#136 in Science & math books
Reddit mentions of Real and Complex Analysis (Higher Mathematics Series)
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Reddit mentions: 11
We found 11 Reddit mentions of Real and Complex Analysis (Higher Mathematics Series). Here are the top ones.
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Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.
Amazon search for Dover Books on mathematics
There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm
Pre-Calculus / Problem-Solving
Calculus
Linear Algebra
Differential Equations
Number Theory
Proof-Writing
Analysis
Complex Analysis
Functional Analysis
Partial Differential Equations
Higher-dimensional Calculus and Differential Geometry
Abstract Algebra
Geometry
Topology
Set Theory and Logic
Combinatorics / Discrete Math
Graph Theory
P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
It's hard to give an objective answer, because any sufficiently advanced book will be bound to not appeal to everyone.
You probably want Daddy Rudin for real analysis and Dummit & Foote for general abstract algebra.
Mac Lane for category theory, of course.
I think people would agree on Hartshorne as the algebraic geometry reference.
Spanier used to be the definitive algebraic topology reference. It's hard to actually use it as a reference because of the density and generality with which it's written.
Spivak for differential geometry.
Rotman is the group theory book for people who like group theory.
As a physics person, I must have a copy of Fulton & Harris.
You've taken some sort of analysis course already? A lot of real analysis textbooks will cover Lebesgue integration to an extent.
Some good introductions to analysis that include content on Lebesgue integration:
Walter Rudin, principle of mathematical analysis, I think it is heavily focused on the real numbers, but a fantastic book to go through regardless. Introduces Lebesgue integration as of at least the 2nd edition (the Lebesgue theory seems to be for a more general space, not just real functions).
Rudin also has a more advanced book, Real and Complex Analysis, which I believe will cover Lebesgue integration, Fourier series and (obviously) covers complex analysis.
Carothers Real Analysis is the book I did my introductory real analysis course with. It does the typical content (metric spaces, compactness, connectedness, continuity, function spaces), it has a chapter on Fourier series, and a section (5 chapters) on Lebesgue integration.
Royden's real analysis I believe covers very similar topics and again has a long and detailed section on Lebesgue integration. No experience with it, recommended for my upcoming graduate analysis course.
Bartle, Elements of Integration is a full book on Lebesgue integration. Again, haven't read it yet, recommended for my upcoming course. It is supposed to be a classic on the topic from what I've heard.
Machine learning is largely based on the following chain of mathematical topics
Calculus (through Vector, could perhaps leave out a subsequent integration techniques course)
Linear Algebra (You are going to be using this all, a lot)
Abstract Algebra (This isn't always directly applicable but it is good to know for computer science and the terms of groups, rings, algebras etc will show up quite a bit)
General Topology (Any time we are going to deal with construction of a probability space on some non trivial manifold, we will need this. While most situations are based on just Borel sets in R^n or C^n things like computer vision, genomics, etc are going to care about Random Elements rather than Random Variables and those are constructed in topological spaces rather than metric ones. This is also helpful for understanding definitions in well known algorithms like Manifold Training)
Real Analysis (This is where you learn proper constructive formulations and a bit of measure theory as well as bounding theorems etc)
Complex Analysis (This is where you will get a proper treatment of Hilbert Spaces, Holomorphic functions etc, honestly unless you care about QM / QFT, P-chem stuff in general like molecular dynamics, you are likely not going to need a full course in this for most ML work, but I typically just tell people to read the full Rudin: Real and Complex Analysis. You'll get the full treatment fairly briefly that way)
Probability Theory (Now that you have your Measure theory out of the way from Real Analysis, you can take up a proper course on Measure Theoretic Probability Theory. Random Variables should be defined here as measurable functions etc, if they aren't then your book isn't rigorous enough imho.)
Ah, Statistics. Statistics sits atop all of that foundational mathematics, it is divided into two main philosophical camps. The Frequentists, and the Bayesians. Any self respecting statistician learns both.
After that, there are lots, and lots, and lots, of subfields and disciplines when it comes to statistical learning.
A sample of what is on my reference shelf includes:
Real and Complex Analysis by Rudin
Functional Analysis by Rudin
A Book of Abstract Algebra by Pinter
General Topology by Willard
Machine Learning: A Probabilistic Perspective by Murphy
Bayesian Data Analysis Gelman
Probabilistic Graphical Models by Koller
Convex Optimization by Boyd
Combinatorial Optimization by Papadimitriou
An Introduction to Statistical Learning by James, Hastie, et al.
The Elements of Statistical Learning by Hastie, et al.
Statistical Decision Theory by Liese, et al.
Statistical Decision Theory and Bayesian Analysis by Berger
I will avoid listing off the entirety of my shelf, much of it is applications and algorithms for fast computation rather than theory anyway. Most of those books, though, are fairly well known and should provide a good background and reference for a good deal of the mathematics you should come across. Having a solid understanding of the measure theoretic underpinnings of probability and statistics will do you a great deal--as will a solid facility with linear algebra and matrix / tensor calculus. Oh, right, a book on that isn't a bad idea either... This one is short and extends from your vector classes
Tensor Calculus by Synge
Anyway, hope that helps.
Yet another lonely data scientist,
Tim.
Awesome! As mentioned, Rudin, Folland, and Royden are the gold standards of measure theory, at least from what I have heard from professors and the internet. I'm sure other people have found other good ones! Another few I somewhat enjoy are Capinski and Kopp and Dudley, as those are more based on developing probability theory. Two of my professors also suggested Billingsley, though I have not really had a good chance to look at it yet. They suggested that one to me after I specifically told them I want to learn measure theory for its own right as well as onto developing probability theory. What is your background in terms of analysis/topology? Also, I am teaching myself basic measure theory (measures, integration, L^p spaces), then I think that should be enough to look into advanced probability. Feel free to PM me if you need some help finding some of these books! I prefer approaching this from the pure math side, so mathematical statistics gets a bit too dense for me, but either way, I would look at probability then try to apply it to statistics, especially at a graduate level. But who am I to be doling out advice?!
*Edit: supplied a bit more context.
This is all assuming you know these subjects already, having a list of theorems is useless unless you know how the subject works, what the context is and understand how the proofs are done. If you are unfamiliar with these subjects, get Dummit & Foote for Algebra, Munkres for Topology and Baby Rudin for Analysis. Those three subjects are the building blocks for the rest of mathematics, basic knowledge (experience and proof techniques) of these three subjects is vital no matter what field you need to study. Especially in Mathematical Physics.
For functions of a single variable, Lebesgue integration is really just chopping up the y-axis, e.g. see Folland, Rudin, or almost any elementary treatment of real analysis. When extending to multiple dimensions, you must consider product spaces, and here the intuitive comparison to Riemann integration is not so clear.
Rudin covers Hilbert spaces and Banach spaces in his Real and Complex Analysis, which is why he jumps straight into topological vector spaces in his book on functional analysis. So perhaps you could read those chapters from Real and Complex Analysis. Alternatively, check out the classic Functional Analysis by Reed and Simon or Conway's book. The reviews published by the MAA might also be interesting to you. And of course, there are many lecture notes available on the web. :-)
Well, there's here, of course. Hilbert spaces are a topic in analysis. I've heard good things about this book, which comes at it from a physics perspective.
If your background in analysis is up for it, they are covered in Rudin. This book is pretty intense.
RPCV checking in. This is a good idea... you're going to have a lot of downtime and it's a great opportunity to read all the things you've wanted to but haven't yet found the time for. That could mean math, or languages, or just old novels.
When I was learning functional analysis, if found this book by Bollobas to be incredibly helpful. Of course, the only real analysis reference you need is Baby Rudin, but if you want to learn measure theory you may want his Real & Complex Analysis instead.
For texts on the other subjects, take a look at this list. You should be able to find anything you need there.
If you have any questions about Peace Corps, feel free to PM me. Good luck!
I have heard from professors that Rudin's Real and Complex Analysis is the go to book for analysis. I've also heard its a bit of a tough book to get through, but the understanding it provides is worth it.
If you end up using it let me know what you think! I'll be taking analysis next year.