Reddit reviews Functional Analysis
We found 6 Reddit comments about Functional Analysis. Here are the top ones, ranked by their Reddit score.
Reddit reviews Functional AnalysisWe found 6 Reddit comments about Functional Analysis. Here are the top ones, ranked by their Reddit score.
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"
Have you done harmonic analysis? That's a good additional skill to add to your PDEs repertoire. I'm a huge fan of the introductory text Fourier Analysis by Javier Duoandikoetxea.
On the subject of PDEs, I think the natural extension of Evans's treatment is the three-volume series Partial Differential Equations I: Basic Theory, Partial Differential Equations II: Qualitative Studies of Linear Equations, and Partial Differential Equations III: Nonlinear Equations by Michael Taylor. Some of the old ground you already know is rehashed, e.g. Sobolev spaces and functional analysis. But you'll also do a whole lot of differential geometry, Lie theory, operator algebra, spectral theory, scattering theory, index theory, etc. Its coverage of both linear and nonlinear PDEs is also very comprehensive, probably the best you can get outside of collecting volumes of monographs.
If you've already worked through Walter Rudin's Functional Analysis or equivalent, you can also look into operator algebra. The commonly recommended text is Bruce Blackadar's Operator Algebras.
My specific area is SDEs. If you're interested in them but know little about the martingale treatment of probability, I suggest starting probability theory with Erhan Cinlar's Probability and Stochastics, learning stochastic calculus with Ioannis Karatzas and Steven Shreve's Brownian Motion and Stochastic Calculus, and then doing SDEs with Bernt Oksendal's Stochastic Differential Equations.
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.
Rudin's Functional Analysis might have what you're looking for.
Functional Analysis out of this monster
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.