The Big Book of Real Analysis : From Numbers to Measures / by Syafiq Johar.

Format:

Book

Author/Creator:

Johar, Syafiq.

Language:

English

Subjects (All):

Mathematics.

Mathematical analysis.

Sequences (Mathematics).

Differential equations.

Measure theory.

Functions of real variables.

Analysis.

Sequences, Series, Summability.

Differential Equations.

Measure and Integration.

Real Functions.

Local Subjects:

Mathematics.

Analysis.

Sequences, Series, Summability.

Differential Equations.

Measure and Integration.

Real Functions.

Physical Description:

1 online resource (950 pages)

Edition:

1st ed. 2023.

Place of Publication:

Cham : Springer Nature Switzerland : Imprint: Springer, 2023.

Summary:

This book provides an introduction to real analysis, a fundamental topic that is an essential requirement in the study of mathematics. It deals with the concepts of infinity and limits, which are the cornerstones in the development of calculus. Beginning with some basic proof techniques and the notions of sets and functions, the book rigorously constructs the real numbers and their related structures from the natural numbers. During this construction, the readers will encounter the notions of infinity, limits, real sequences, and real series. These concepts are then formalised and focused on as stand-alone objects. Finally, they are expanded to limits, sequences, and series of more general objects such as real-valued functions. Once the fundamental tools of the trade have been established, the readers are led into the classical study of calculus (continuity, differentiation, and Riemann integration) from first principles. The book concludes with an introduction to the study ofmeasures and how one can construct the Lebesgue integral as an extension of the Riemann integral. This textbook is aimed at undergraduate students in mathematics. As its title suggests, it covers a large amount of material, which can be taught in around three semesters. Many remarks and examples help to motivate and provide intuition for the abstract theoretical concepts discussed. In addition, more than 600 exercises are included in the book, some of which will lead the readers to more advanced topics and could be suitable for independent study projects. Since the book is fully self-contained, it is also ideal for self-study.

Contents:

Preface

1. Logic and Sets

2. Integers

3. Construction of the Real Numbers

4. The Real Numbers

5. Real Sequences

6. Some Applications of Real Sequences

7. Real Series

8. Additional Topics in Real Series

9. Functions and Limits

10. Continuity

11. Function Sequences and Series

12. Power Series

13. Differentiation

14. Some Applications of Differentiation

15. Riemann and Darboux Integration

16. The Fundamental Theorem of Calculus

17. Taylor and MacLaurin Series

18. Introduction to Measure Theory

19. Lebesgue Integration

20. Double Integrals

Solutions to the Exercises

Bibliography

Index.

Other Format:

Print version: Johar, Syafiq The Big Book of Real Analysis

ISBN:

9783031308321

3031308328

OCLC:

1417758294