Ask HN: Learning advanced math

For real analysis, I suggest Bartle's "Introduction...". I had a horrible teacher (well, I loved him, but he never prepped for class), and was able to learn everything by working through the proofs in the text. Bartle has a multi-dimensional version too.<p>For linear algebra, I was suggest Strang's "Linear Algebra with applications", 3rd edition. Then "Linear Algebra Done Right".<p>While you are at it, get Hungerfords "Abstract Algebra: An Introduction"; you will need an easy reference to fields and groups and polynomials.<p>All these books require an hour per page, but they lay it all out for you if you work for it. These are definitely undergrad books, but that is their beauty.<p>Bartle also has "Elements of integration and lebesgue measure" -- I bet it is great, but I haven't used it.<p>And if you find a good probability book, please post the title ;)

Advanced Linear Algebra, Roman; Linear Algebra, Hoffman &#38; Kunze; Matrix Analysis, Horn &#38; Johnson; Principles of Mathematical Analysis, Rudin; Real Analysis, Royden.<p>Miscellaneous comments:<p>- Reading pure abstract algebra (e.g. Dummit &#38; Foote) isn't a good use of time if you intend to go into statistics, since it only shows up in a few very special subareas. If you decide to go into one of these areas, you can learn this later.<p>- More advanced books on linear algebra usually emphasize the abstract study of vector spaces and linear transformations. This is fine, but you also need to learn about matrix algebra (some of which is in that Horn &#38; Johnson book) and basic matrix calculus, since in statistics, you'll frequently be manipulating matrix equations. The vector space stuff generally does not help with this, and this material isn't in standard linear algebra books. (Similarly, you should learn the basics of numerical linear algebra and optimization -- convex optimization in particular shows up a lot in statistics.)<p>- People have different opinions on books like Rudin, but you need to learn to read material like this if you're going into an area like probability. It's also more or less a de facto standard, so it is worth reading partly for that reason as well. So read Rudin/Royden (or equivalent, there are a small handful of others), but supplement them with other books if you need (e.g. 'The Way of Analysis' is the complete opposite of Rudin in writing style). It helps to read a few different books on the same topic simultaneously, anyway.<p>- Two books on measure-theoretic probability theory that are more readable than many of the usual suspects are "Probability with Martingales" by Williams and "A User's Guide to Measure-Theoretic Probability" by Pollard. There is also a nice book called "Probability through Problems" that develops the theory through a series of exercises.

The roadmap would be to first learn analysis (advanced calculus), then measure theory, then get a measure theory probability book.<p>I'll assume you are teaching this to yourself.<p>For analysis, I wouldn't get Rudin's book (concepts are poorly motivated). There are plenty of good Dover books. But I haven't read them because, well, I learned from Rudin.<p>For measure theory, I'd read Kolmogorov and Fomin's book. Rudin also has a measure theory book, which is much better than his analysis book, but it's hefty. Good problems though.<p>For a book on Probability, we read A Probability Path at my university. I wasn't fond of it. Someone referred me to Probability with Martingales, and though I didn't read it, it looked very good.

One of my favorite not-much-talked about analysis books is "A Radical Approach to Real Analysis" by Bressoud. Must read if you enjoy reading about the history of mathematics and famous mathematicians. A lot of people here recommend Rudin or Royden, but I have seen many people become turned off to the whole field of analysis because of how terse and user-unfriendly those are. In that sense, ARTRA is the polar opposite. The follow-up, A Radical Approach to Lebesgue's Theory of Integration, is also superb.

Baby Rudin is a great choice, especially for self study, since there are a number of study guides for it that go through solutions, additional explanation of material, etc. I'd recommend against the chapters on differential forms, though, just because the treatment is outdated. Marsden and Hoffman's approach is to focus on explanations for why theorems are true before giving proofs, which some people find useful.<p>Axler is a good choice for linear algebra. Dummit and Foote is the standard choice for algebra generally. I'm of the opinion that we should teach algebra before linear algebra in general, but this seems like a minority view.

For linear algebra, Sheldon Axler's _Linear Algebra Done Right_. For analysis, Stephen Abbott's _Understanding Analysis_.<p>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 Mathematical Analysis_), but it can be too dry to many people.

Trefethen and Bau is a good numerical linear algebra text, a fair bit of statistics can be understood in the linear algebra setting and this book helps teach how to think about linear algebra in a very useful way. I don't have a great reference, but stochastic PDEs is definitely a hot topic and promises to remain as such for a long time.

I would recommend video lectures from the Khan Academy (<a href="http://www.khanacademy.org/" rel="nofollow">http://www.khanacademy.org/</a>). They are organized in nice little chucks of topics and have a large range of topics in math.

Michaek Spivak's "Calculus" is pretty great for learning real analysis. The other canonical text is "Principles of Mathematical Analysis" by Walter Rudin, which I've never used, but is purported to be of the same quality.

Walter Rudin's book is a masterpiece. I read this book 35 years ago for my comps, and remember it as one of the best. Bear in mind there's no easy way to learn mathematics - it's going to be hard no matter what approach you take because that's the nature of the subject. My advice to undergrads is to skip computing completely because it's much easier to learn when you need it (unlike mathematics). You'll build your first computer program long before you prove your first theorem.

I've had "The Princeton Companion to Mathematics" sitting in my Amazon wish list for quite a while now, more or less for this purpose - I think on a HN recommendation.<p>Any thoughts on this book?

I highly recommend this book for Real Analysis <a href="http://www.amazon.com/Introduction-Analysis-Maxwell-Rosenlicht/dp/0486650383/ref=pd_sim_b_5" rel="nofollow">http://www.amazon.com/Introduction-Analysis-Maxwell-Rosenlic...</a><p>We used it at the University of Texas at Austin for the first semester in Real Analysis. I found it very clear and easy to follow.

Rosenlicht. Linear Algbera done Right. Then Finite Dimensional Vector Spaces. Halmos' Measure Theory. Awesome.

"Naive Set Theory" by Halmos is a true gem, and readable, too.

You will find course material from MIT OCW mathematics to be very useful:<p><a href="http://ocw.mit.edu/courses/mathematics/" rel="nofollow">http://ocw.mit.edu/courses/mathematics/</a>

iTunes U has some amazing lectures. Gilbert Strang (MIT) has some video lectures (MIT OCW) on linear algebra. They are a bit dull, but they are pretty decent.<p>I also recommend academicearth.org.<p>If you are near a university, take a class "mathematical physics". These kind of classes usually cover a lot of undergraduate material in a semester and are offered by many physics department. They usually use "Mathematical Methods" by Boas as a text.

Polya and Szego's "Problems and Theorems in Analysis I &#38; II" are classics. If you can work through any part, you will have learned a lot.

How can nobody here mention Courant and John's <i>Introduction to Calculus and Analysis</i>? I think I'm going to cry.

Can anyone suggest a good probability or statistics book?

Your<p>"I'm more than willing to spend an hour per page and do all the exercises, but I would like good exposition."<p>is essentially necessary and sufficient.<p>If an exercise takes more than two hours, then swallow your pride and skip the exercise (it may be <i>misplaced</i>, have an error, or just be way too difficult for effective education).<p>For linear algebra:<p>(1) Work through a few introductory texts.<p>(2) Work carefully through the long time, unchallenged, world-class classic,<p>Halmos, <i>Finite Dimensional Vector Spaces</i>.<p>and there note near the back his cute ergodic convergence theorem.<p>The glory here is the polar decomposition.<p>(3) Get some contact with some applications, including in elementary multi-variate statistics, numerical techniques, optimization, etc.<p>(4) Pick from<p>Horn and Johnson, <i>Matrix Analysis.</i><p>and<p>Horn and Johnson, <i>Topics in Matrix Analysis.</i><p>My first course was an "advanced course" from one of Horn, Johnson, and I knocked the socks off all the other students. How'd I do that? Brilliant? Worked hard? Learned a lot? Nope. Instead the key was just my independent work with (1) -- (3).<p>So if you do (1) -- (4), then you will be fine.<p>For analysis, (Baby Rudin)<p>Walter Rudin, <i>Principles of Mathematical Analysis.</i><p>Note in the back that a function is Riemann integrable if and only if it is continuous everywhere except on a set of Lebesgue measure 0.<p>Also know cold that a uniform limit of continuous functions is continuous.<p>Royden, <i>Real Analysis.</i><p>and the first, real, half of (Papa Rudin)<p>Rudin, <i>Real and Complex Analysis.</i><p>Of course, emphasize the Radon-Nikodym theorem; I like the easy steps in Royden and Loeve (below), but see also the von Neumann proof in Papa Rudin.<p>For probability based on measure theory and the limit theorems,<p>Breiman, <i>Probability.</i><p>Note his result on regular conditional probabilities.<p>Neveu, <i>Mathematical Foundations of the Calculus of Probability.</i><p>If you can work all the Neveu exercises, then someone should buy you a <i>La Tache</i> 1961.<p>Loeve, <i>Probability Theory.</i><p>Note the classic Sierpinski counterexample exercise on regular conditional probabilities (also in Halmos, <i>Measure Theory</i>),<p>Cover the Lindeberg-Feller version of the central limit theorem as well as simpler versions. Do the weak law of large numbers as an easy exercise. Cover the martingale convergence theorem (I like Breiman here) and use it to give the nicest proof of the strong law of large numbers. Cover the ergodic theorem (Garcia's proof) and its (astounding) application to Poincare recurrence. Cover the law of the iterated logarithm and its (astounding) application to the growth of Brownian motion.<p>Of course apply the Radon-Nikodym theorem and conditioning to sufficient statistics and note that order statistics are always sufficient. Show that sample mean and variance are sufficient for i.i.d. Gaussian samples and extend to the exponential family.<p>Give yourself an exercise: In Papa Rudin, just after the Radon-Nikodym theorem, note the Hahn decomposition and use it to give a quite general proof of the Neyman-Pearson lemma.<p>To appreciate the law of large numbers in statistics, read the classic Halmos paper on minimum variance, unbiased estimation.<p>For tools for research in statistics, might want to get going in stochastic processes. So, for elementary books, look for authors Karlin, Taylor, and Cinlar and touch on some applications, e.g., Wiener filtering and power spectral estimation. Note the axiomatic derivation of the Poisson process and the main convergence theorem in finite Markov chains (also a linear algebra result). Then for more, note again the relevant sections of Breiman and Loeve and then:<p>Karatzas and Shreve, <i>Brownian Motion and Stochastic Calculus.</i>