Best functional analysis mathematics books according to redditors

We found 43 Reddit comments discussing the best functional analysis mathematics books. We ranked the 16 resulting products by number of redditors who mentioned them. Here are the top 20.

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Top Reddit comments about Functional Analysis Mathematics:

u/Lhopital_rules · 64 pointsr/AskScienceDiscussion

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

u/maruahm · 12 pointsr/math

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.

u/mathwanker · 5 pointsr/math

These were the most enlightening for me on their subjects:

u/dogdiarrhea · 5 pointsr/math

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.

u/darf · 5 pointsr/math

I like Kolmogorov as well.

You might try Reed and Simon which is aligned more closely to physics than to pure mathematics, but has all the rigor you would hope for.

u/bluecoffee · 5 pointsr/math

Before google, the standard reference for applied maths was Abramowitz & Stegun, which weighs in at a thousand pages, and even that doesn't have "most".

I'm afraid to say that there are a lot of formulas in mathematics. Really if you want a formula sheet, you have to restrict yourself to a single topic.

u/DFractalH · 5 pointsr/math

Sure, but they're in German. I don't know if there is an English translation.

  1. Königsberger: Analysis 1, Analysis 2
  2. Forster: Analysis 2
  3. Hildebrandt: Analysis 2

    I noticed that one book that I thought had ODEs didn't, which is rather akward, since that's what we're officially using in our lecture .. in which we're doing ODEs at the moment. I'm using a different book, though.

    Edit: I browsed through many books on amazon.co.uk, and it appears that most books don't introduce DEs of any kind. So it's definitely just my subjective experience that made me talking. :>

    I also stumbled upon this. I am convinced that one can woo mathematicians just by walking around with this badboy looking all suave and shit.
u/a_contact_juggler · 5 pointsr/math

There is an excellent series of Counterexamples in ... books which might be relevant to this thread:

counterexamples in...

u/timshoaf · 4 pointsr/statistics

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.

u/rcmomentum · 4 pointsr/math

If you're aiming to do nonlinear PDE, then see Brezis for basic functional analysis with some nonlinear topics scattered throughout the text (and very hard exercises and problems), or Lax which will take you from basic to advanced topics relevant for PDEs.

u/grandnational · 4 pointsr/puremathematics

I'm a big fan of
Strocchi's Introduction to the Mathematical Structure of Quantum Mechanics where the first couple of chapters give a very nice, concise introduction to (and derivation of!) the C*-algebraic background to quantum mechanics. If you really want to do things rigorously, you'll of course end up with the four volumes of Reed and Simon... of course, these sources are for quantum mechanics, not QFT per se, which is a different kettle of fish entirely.

If you want to carry on with the algebraic formalism in QFT you'll end up with local QFT which involves assigning an algebra of observables to regions of spacetime.

u/Hilbert84 · 3 pointsr/math

If you enjoy analysis, maybe you'd like to learn some more?

I really enjoyed learning introductory functional analysis, which is presented incredibly well in Kreyszig's book Introductory Functional Analysis with Applications. It's very easy to read, and covers a lot and assumes very little on the part of the reader (basic concepts from analysis and linear algebra). This will teach you about doing analysis on finite and infinite dimensional spaces and about operators between such spaces. It's incredibly interesting, and I highly recommend it if you enjoy analysis and linear algebra.

Another great analysis topic is Fourier Analysis and wavelets. I enjoyed the books by Folland Fourier Analysis and Its Applications. I don't believe that book has any wavelets in it, so if you're interested in learning Fourier analysis plus wavelet theory, then I highly recommend the very approachable and fun book by Boggess and Narcowich A First Course in Wavelets with Fourier Analysis. If you have any interest at all in applications (like signals processing), this subject is fundamental.

u/mugged99 · 3 pointsr/math

Functional Analysis by Lax is the most famous one I know of: http://www.amazon.com/Functional-Analysis-Peter-D-Lax/dp/0471556041.
Not sure about gentle though.

Do you have a specialization? You might be able to get your feet wet with a book that deals with functional analysis applied to your topic of interest. Me personally, I did PDEs and there are some neat books that have both PDEs and how functional analysis is used for them.

u/G-Brain · 3 pointsr/math

Rudin's Functional Analysis might have what you're looking for.

u/microwave_safe_bowl · 2 pointsr/Physics

I am a PhD candidate in Applied Mathematics and Engineering Science in an engineering school at a pretty well known university. we do almost exclusively mathematical physics. These may be too advanced but it's worth a shot

  1. this

  2. this may not seem directly relevant but asymptotics is absolutely critical to mathematical physics

  3. both volumes of this
u/lurking_quietly · 2 pointsr/math

>what is the difference really between 'calculus' and 'real analysis'

At the undergraduate level, "calculus" typically means the what. For example: what is this limit? What is the derivative of a given function? What is the value of this integral?

"Analysis" more typically gets into the why behind calculus. Why does this function have a limit? Justify why the typical rules for differentiation—product rule, chain rule, etc.—are valid. Define what it means for a function to be integrable over a given interval, and justify your computation of a given integral.

There's a lot more going on than just that, but to first approximation, making the distinction between the what of calculus and the why of analysis is a good starting point.

---

I don't have a copy of Kolmogorov's text, so I'm at a disadvantage. I assume you mean something like this book in the Dover series? If so, then the table of contents suggests it's a pretty ambitious book, at least for typical undergraduates—and especially if it's one's introduction to the subject matter. That text by Kolmogorov covers some of both metric space topology and point-set topology, as well as linear algebra, measure theory, integration, and differentiation (itself in the context of Lebesgue integration). I'm no expert on the matter, but Kolmogorov's (and Fomin's) text seems more representative of what's often called "functional analysis" rather than just "real analysis". I suspect that pedagogically, you might benefit from a more "concrete" introduction to real analysis before tackling something like this textbook.

As for the inverse and implicit function theorems, there are a handful of ways to approach those results. One way is to show that the two theorems are equivalent: the inverse function theorem is true if and only if the implicit function theorem is true. The way a lot of books proceed is to establish the inverse function theorem by making some suitable simplifications—e.g., that the derivative map is being evaluated at the origin, and that this derivative map is the identity map—then apply the contraction mapping theorem. (Of course, the two theorems are equivalent, so one could instead prove the implicit function theorem first, instead.)

Rudin is emphatically not the only suitable textbook for something like this, but nearly any such "suitable" textbook will inevitably be challenging. It will help you considerably to have already had linear algebra, at least, especially if you turn to a textbook that presupposes linear algebra as a prerequisite. I'm not sure what to recommend to you, but here are a few textbooks I've used over the years (in addition to those already mentioned above):

u/TheAntiRudin · 2 pointsr/math
u/DepravedManInky · 2 pointsr/math

I'm not really sure what you mean by 'algebraic'. Quantum mechanics is inherently an analytical subject.

That said, I found

http://www.amazon.com/Mathematical-Concepts-Quantum-Mechanics-Universitext/dp/3642218652

to be a good, rigorous introduction.

u/CunningTF · 2 pointsr/math

I personally think the book by Priestly is quite underrated. I learnt CA from this book, and would happily recommend it to anyone taking CA for the first time. Doesn't go as in depth as Ahlfors though, it's a more elementary introduction. But if you're studying the material for the first time as an undergrad, I'd defintitely choose this one.

u/FrijjFiji · 2 pointsr/math

Overe here in the UK, the standard book on complex analysis is Priestley's An Introduction to Complex Analysis.

Fully rigorous and easy to read, probably one of the best text books I've used.

u/sakattack · 2 pointsr/math

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.

u/phillyfanjd · 2 pointsr/books

Thanks for compiling all the links to those books. As for the two you couldn't find here are my best guesses:

Cancer

u/ArthurAutomaton · 2 pointsr/learnmath

I think Methods of Modern Mathematical Physics by Reed and Simon is the standard reference for this. Some other books I've seen recommended are Linear Operators in Hilbert Spaces by Weidmann and Unbounded Self-adjoint Operators on Hilbert Space by Schmüdgen.

u/poincareDuality · 2 pointsr/math

Functional Analysis out of this monster

u/functor7 · 2 pointsr/math
  • Lang for Algebra.

  • Hartshorne for Algebraic Geometry.

  • Hatcher for Algebraic Topology (you can just state most point-set things as fact, no need to reference anything).

  • Rudin for Real and Complex Analysis.

  • Rudin again for Functional Analysis.

  • Jech for Set Theory (unless you are talking about large cardinals, models, forcing or any other non-intuitive subject from set theory, you can just state things as fact).

  • I don't really have anything for Differential Geometry, maybe Hirsch? Not sure though, DG ain't my thing.

    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.
u/doctorbong · 2 pointsr/learnmath

Pick up any decent book in complex analysis. I'd strongly recommend either Brown & Churchill or Ahlfors; the second is better if you already know some complex analysis, though.

u/math_SS · 1 pointr/SubredditSimulator

Ahlfors is proof-intensive and can be used in lots of ways but not always easy. http://www.amazon.com/Complex-Analysis-Lars-Ahlfors/dp/0070006571.

u/WhackAMoleE · 1 pointr/math

Ahlfors is proof-intensive and can be used at the upper division or grad level. Great book in many ways but not always easy.

http://www.amazon.com/Complex-Analysis-Lars-Ahlfors/dp/0070006571

u/yo_mommas_lemma · 1 pointr/math

Here's a book that I've read a bit of. It's a bit like algebra: chapter 0, in that it takes a very categorical approach. It's actually pretty interesting that a lot of things in functional analysis can be stated pretty clearly as colimits.
https://www.amazon.com/dp/0821840983/?tag=stackoverfl08-20


u/frustumator · 1 pointr/math

Unfortunately I don't know of any good books firsthand - my knowledge comes from learning physics and then learning analysis =P

A great reference (albeit rather heavy for an intro) is Reed & Simon's Functional Analysis, first in a 4-part series. Volumes 2 and 4 look like they would be more physics-oriented, though I haven't read them myself.

u/JoinXorDie · 1 pointr/datascience

If you want theoretical / mathematical I would suggest reading a few math, stats or engineering books.

Dover is a great place to find some cheaper reading material. They republish old scientific and math texts that were popular in their time in a smaller sized paperback. They're a nice size to bring around with you and they don't cost much.

Math and stats findings of today build on this knowledge, and much of it is still used in state-of-the-art applications. Or, that math/stats is used as part of some state-of-the-art algorithm. Lots of the newest ML algorithms are blending math from a variety of areas.

Statistical analysis of experimental data

Principals of Statistics

Information Theory

Statistics Manual

Some theory of sampling

Numerical Methods for Scientists and Engineers (Hamming)

Mathematical Handbook for Scientists Engineers

Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables

==

There is also the Data-Science Humble Bundle for more technical / practical skill building.