Real Analysis

The typical introductory real analysis text starts with an analysis of the real number system and uses this to develop the definition of a limit, which is then used as a foundation for the definitions encountered thereafter. While this is certainly a reasonable approach from a logical point of view, it is not how the subject evolved, nor is it necessarily the best way to introduce students to the rigorous but highly non-intuitive definitions and proofs found in analysis.

Analysis Mathematics

Tác giả: Eugene Boman & Robert Rogers
Nhà xuất bản: Pennsylvania State University & SUNY Fredonia
Nguồn: LibreTexts
Giấy phép: CC BY-NC-SA
Ngôn ngữ: EN
Danh mục: Khoa học cơ bản

Đọc trực tuyến Tải PDF (nguồn mở)

Mục lục· 45 mục

01: Numbers - Real () and Rational ()
1.01: Real and Rational Numbers
1.E: Numbers - Real (ℝ) and Rational (ℚ) (Exercises)
02: Calculus in the 17th and 18th Centuries
2.01: Newton and Leibniz Get Started
2.02: Power Series as Infinite Polynomials
2.E: Calculus in the 17th and 18th Centuries (Exercises)
03: Questions Concerning Power Series
3.01: Taylor’s Formula
3.02: Series Anomalies
3.E: Questions Concerning Power Series (Exercises)
04: Convergence of Sequences and Series
4.01: Sequences of Real Numbers
4.02: The Limit as a Primary Tool
4.03: Divergence of a Series
4.E: Convergence of Sequences and Series (Exercises)
05: Convergence of the Taylor Series- A Tayl of Three Remainders
5.01: The Integral Form of the Remainder
5.02: Lagrange’s Form of the Remainder
5.03: Cauchy’s Form of the Remainder
5.E: Convergence of the Taylor Series- A “Tayl” of Three Remainders (Exercises)
06: Continuity - What It Isnt and What It Is
6.01: An Analytic Deﬁnition of Continuity
6.02: Sequences and Continuity
6.03: The Deﬁnition of the Limit of a Function
6.04: The Derivative - An Afterthought
6.E: Continuity - What It Isn’t and What It Is (Exercises)
07: Intermediate and Extreme Values
7.01: Completeness of the Real Number System
7.02: Proof of the Intermediate Value Theorem
7.03: The Bolzano-Weierstrass Theorem
7.04: The Supremum and the Extreme Value Theorem
7.E: Intermediate and Extreme Values (Exercises)
08: Back to Power Series
8.01: Uniform Convergence
8.02: Uniform Convergence- Integrals and Derivatives
8.03: Radius of Convergence of a Power Series
8.04: Boundary Issues and Abel’s Theorem
09: Back to the Real Numbers
9.01: Trigonometric Series
9.02: Infinite Sets
9.03: Cantor’s Theorem and Its Consequences
10: Epilogue to Real Analysis
10.01: On the Nature of Numbers
10.02: Building the Real Numbers