Real Analysis and Foundations 4th Edition by Steven G Krantz ISBN 1498777686 9781498777681

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ISBN 10: 1498777686 
ISBN 13: 9781498777681
Author: Steven G Krantz

The first three editions of this popular textbook attracted a loyal readership and widespread use; students find the book concise, accessible, and complete, while instructors find it clear, authoritative, and dependable. The goal of this new edition is to make real analysis relevant and accessible to a broad audience of students with diverse backgrounds. Real analysis is a basic tool for all mathematical scientists, ranging from mathematicians to physicists, engineers, and researchers in the medical profession. This text aims to be the generational touchstone for the subject and the go-to text for developing young scientists. In this new edition, we endeavor to make the book accessible to a broader audience by including more explanation, more elementary examples, and stepladdered exercises; figures are updated and clarified, sections are more concise, and overly technical details are omitted. We have updated and augmented the multivariable material to highlight the geometric nature of the topic, enhancing and fleshing out the figures. Features include a renewed enthusiasm for the topic shown in a revised presentation, a new organization that removes some advanced topics while retaining related ones, more tiered exercises for a more accessible course, and key sections revised for brevity.

Real Analysis and Foundations 4th Table of contents:

1. The Real Number System

• 1.1 Ordered Sets and Fields

• 1.2 The Completeness Axiom

• 1.3 The Archimedean Property

• 1.4 The Density of the Rational Numbers

2. Sequences and Series

• 2.1 Convergence of Sequences

• 2.2 Subsequences and Limit Points

• 2.3 Monotone Sequences and the Monotone Convergence Theorem

• 2.4 Cauchy Sequences and Completeness

• 2.5 Infinite Series and Tests for Convergence

3. Topology of the Real Line

• 3.1 Open and Closed Sets

• 3.2 Compact Sets and Heine-Borel Theorem

• 3.3 Perfect Sets and Connectedness

• 3.4 The Cantor Set

4. Limits and Continuity

• 4.1 Limits of Functions

• 4.2 Continuity and Discontinuities

• 4.3 Uniform Continuity

• 4.4 Properties of Continuous Functions on Compact Sets

5. Differentiation

• 5.1 The Derivative and Differentiability

• 5.2 Mean Value Theorems

• 5.3 Higher-Order Derivatives

• 5.4 Taylor’s Theorem and Applications

6. The Riemann Integral

• 6.1 Definition and Properties of the Riemann Integral

• 6.2 Integrability Criteria

• 6.3 Fundamental Theorem of Calculus

• 6.4 Improper Integrals

7. Sequences and Series of Functions

• 7.1 Pointwise and Uniform Convergence

• 7.2 Power Series

• 7.3 Weierstrass Approximation Theorem

• 7.4 Equicontinuity and Arzelà-Ascoli Theorem

8. Metric Spaces and General Topology

• 8.1 Definition and Examples of Metric Spaces

• 8.2 Open and Closed Sets in Metric Spaces

• 8.3 Completeness and Completion

• 8.4 Compactness and Sequential Compactness

• 8.5 Connectedness

9. Lebesgue Measure and Integration (Optional/Advanced)

• 9.1 Outer Measure and Measurable Sets

• 9.2 Lebesgue Measure on the Real Line

• 9.3 Lebesgue Integration

• 9.4 Convergence Theorems

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Tags: Steven G Krantz, Real Analysis, Foundations