Real Analysis

Real analysis is, in a nutshell, the rigorous foundation of calculus. However, it is often taught in a more general setting, using more general metric spaces or complex numbers instead of always sticking to the reals. It also extends calculus in ways that cannot really be done without a rigorous foundation. A good example of this is studying sequences and series of functions instead of simply real numbers.

The text from which I studied this topic is “Principles of Mathematical Analysis” by Walter Rudin, which is affectionately known as “Baby Rudin” in math circles. This is considered a very good real analysis text, but is sometimes not recommended as a first text. This is primarily due to the very dry presentation, often with little exposition or motivation. Many times it’s just a definition, theorem, proof, theorem, proof, etc. with no exposition in between. Additionally the exercises can be quite difficult, and I freely admit that I had more trouble with them than with the other texts I have studied and often had to eventually look up solutions. Having a solid grasp of logic and proofs and basic set theory is critical before tackling this text.

I studied this text as part of a study group that formed on the /r/math subreddit. However, as people studied at their own pace, interaction mainly consisted of the occasional post with questions about concepts or exercises. The subreddit still exists as /r/babyrudin, which I recommend viewing in the old reddit style as linked because the sidebar has some useful resources but is not shown in the new layout.

It is also worth noting that this text has delves into more advanced analysis in the later chapters. However, I only studied up through chapter 8 because the consensus on reddit was that this material is treated better in other, more advanced texts.

Resources

  • Buy the text at Amazon (affiliate link)
  • We developed a community solutions manual as part of the study group but it seems that, presently it is no longer available. Check back soon because I am working on getting this back online.
  • I also used this nice solutions manual for reference
  • Francis Su’s Real Analysis Lectures – This a a complete set of lectures for the Real Analysis course at Harvey Mudd College. The audiovisual quality is not stellar but it’s perfectly understandable. He teaches out of Baby Rudin but takes a couple of interesting digressions to topics not in the book. He also offers insight into many of the concepts that can’t be found in the text. I watched these videos as I worked through chapters in the text and found them quite useful in cementing understanding.

Other Texts

Below are some other texts that I see recommended all the time as perhaps more gentle introductions to real analysis. I cannot vouch for these texts, however, as I have not studied them myself.