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Designed for courses in advanced calculus and introductory real analysis, Elementary Classical Analysis strikes a careful balance between pure and applied mathematics with an emphasis on specific techniques important to classical analysis without vector calculus or complex analysis. Intended for students of engineering and physical science as well as of pure mathematics. Copyright © Libri GmbH. All rights reserved.

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• 1 Introduction: Sets and Functions
• Supplement on the Axioms of Set Theory
• 2 The Real Line and Euclidean Space
• Ordered Fields and the Number Systems
• Completeness and the Real Number System
• Least Upper Bounds
• Cauchy Sequences
• Cluster Points: lim inf and lim sup
• Euclidean Space
• Norms, Inner Products, and Metrics
• The Complex Numbers
• 3 Topology of Euclidean Space
• Open Sets
• Interior of a Set
• Closed Sets
• Accumulation Points
• Closure of a Set
• Boundary of a Set
• Sequences
• Completeness
• Series of Real Numbers and Vectors
• 4 Compact and Connected Sets
• Compacted-ness
• The Heine-Borel Theorem
• Nested Set Property
• Path-Connected Sets
• Connected Sets
• 5 Continuous Mappings
• Continuity
• Images of Compact and Connected Sets
• Operations on Continuous Mappings
• The Boundedness of Continuous Functions of Compact Sets
• The Intermediate Value Theorem
• Uniform Continuity
• Differentiation of Functions of One Variable
• Integration of Functions of One Variable
• 6 Uniform Convergence
• Pointwise and Uniform Convergence
• The Weierstrass M Test
• Integration and Differentiation of Series
• The Elementary Functions
• The Space of Continuous Functions
• The Arzela-Ascoli Theorem
• The Contraction Mapping Principle and Its Applications
• The Stone-Weierstrass Theorem
• The Dirichlet and Abel Tests
• Power Series and Cesaro and Abel Summability
• 7 Differentiable Mappings
• Definition of the Derivative
• Matrix Representation
• Continuity of Differentiable Mappings; Differentiable Paths
• Conditions for Differentiability
• The Chain Rule
• Product Rule and Gradients
• The Mean Value Theorem
• Taylor's Theorem and Higher Derivatives
• Maxima and Minima
• 8 The Inverse and Implicit Function Theorems and Related Topics
• Inverse Function Theorem
• Implicit Function Theorem
• The Domain-Straightening Theorem
• Further Consequences of the Implicit Function Theorem
• An Existence Theorem for Ordinary Differential Equations
• The Morse Lemma
• Constrained Extrema and Lagrange Multipliers
• 9 Integration
• Integrable Functions
• Volume and Sets of Measure Zero
• Lebesgue's Theorem
• Properties of the Integral
• Improper Integrals
• Some Convergence Theorems
• Introduction to Distributions
• 10 Fubini's Theorem and the Change of Variables Formula
• Introduction
• Fubini's Theorem
• Change of Variables Theorem
• Polar Coordinates
• Spherical Coordinates and Cylindrical Coordinates
• A Note on the Lebesgue Integral
• Interchange of Limiting Operations
• 11 Fourier Analysis
• Inner Product Spaces
• Orthogonal Families of Functions
• Completeness and Convergence Theorems
• Functions of Bounded Variation and Fejã©r Theory (Optional)