Name: Understanding Analysis
ISBN: 9780387950600

Understanding Analysis

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 rational numbers be written as a countable intersection of open sets? Is an infinitely differentiable function necessarily the limit of its Taylor series? Giving these topics center stage, the motivation for a rigorous approach is justified by the fact that they are inaccessible without it.

What Hacker News says

turzmoAug 2026

Baby Rudin is a special textbook: it comes from a time when mathematics textbooks were not at all judged by 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 we’ve seen, LLMs seem to have absolutely no understanding or intuition. They are much better at things that have been expounded at length by humans before.

I would argue that if you’re going to use an LLM to make Rudin easier to understand, you are not learning how to absorb difficult material, nor are you learning analysis efficiently.

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seanhunterAug 2025

There is no royal road to mathematics[1], and it's incredibly arrogant to think that any person can provide a single optimal path. For me for example the next steps are Axler, Abbott and Herstein[2]. That's where I am at the moment, and it's way earlier than the books listed here. It would be far from optimal for me to try to bang my head stubbornly on this list. Mathematics demands you put in the work to build a foundation - you cant just skip steps. For some people those books I listed are very rudimentary. For others they are definitely too advanced for where they are and they'll need something else.

Even more so is the idea that you can actually cover the material listed in that page in 3 years. If you were to blast through it in that time you would only be skimming the very surface of the topics. There's simply no way you could possibly do all of those subjects justice in that time.

[1] As Euclid is supposed to have said about geometry to the Pharoah Ptolemy when Ptolemy said he wanted to learn geometry but because of all the concerns of his kingdom he didn't have time to read the Elements.

[2] "Linear Algebra done Right" by Sheldon Axler

"Understanding Analysis" by Stephen Abbott

"Topics in Algebra" by Herstein. this is a lovely book and beautifully written but some of the notation is a bit dated. I have two more recent algebra books but they are a bit advanced for me until I work through Herstein. They are Aluffi "Algebra Chapter 0" which is a good modern algebra book which introduces category theory at the start and Hien I forget the title but it's a springer one that he claims is good for an introduction but it's definitely not. It assumes you know a lot. It's very good though.

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laichzeit0Jan 2024

Yeah I laughed a bit when I saw Rudin’s book on that list. Understanding Analysis would be a better choice, even the end chapters of Spivak’s Calculus are a great into to Real Analysis.

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gmadsenJan 2024

Sure, texts like "linear algebra done right" or "Understanding Analysis" do a unusually good job of integrating large multi part examples where the reader works through them and proves the theorems themselves before they are explained in the book. The nominal case for most math texts is definition, lemmas, theorems, and sometimes they provide examples, and other times, readers are expected to make their own examples. For example, basic topology really only needs 10-20 pages to completely define, but to really understand why the axioms are chosen and the implications, you must work through examples, it is the only way. In fact there is a good book on topology specifically due to this " Counterexamples in Topology"

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blahblahblah10Jan 2023

Out of this list, the books I am familiar with, are great (Hilbert-Courant, Spivak, Korner's books). At the same time, even with extensive mathematical training, I haven't read them from start to finish. I wouldn't even like to say "read". For someone who's not used to mathematical reading, some of these books require careful study. That means generating examples to understand results (theorems), trying your own conjectures, proving things yourself etc. Over time, one becomes familiar with most/all the material in a book but the knowledge might have been acquired through various books (and courses) over time.

Also, mathematics is a massive field. The first question would be what kinds of mathematics would you like to get better at. There are great books in analysis. If you are starting out with a solid calculus knowledge, try Abbott's Understanding Analysis [1] or Duren's Invitation to Classical Analysis [2]. For asymptotic methods in PDEs, try Bender and Orszag [3] which is a wonderful book. But again, this might not be your cup of tea at all and there are more abstract or formal books like Rudin's.

If you want to approach fields without a lot of machinery, graph theory books by Bollobas are great (but difficult). See his Modern Graph Theory book [4] as an example.

For linear algebra, one of my favorites (but it was after I already learned the subject) is Trefethen's Numerical Linear Algebra book [5]. Another beautiful topic is at the intersection of linear algebra and combinatorics. See Babai and Frankl's lectures freely available online.

Then there are wonderful topics in geometry. A massive mountain to climb would be algebraic geometry. For one starting point, see [6]. Differential geometry (Spivak's multi-volume work or Needham's differential forms book) is another wonderful area. I would recommend Crane's discrete differential geometry course at Carnegie Mellon [7] if you want a concrete introduction.

You might want to demystify a topic you have heard about. E.g. Galois theory and the unsolvability of quintic equations. You could look at [8] which guides your way through wonderful problems.

We haven't even touched huge swathes of mathematics including anything topological or number theory. Even within the topics mentioned above, once you start, your journey will take a life of its own and you'll encounter multiple books and papers opening up new sub-fields.

The only approach that worked well for me in the past was to get completely consumed by what one topic one was studying. This meant not getting distracted by multiple topics. Once one enters the workforce, this is very hard (or at least has been for me). Without knowing someone, it's hard to recommend anything but the advantage with topics like graph theory and combinatorics is that one needs less machinery (as opposed to something like algebraic geometry). These fields lead you to interesting problems very rapidly and one can wrestle with them part-time.

[1] https://www.amazon.com/Understanding-Analysis-Undergraduate-...

[2] https://www.amazon.com/Invitation-Classical-Analysis-Applied...

[3] https://www.amazon.com/Advanced-Mathematical-Methods-Scienti...

[4] https://www.amazon.com/Modern-Graph-Theory-Graduate-Mathemat...

[5] https://www.amazon.com/Numerical-Linear-Algebra-Lloyd-Trefet...

[6] https://www.amazon.com/Algebraic-Geometry-Approach-Mathemati...

[7] https://www.cs.cmu.edu/~kmcrane/Projects/DDG/

[8] https://www.amazon.com/Through-Exercises-Springer-Undergradu...

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ssequeiraNov 2022

Understanding Analysis by Stephen Abbott and Algebra by Michael Artin.

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