Real Mathematical Analysis
by Charles Chapman Pugh · first published 2003
89% recommend11% neutral
Why HN recommends it
Rigorous yet approachable introduction to real analysis for mathematically inclined readers moving from calculus to proofs; praised as a good middle step between Spivak and Rudin.
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I absolutely, positively second the recommendation of "Real Mathematical Analysis" by Charles Pugh (don't miss the advice he relates from his colleague, on pages 9&10, with the heading "Metaphor and Analogy", which could easily form the basis for a dissertation on the psychology of…
on “Ask HN: How or where to begin learning mathematics from first principles?” ↗…Spivak's Calculus. Some people start with Topology by Munkres, which is not a difficult book but is very abstract and rigorous so makes a good introduction. If you feel like you have ok calculus chops, maybe Real Mathematical Analysis by Charles Pugh. Other good books are Linear Algebra Done Right by Axler, or the linear algebra book by Friedberg, Insel, and Spence. Maybe even learn linear algebra first. It's so useful. Do plenty…
on “Ask HN: How or where to begin learning mathematics from first principles?” ↗…Download the Book of Proof: http://www.people.vcu.edu/~rhammack/BookOfProof/ You read through it and do all the odd numbered exercises (the solutions are at the end of the book). Step 3: Get a book called Real Mathematical Analysis by Charles Pugh and you work through that and attempt as many problems as you can, with a view not to rush through it, but to expand your mind through each problem. Step 4: Pick any of these books that…
on “Mathematicians are chronically lost and confused (2014)” ↗(1) Real Mathematical Analysis, by Charles Pugh is a wonderful introduction to pure mathematics for a mathematically inclined engineer. The back cover starts like this > Was plane geometry your favourite math course in high school?…
on “Ask HN: Which book can attract anyone towards your field of study?” ↗A lot of recommendations depend on what you're trying to learn. But I've enjoyed the following texts to a larger extent than others: - Algebra: Chapter 0 (Aluffi) - Real Mathematical Analysis (Pugh) - Mathematics and its History (Stillwell) - An Introduction to Manifolds (Tu) - Gauge Fields, Knots and Gravity (Baez) - A First Look at Rigorous Probability Theory (Rosenthal) - All of Statistics…
on “Ask HN: Math books that made you significantly better at math?” ↗…Basic 1. Chapter Zero (Carol Schumacher) 2. Naive Set Theory (Paul Halmos) Linear Algebra 1. Finite-Dimensional Vector Space (Paul Halmos) 2. Linear Algebra Done Right (Steven Axler) Real Analysis 1. Real Mathematical Analysis (Charles Pugh) 2. Introduction to Analysis (Maxwell Rosenlicht) Algebra 1. A first course in abstract algebra (John Fraleigh) These books are better for self-directed reading compared to some of the…
on “Ask HN: Good Reading/Immersion in Mathematics” ↗…http://www.math.uh.edu/~tomforde/Books/Halmos-How-To-Write.p... http://www.math.northwestern.edu/graduate/Forum/HALMOS.html collected in his Selecta of expository writing. Some other books I like are Sheldon Axler's Linear Algebra Done Right, Real Mathematical Analysis by Charles Pugh, and the three books on manifolds (Topological Manifolds, Smooth Manifolds and Reimannian Manifodls) by John Lee, but there are many other well-known books that are clear in their exposition.…
on “The pitfalls of jargon and the damage it does to mathematics” ↗…Mathematical Analysis" by Burkill. Other viable options include "Calculus" by Spivak (it's really a book on real analysis at a level somewhere between a typical freshman calculus book and "baby Rudin"), "Real Mathematical Analysis" by Pugh, or "Yet Another Introduction to Analysis" by Bryant.
on “Math Books Organized by Area of Mathematics” ↗As mentioned in another subthread, Pugh's Real Mathematical Analysis is great, and better motivated than baby Rudin. Wilf - Generatingfunctionology (CRC 3rd ed.; free 2nd ed. pdf[1]) Lovasz - Combinatorial Problems and Exercises (AMS Chelsea) The classic probability…
on “Ask HN: Math books like SICP?” ↗…self-motivation. In addition to Abbott which you mentioned, "How to Think About Analysis" (Alcock), good gentle introduction for anyone who tries to dive into an analysis textbook and hits concrete. "Real Mathematical Analysis" (Pugh), good, rigorous approach that doesn't require quite as much effort to power through as Rudin does, although some of the exercises are tough. "Analysis I" and II (Tao), is good and starts from…
on “The Hacker News Top books of 2023” ↗…of many beginning mathematics students' misery). Anyone that's worked through this book knows the world of pain you're about to enter. I'd love if there was a similar recording for say, Charles Pugh's real mathematical analysis. I mean he's still alive, dunno if he still lectures from his book, but why not record one of those? It would be game changing for those who self-study. Here's another request: Terrence Tao does a…
on “Ask HN: Is there a website specialized in mathematics courses?” ↗Thank you very much for taking time to post many elaborate, useful comments! I straight away went and purchased "Real Mathematical Analysis" thanks to your recommendation. > One thing to keep in mind: do not be mislead by similar notation between mathematics and physics: they are very, very different subjects. This has always bothered me…
on “Ask HN: How or where to begin learning mathematics from first principles?” ↗As an opposing view, I offer you the following analogy, by mathematician Charles C. Pugh, which he draws between conceptions of metaphor as understood in natural versus mathematical language: From "Real Mathematical Analysis" 1st edition, p. 9: Metaphor and Analogy In high school English, you are taught that a metaphor is a figure of speech in which one idea or word is substituted for another to suggest a likeness or similarity.…
on “Calculus Made Easy (1914) [pdf]” ↗