یادداشت‌های همراه: یک سفر کاری برای همراهی با بیبی رادین

"These notes have been prepared to assist students who are learning Advanced Calculus/Real Analysis for the first time in courses or self-study programs that are using the text Principles of Mathematical Analysis (3rd Edition) by Walter Rudin." Contents Preface 0.0.1 About the Organization of the Material 0.0.2 About the Errors Chapter 1: The Field of Reals and Beyond 1.1 Fields 1.2 Ordered Fields 1.2.1 Special Subsets of an Ordered Field 1.2.2 Bounding Properties 1.3 The Real Field 1.3.1 Density Properties of the Reals 1.3.2 Existence of nth Roots 1.3.3 The Extended Real Number System 1.4 The Complex Field 1.4.1 Thinking Complex 1.5 Problem Set A Chapter 2: From Finite to Uncountable Sets 2.1 Some Review of Functions 2.2 A Review of Cardinal Equivalence 2.2.1 Denumerable Sets and Sequences 2.3 Review of Indexed Families of Sets 2.4 Cardinality of Unions Over Families 2.5 The Uncountable Reals 2.6 Problem Set B Chapter 3: METRIC SPACES and SOME BASIC TOPOLOGY 3.1 Euclidean n-space 3.2 Metric Spaces 3.3 Point Set Topology on Metric Spaces 3.3.1 Complements and Families of Subsets of Metric Spaces 3.3.2 Open Relative to Subsets of Metric Spaces 3.3.3 Compact Sets 3.3.4 Compactness in Euclidean n-space 3.3.5 Connected Sets 3.3.6 Perfect Sets 3.4 Problem Set C Chapter 4: Sequences and Series–First View 4.1 Sequences and Subsequences in Metric Spaces 4.2 Cauchy Sequences in Metric Spaces 4.3 Sequences in Euclidean k-space 4.3.1 Upper and Lower Bounds 4.4 Some Special Sequences 4.5 Series of Complex Numbers 4.5.1 Some (Absolute) Convergence Tests 4.5.2 Absolute Convergence and Cauchy Products 4.5.3 Hadamard Products and Series with Positive and Negative Terms 4.5.4 Discussing Convergence 4.5.5 Rearrangements of Series 4.6 Problem Set D Chapter 5: Functions on Metric Spaces and Continuity 5.1 Limits of Functions 5.2 Continuous Functions on Metric Spaces 5.2.1 A Characterization of Continuity 5.2.2 Continuity and Compactness 5.2.3 Continuity and Connectedness 5.3 Uniform Continuity 5.4 Discontinuities and Monotonic Functions 5.4.1 Limits of Functions in the Extended Real Number System 5.5 Problem Set E Chapter 6: Differentiation: Our First View 6.1 The Derivative 6.1.1 Formulas for Differentiation 6.1.2 Revisiting A Geometric Interpretation for the Derivative 6.2 The Derivative and Function Behavior 6.2.1 Continuity (or Discontinuity) of Derivatives 6.3 The Derivative and Finding Limits 6.4 Inverse Functions 6.5 Derivatives of Higher Order 6.6 Differentiation of Vector-Valued Functions 6.7 Problem Set F Chapter 7: Riemann–Stieltjes Integration 7.1 Riemann Sums and Integrability 7.1.1 Properties of Riemann–Stieltjes Integrals 7.2 Riemann Integrals and Differentiation 7.2.1 Some Methods of Integration 7.2.2 The Natural Logarithm Function 7.3 Integration of Vector-Valued Functions 7.3.1 Rectifiable Curves 7.4 Problem Set G Chapter 8: Sequences and Series of Functions 8.1 Pointwise and Uniform Convergence 8.1.1 Sequences of Complex-Valued Functions on Metric Spaces 8.2 Conditions for Uniform Convergence 8.3 Property Transmission and Uniform Convergence 8.4 Families of Functions 8.5 The Stone–Weierstrass Theorem 8.6 Problem Set H Chapter 9: Some Special Functions 9.1 Power Series Over the Reals 9.2 Some General Convergence Properties 9.3 Designer Series 9.3.1 Another Visit With the Logarithm Function 9.3.2 A Series Development of Two Trigonometric Functions 9.4 Series from Taylor’s Theorem 9.4.1 Some Series To Know & Love 9.4.2 Series From Other Series 9.5 Fourier Series 9.6 Problem Set I Index