Principles of Mathematical Analysis

The three for me were: - Principles of Mathematical Analysis by Walter Rudin (aka “Little Rudin”) - Linear Algebra and its Applications by David Strang - Elementary Differential Equations and Boundary Value Problems by Boyce and Diprima

I jumped into the first analysis class (using baby Rudin) completely unprepared and this course saved me!

It’s true that most people will probably drown if you throw them into Rudin first thing (although I’ve known a couple who held their breath, swam the whole length, jumped out and asked for more). The thing is, “analysis” in (English) mathematical vernacular covers a lot more than were dreamt of in Newton’s philosophy, and that in turn is a lot more than is habitually included in a course entitled “calculus”. On the other hand, most calculus courses cover (badly and shallowly) many things that are properly from other places (commutative algera [field axioms], order theory [Dedekind completion], general topology [limits and opens], set theory [cardinals]) but just can’t be avoided when talking about the reals. So, what do you not get in a standard first course in calculus that still goes under “analysis” (but is not a research topic)? - Filters and/or nets (a coherent viewpoint on all the limits) - (Multiple) derivatives as objects of (multi)linear algebra (no more horrific “Jacobians” and “Hessians”) - Implicit / inverse function theorem (local normal form under smooth change of coordinates, cf Morse’s lemma as well)…

+1, but nitpicking: There are situations where baby Rudin ("Principles of Mathematical Analysis") can be very good for a first course...but those are corner cases, that you'd want a really good, experienced instructor to sign off on.

Books that are given nicknames by grad students. :) For a foundational understanding of Calculus, and just becoming a better mathematician, a good start would be -> Baby Rudin (Principles of Mathematical Analysis, Walter Rudin) Papa Rudin (Real and Complex Analysis, Walter Rudin) Grandpa Rudin (Functional Analysis, Walter Rudin)

Bluntly, with only occasional exceptions, have to learn from the books. Period. No biggie: Once are in grad school or a prof, have to learn from papers, and the good books are typically MUCH easier to read than the papers! For first calculus, read one of the well respected books. In my experience and opinion, don't need high school advanced placement calculus -- regard it as a waste of time. Also the last time I looked at Khan Academy, I concluded that it was a waste or worse and by people who didn't have a good background in calculus. Generally the Internet sources are too elementary. Solution: Just use some good books. Protter and Morrey is good (I taught from it), but it is a bit elementary. I learned from Johnson and Kiokmeister; I like it. At the time it was also being used at Harvard. The exercises are especially good. It's old, but calculus hasn't changed much since 1950 or so about when it gave up on teaching infinitesimals. But might check out the Maxima software for symbolic indefinite integration -- as I recall, its algorithms are so good they do any indefinite integration that can be done in closed form.…

Agree, Rudin is excellent as a second course on real analysis, but as a first one it is absolutely terrible.

Came here to say Rudin. His presentation is “odd” in that he introduces powerful concepts and whacks large numbers of problems on the head with them, whereas the usual pedagogical approach is to teach techniques that correspond to these concepts but never explain the concept. I also really liked Herstein on algebra but I believe it’s out of print now.

Baby Rudin is indeed awesome. I feel like Linear Algebra Done Right is a great book, but it is a better second book about Linear Algebra, rather than a good first book. Actually, I think the whole subject deserves a fairly mechanical treatment first, and only then an explanation of the underlying math. I think it would make sense to introduce Matrix multiplication and related concepts to students in high school, so they become completely mechanical - much like we teach students how to differentiate functions as a purely mechanical process.

Baby Rudin, Principles of Mathematical Analysis, was that book for me. It was that book for a lot of people, which I guess means it felt "perfect" for people with a broad range of backgrounds. I never found a book like that for algebra. Algebra felt opaque, simultaneously hard and boring, and linear algebra felt dry and mechanical, more like a vocational skill than like mathematics. If I had encountered Axler's Linear Algebra Done Right in undergrad, I might have been more interested in algebra. By the time I encountered "computer science" in my mid-twenties, I don't think the quality of textbook mattered as much to me, but I wish I could remember the name of the formal languages and automata textbook I used to have. It was short and probably not very advanced, but the concepts are still vivid in my head. I see state machines everywhere, and things feel easy to me when I can tackle them as state machines.

+1 for baby Rudin, while I wouldn't say it was THE book for me, it was certainly one that gave me a proper understanding of mathematical proofs as opposed the mroe applied mathematics books I had read until then. And the exercises are excellent!

SICP for CS and programming. That book (and its exercises) really instilled in me the thought patterns that drastically improved my programming skills and made me realize how important fundamental CS knowledge is to being a good programmer. What I learned from that book was: becoming comfortable with recursion, realizing the importance of abstractions, functional programming concepts and it gave me insights into how interpreters and the machine work. I studied math in university and the book that really improved me mathematically was baby rudin. I really struggled with this the first time I worked through it and had to also fall back on Abott's analysis book, but after my first analysis course I worked through Rudin again and it just clicked. The exercises are really well chosen and the text is just so on point. Every sentence in the book is extremely carefully chosen and of fundamental importance to what Rudin wants to teach you. It's really rare that you find textbooks that almost kind of have a certain magic to them, like you are directly taught by one of the greatest minds in that field. These two books definitely meet that criteria.

> You can work through the concepts and delude yourself into thinking you understand something when really you're just hand waving it I keep seeing this and I don't quite understand how this one works. If I were studying math on my own (which I've done and still do), I'd do the following: 1. Pick a book. Say, Rudin's Principles of Mathematical Analysis[0]. Read a section, then attempt problems. Pick a problem. Say, "prove {1/n: n is natural} U{0} is compact directly from the definition(not using Heine-Borel)". It's guaranteed your "proof" is not a proof. 2. Compare your solution to existing solution manuals or ask a question on MSE[1]. Since the given book by Rudin is super-massively famous, each question has probably been asked/answered about a bagillion times each on MSE, so just searching MSE alone would likely to spit out many answers to your questions. People on MSE will tell you exactly why your "solution" is wrong and where you tripped up. Sometimes even the clarifying answers are hard to understand. But then you can ask new questions, think more, correct your misconceptions until it all finally clicks. Do that with all the rest of the problems[2].…

Linear algebra, and more than one such book. IMHO long and still the best linear algebra book is Halmos, Finite Dimensional Vector Spaces (FDVS). It was written in 1942 when Halmos was an "assistant" to John von Neumann at the Institute for Advanced Study. It is intended to be finite dimensional vector spaces but done with the techniques of Hilbert space. The central result in the book, according to Halmos, is the spectral decomposition. One result at a time, the quality of von Neumann comes through. Commonly physicists have been given that book as their introduction to Hilbert space for quantum mechanics. But FDVS is a little too much for a first book on linear algebra, or maybe even a second book, should be maybe a third one. Also high quality is Nering, Linear Algebra and Matrix Theory. Again, the quality comes through: Nering was a student of Artin at Princeton. There Nering does most of linear algebra on just finite fields, not just the real and complex fields; finite fields in linear algebra are important in error correcting codes. So, that finite field work is a good introduction to abstract algebra.…

I think that those who are down-voting people suggesting Rudin are missing the forest for the trees. The level is not as low as Feynman's lectures, but if you want a text that has the hallmarks of a true master at work, Rudin fits the bill. I personally always felt that what made Feynman's lectures what they are were, was that the man actually understood the subjects on such a deep level that his mind was able to hone in on those little simple thought experiments and ways of looking at things that gave you huge insight into what he was lecturing on. Rudin comes at analysis with the same level of understanding and that's why Real & Complex and Baby Rudin are still the gold standard for analysis texts. It is impossible to convey the sense of elegance that mathematicians speak about until you've seen it and been blown away by it. Rudin will do that and leave you speechless, he's the master. I can tell you that my first encounter with Real & Complex leaving me thinking that Rudin was on a fundamentally higher plane than any author I had read before, it was like watching a magician. If you're looking for a real Feynman in mathematics, IMHO Rudin is your man.

Principles of Mathematical Analysis by Walter Rudin

I'm a big fan of SICP. I'd describe it as "hard-core": requiring hard work, but extremely rewarding for the people who take the time and effort to go through it. I'd recommend the following three books to similarly cover the three main areas of higher mathematics (analysis, algebra, and topology): 1. Principles of Mathematical Analysis by Rudin (http://www.amazon.com/Principles-Mathematical-Analysis-Inter... (http://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X)) 2. Algebra by Artin (http://www.amazon.com/Algebra-2nd-Featured-Titles-Abstract/d... (http://www.amazon.com/Algebra-2nd-Featured-Titles-Abstract/dp/0132413779)) 3. Topology by Munkres (http://www.amazon.com/Topology-2nd-James-Munkres/dp/01318162... (http://www.amazon.com/Topology-2nd-James-Munkres/dp/0131816292))…

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.

It's not nearly as easy to work through as SICP is but Principles of Mathematical Analysis by Walter Rudin (sometimes referred to as little Rudin) is a great place to start if you're interested in analysis. It's a hard book, but it's pretty much the standard for undergrad analysis.

"Can you please suggest on-line resource to learn great beginning and advanced probability concepts?" Yup! The subject at the level you ask is a major topic in applied math but is not very popular in US universities. So, you will not get many good answers. In particular, the 'computer science' community, with 'machine learning' and 'artificial' this, Bayesian that, likely won't have good answers. Your "on-line" part is asking a bit much; I can give you references to books but not all on-line. There may be some PDF files on-line, from TeX, that have such material; try some Google searching with the keywords used here. The intuitive foundations of probability went back to gambling. 'Probability' is a field of 'applied' math and as such is well defined: About 100 years ago, E. Borel student H. Lebesgue invented 'measure theory' which essentially 'rewrote' classic calculus, especially the part about integration. For the simple cases, what Lebesgue did gets the same numerical values as the classic Riemann integral. The difference is that in theoretical work Lebesgue's integral is much more general and much better 'behaved'.…