Essential Real Analysis

1. Front Matter

   Pages i-xvii

   PDF

2. Sets, Functions and the Real Numbers

   • Michael Field

   Pages 1-29

3. Basic Properties of Real Numbers, Sequences and Continuous Functions

   • Michael Field

   Pages 31-90

4. Infinite Series

   • Michael Field

   Pages 91-127

5. Uniform Convergence

   • Michael Field

   Pages 129-159

6. Functions

   • Michael Field

   Pages 161-210

7. Topics from Classical Analysis: The Gamma-Function and the Euler–Maclaurin Formula

   • Michael Field

   Pages 211-243

8. Metric Spaces

   • Michael Field

   Pages 245-328

9. Fractals and Iterated Function Systems

   • Michael Field

   Pages 329-347

10. Differential Calculus on \(\mathbb{R}^{m}\)

    • Michael Field

    Pages 349-442

11. Back Matter

    Pages 443-450

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This book provides a rigorous introduction to the techniques and results of real analysis, metric spaces and multivariate differentiation, suitable for undergraduate courses.

Starting from the very foundations of analysis, it offers a complete first course in real analysis, including topics rarely found in such detail in an undergraduate textbook such as the construction of non-analytic smooth functions, applications of the Euler-Maclaurin formula to estimates, and fractal geometry. Drawing on the author’s extensive teaching and research experience, the exposition is guided by carefully chosen examples and counter-examples, with the emphasis placed on the key ideas underlying the theory. Much of the content is informed by its applicability: Fourier analysis is developed to the point where it can be rigorously applied to partial differential equations or computation, and the theory of metric spaces includes applications to ordinary differential equations andfractals.

Essential Real Analysis will appeal to students in pure and applied mathematics, as well as scientists looking to acquire a firm footing in mathematical analysis. Numerous exercises of varying difficulty, including some suitable for group work or class discussion, make this book suitable for self-study as well as lecture courses.

• use of euler maclaurin formula
• Euler-Maclaurin formula
• metric space theory
• metric space in real analysis
• fourier series
• power series
• infinite series and products
• smooth and analytic functions
• uniform approximation
• existence theorem for ODEs
• derivative of vector-valued map
• implicit function theorem
• smooth functions theory
• real analysis techniques
• real analysis books
• 26-01, 40-01, 26Axx, 26Bxx, 26B05, 26B10, 26Exx, 26E05, 26E10
• 33Bxx, 33B15, 34A12, 40Axx, 42Axx, 42A10, 54Exx, 54-01, 54E35

“This is a well written text on Real Analysis that may be used for a course in Advanced Calculus. It can also serve as a reference for advanced topics in Real Analysis.” (Charles Traina, MAA Reviews, January 4, 2020)

“This book contains a reasonably complete exposition of real analysis which is needed for beginning undergraduate-level students. … This is a well-written textbook with an abundance of worked examples and exercises that are intended for a first course in analysis. This book offers a sound grounding in analysis. In particular, it gives a solid base in real analysis from which progress to more advanced topics may be made.” (Teodora-Liliana Rădulescu, zbMATH 1379.26001, 2018)

• Engineering Mathematics Department, Merchant Venturers School of Engineering, Bristol University, United Kingdom

  Michael Field

Michael Field has held appointments in the UK (Warwick University and Imperial College London), Australia (Sydney University) and the US (the University of Houston and Rice University) and has taught a wide range of courses at undergraduate and graduate level, including real analysis, partial differential equations, dynamical systems, differential manifolds, Lie groups, complex manifolds and sheaf cohomology. His publications in the areas of equivariant dynamical systems and network dynamics include nine books and research monographs as well as many research articles. His computer graphic art work, based on symmetric dynamics, has been widely exhibited and is on display at a number of universities around the world.

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