Christopher E. Heil

From the preface: This text is an introduction to real analysis. There are several classic analysis texts that I keep close by on my bookshelf and refer to often. However, I find it difficult to use any of these as the textbook for teaching a first course on analysis. They tend to be dense and, in the classic style of mathematical elegance and conciseness, they develop the theory in the most general setting, with few examples and limited motivation. These texts are valuable resources, but I suggest that they should be the second set of books on analysis that you pick up. I hope that this text will be the analysis text that you read first. The definitions, theorems, and other results are motivated and explained; the why and not just the what of the subject is discussed. Proofs are completely rigorous, yet difficult arguments are motivated and discussed ...

      

• Table of contents.
• Preface.
  
  
• See the book at:
  • Springer.
    
  • Amazon.
    
  • Google.
  
  
• Book Reviews:
  • Mathematical Association of America. Quote: "Heil has understood that what does not draw attention has little chance of being thought about, and what is not thought about cannot possibly be learned." Quote: "This challenge has been met hands down."
    
    
  • MathSciNet. Quote: "This book gives an accessible introduction to real analysis that is suitable for first-year graduate students."
    
    
  • Zentralblatt Math. Quote: "The main aim of the author is to clarify the central ideas of modern analysis."
    
    
  • Amazon customer review. Quote: "This book is amazing."
  
  
• Related Video:
  • Lecture on Absolute Continuity and the Banach-Zaretsky Theorem, presented at the Faraway Fourier Talks on March 29, 2021.
  
  
• Extra Text Resources:
  • An Instructor's Guide, with course outlines, commentary, and extra problems (updated October 2021). This material may also be useful for students reading the text.
    
    
  • A paper on Absolute Continuity and the Banach-Zaretsky Theorem. This article gives a streamlined presentation of the main ideas from Chapters 5 and 6 of the text.
    
    
  • Extra online Chapter 0: Expanded Notation and Preliminaries. This is a greatly expanded version of the "Preliminaries" section in the main text.
    
    
  • Alternative Chapter 1: An Introduction to Norms and Banach Spaces. The Chapter 1 that appears in the main text only presents a compressed review of metrics and norms; the expanded Alternative Chapter 1 covers this material in much greater detail and with discussion and exercises, focused on the setting of normed spaces.
    
    
  • Extra online Chapter 10: Abstract Measure Theory, presenting an introduction to abstract measures.
    
    
  • Selected Solutions for Students, containing a worked solution to approximately one exercise or problem from each section of the text.
    
    
  • A Solutions Manual for Instructors is available for instructors who register at the Birkhäuser website.
    
    
  • The current Errata List (updated December 2023). Please email me with any typos or errors that you may find!