A Guide to Real Variables

A Guide to Real Variables provides 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,...

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Hlavní autor: Krantz, Steven G.
Médium: E-kniha Kniha
Jazyk:angličtina
Vydáno: Providence, Rhode Island American Mathematical Society 2009
Mathematical Association of America
Vydání:1
Edice:Dolciani Mathematical Expositions
Témata:
Functions of real variables
Mathematics
ISBN:0883853442, 9780883853443
On-line přístup:Získat plný text
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Obsah:
  • Preface -- Basics -- Sequences -- Series -- The Topology of the Real Line -- Limits and the Continuity of Functions -- The Derivative -- The Integral -- Sequences and Series of Functions -- Advanced Topics -- Glossary of Terms from Real Variable Theory -- Bibliography
  • Front Matter Table of Contents Preface CHAPTER 1: Basics CHAPTER 2: Sequences CHAPTER 3: Series CHAPTER 4: The Topology of the Real Line CHAPTER 5: Limits and the Continuity of Functions CHAPTER 6: The Derivative CHAPTER 7: The Integral CHAPTER 8: Sequences and Series of Functions CHAPTER 9: Advanced Topics Glossary of Terms from Real Variable Theory Bibliography Index Back Matter
  • 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
  • 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