exams will find this book to be a concise, focused and informative resource. Professional mathematicians who need a quick review of the subject, or need a place to look up a key fact, will find this book to be a useful resource too.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.