Understanding Analysis
by Stephen Abbott · first published 2001
82% recommend18% neutral
Why HN recommends it
Rigorous but friendly real analysis intro with a narrative style that explains the problem and motivation, clear intuition, and well-chosen exercises; suits self-study and as an easier on-ramp before Rudin.
Summarised by Claude from recommending comments
Introduction to the Problems in Analysis outlines an elementary, one semester course which exposes students to both the process of rigor, and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable. The aim of a course in real analysis should be to challenge and improve mathematical intuition rather than to verify it. The philosophy of this book is to focus attention on questions which give analysis its inherent fascination. Does the Cantor set contain any irrational numbers? Can the set of points where a function is discontinuous be arbitrary? Can the
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…How to Prove It - A Structured Approach - Velleman OR Book of Proof (2nd ed) - Hammack (it's free!) 4. Mathematical Analysis: 4a. Good prep for Analysis: How to Think About Analysis - Alcock 4b. Understanding Analysis (2nd ed) - Abbott OR Yet Another Introduction to Analysis - Bryant (has full solutions) OR The How and Why of One Variable Calculus - Sasane OR Mathematical Analysis - A Straightforward Approach (2nd…
on “Ask HN: How to self-learn math?” ↗…do some exercises from the first 10 chapters. If you can do them easily, move onto analysis, if not, work through those 10 chapters first. 2) The book I personally like best for self-study is Abbott, “Understanding analysis” particularly if you can get the solutions manual, I think the explanations of the proofs are very good. 3) I would also recommend Lara Alcock’s book “How to think about analysis”, which is NOT a textbook,…
on “Ask HN: Resources to learn real analysis?” ↗Echo the above. Pairing little Rudin with a more user-friendly book like Understanding Analysis by Abbott is a great way to learn the material if you get stuck.
on “Ask HN: Math books like SICP?” ↗…in love with absolutely pure mathematics and most likely only in an academic setting, while I am interested in applications and self-studying. From modern texts, given your interests, I would look at "Understanding analysis" by Abbott and "Measure, Integral and Probability" by Capinski, both pleasant to read and together providing a not too steep path toward measure-theoretic probability.
on “Math topics useful for computer science/programming” ↗If you liked Axler, you might check out Abbott's Understanding Analysis, also in the Springer UTM series. I think it covers somewhat less than Rudin (e.g. looking at baby Rudin's contents, I'm pretty sure Abbott doesn't touch Lebesgue integration) but it's a pretty great…
on “A Software Engineer’s Adventures in Learning Mathematics” ↗Right now, I'm self-studying analysis with Stephen Abbott's "Understanding Analysis". I strongly recommend it. It has clear explanations that help to develop the intuition about each concept and what I think are well chosen exercises.
on “Elementary Applied Topology (2014)” ↗…doesn't make sense to choose courses (or even subjects to learn) based on whether video lectures are available. I don't maintain a collection of these, but I can probably recommend * Stephen Abbott, Understanding Analysis, for mathematical analysis and a little bit of real analysis * David Williams, Probability with Martingales, for advanced probability * Dym and McKean, Fourier Series and Integrals Alternatively if…
on “Ask HN: Is there a website specialized in mathematics courses?” ↗…Linear Algebra: Step by Step by Singh). Then: Linear Algebra - Friedberg, et al Linear Algebra Done Right - Axler Linear Algebra - Hoffman & Kunze Real Analysis: How to Think About Analysis - Alcock Understanding Analysis - Abbott Tao's Analysis text Principles of Mathematical Analysis - Rudin Topology: Topology - Munkres Topology A Categorical Approach - Tai-Danae Bradley, Tyler Bryson, and John Terilla Check out…
on “Physics Student Earns PhD at Age 89” ↗Agree that Baby Rudin is VERY difficult to study on its own. I recommend only studying it alongside the other two books I listed: Abbott's Understanding Analysis and Spivak's Calculus (which has a solutions manual). Abbott in particular is very straightforward (at least in comparison with baby Rudin haha)
on “So you want to study mathematics” ↗…study (I'm a bit biased here since I'm really having a hard time with baby Rudin). Two books that are somewhat a more gentle introduction to Analysis are Kenneth A. Ross' Elementary Analysis and Abbott's Understanding Analysis.
on “Ask HN: How to learn mathematical proofs from scratch?” ↗…their pedagogical value, but instead by their aesthetic appeal to mathematicians (a la Bourbaki). If you actually want to learn analysis, there are a nearly infinite number of friendlier resources (e.g., Understanding Analysis by Abbott). Learning to unpack difficult text on one’s own is a valuable skill. Research papers often require a similar amount of suffering, and at the frontier of knowledge, despite all the advances…
on “How I use LLMs to learn complex topics” ↗I'm missing: "Understanding Analysis" S. Abbott - great book on the foundations of Real Analysis, esp. for an honors course "Introductory Functional Analysis" E. Kreyzig
on “Math Books Organized by Area of Mathematics” ↗…analysis in particular, one approach is to start studying it in the order it was discovered. There's a well-regarded book called Analysis by Its History, but I haven't read it. There's another called Understanding Analysis by Stephen Abbott that I recommend if you want to regroup before attacking baby Rudin again.
on “Noted mathematician Rudin dies at 89” ↗For linear algebra, Sheldon Axler's _Linear Algebra Done Right_. For analysis, Stephen Abbott's _Understanding Analysis_. These two books will give you a very solid grounding in the undergraduate linear algebra and analysis. Personally, I also worship the style of Baby Rudin (it's the nickname of his _The Principles of…
on “Ask HN: Learning advanced math” ↗…http://www.youtube.com/playlist?list=PL0E754696F72137EC These could go with the either the book by W. Rudin or (haven't read) A. Browder. These are not easy for a complete beginner; maybe the lectures can provide some motivation. "Understanding Analysis" by S. Abbott is another rigorous and much easier, but very good, introduction. "Advanced calculus" by Shlomo Sternberg ... a work of profound beauty. This was the book used at Harvard in the 60s for…
on “Ask HN: Exploring math without a teacher?” ↗