Essential Real Analysis

  • Michael Field

Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. Michael Field
    Pages 1-29
  3. Michael Field
    Pages 91-127
  4. Michael Field
    Pages 129-159
  5. Michael Field
    Pages 161-210
  6. Michael Field
    Pages 245-328
  7. Michael Field
    Pages 329-347
  8. Michael Field
    Pages 349-442
  9. Back Matter
    Pages 443-450

About this book


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 and fractals.

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

Authors and affiliations

  • Michael Field
    • 1
  1. 1.Engineering Mathematics Department, Merchant Venturers School of EngineeringBristol UniversityUnited Kingdom

Bibliographic information