(Part 2) Best mathematical analysis books according to redditors
We found 510 Reddit comments discussing the best mathematical analysis books. We ranked the 148 resulting products by number of redditors who mentioned them. Here are the products ranked 21-40. You can also go back to the previous section.
/u/3blue1brown Have you seen Wegert’s book Visual Complex Functions (site, gallery, amzn)?
He has a number of nice phase portraits of the Zeta function and of its critical strip, etc.
There are very few true textbooks - i.e. books designed to teach the material to those who don't already know the classical versions - written in this style.
While we're at it, a quick skim through the algebra chapter of Troelstra: Constructivism in Mathematics, vol. 2 should explain why there are no textbooks on abstract algebra written in the purely constructive tradition.
> Mathematical Logic
It's not exactly Math Logic, just a bunch of techniques mathematicians use. Math Logic is an actual area of study. Similarly, actual Set Theory and Proof Theory are different from the small set of techniques that most mathematicians use.
Also, looks like you have chosen mostly old, but very popular books. While studying out of these books, keep looking for other books. Just because the book was once popular at a school, doesn't mean it is appropriate for your situation. Every year there are new (and quite frankly) pedagogically better books published. Look through them.
Here's how you find newer books. Go to Amazon. In the search field, choose "Books" and enter whatever term that interests you. Say, "mathematical proofs". Amazon will come up with a bunch of books. First, sort by relevance. That will give you an idea of what's currently popular. Check every single one of them. You'll find hidden jewels no one talks about. Then sort by publication date. That way you'll find newer books - some that haven't even been published yet. If you change the search term even slightly Amazon will come up with completely different batch of books. Also, search for books on Springer, Cambridge Press, MIT Press, MAA and the like. They usually house really cool new titles. Here are a couple of upcoming titles that might be of interest to you: An Illustrative Introduction to Modern Analysis by Katzourakis/Varvarouka, Understanding Topology by Shaun Ault. I bet these books will be far more pedagogically sound as compared to the dry-ass, boring compendium of facts like the books by Rudin.
If you want to learn how to do routine proofs, there are about one million titles out there. Also, note books titled Discrete Math are the best for learning how to do proofs. You get to learn techniques that are not covered in, say, How to Prove It by Velleman. My favorites are the books by Susanna Epp, Edward Scheinerman and Ralph Grimaldi. Also, note a lot of intro to proofs books cover much more than the bare minimum of How to Prove It by Velleman. For example, Math Proofs by Chartrand et al has sections about doing Analysis, Group Theory, Topology, Number Theory proofs. A lot of proof books do not cover proofs from Analysis, so lately a glut of new books that cover that area hit the market. For example, Intro to Proof Through Real Analysis by Madden/Aubrey, Analysis Lifesaver by Grinberg(Some of the reviewers are complaining that this book doesn't have enough material which is ridiculous because this book tackles some ugly topological stuff like compactness in the most general way head-on as opposed to most into Real Analysis books that simply shy away from it), Writing Proofs in Analysis by Kane, How to Think About Analysis by Alcock etc.
Here is a list of extremely gentle titles: Discovering Group Theory by Barnard/Neil, A Friendly Introduction to Group Theory by Nash, Abstract Algebra: A Student-Friendly Approach by the Dos Reis, Elementary Number Theory by Koshy, Undergraduate Topology: A Working Textbook by McClusckey/McMaster, Linear Algebra: Step by Step by Singh (This one is every bit as good as Axler, just a bit less pretentious, contains more examples and much more accessible), Analysis: With an Introduction to Proof by Lay, Vector Calculus, Linear Algebra, and Differential Forms by Hubbard & Hubbard, etc
This only scratches the surface of what's out there. For example, there are books dedicated to doing proofs in Computer Science(for example, Fundamental Proof Methods in Computer Science by Arkoudas/Musser, Practical Analysis of Algorithms by Vrajitorou/Knight, Probability and Computing by Mizenmacher/Upfal), Category Theory etc. The point is to keep looking. There's always something better just around the corner. You don't have to confine yourself to books someone(some people) declared the "it" book at some point in time.
Last, but not least, if you are poor, peruse Libgen.
Hartshorne's Geometry: Euclid and Beyond is a much more readable book compared to his other well-known work.
In addition to Needham, I've heard very good things about Remmert's Theory of Complex Functions for its use of history and Wegert's Visual Complex Functions for its visual approach to complex analysis, similar to but perhaps more rigorous than Needham. Kenji Ueno's three-volume A Mathematical Gift is similar in its intuitive explanations, but it covers various topics in mathematics as opposed to just complex analysis and can act as a nice introduction or as light reading (yes, he has another three-volume work on AG). I can also recommend Foundations and Fundamental Concepts of Mathematics by Howard Eves for its breezy overview of the foundations of mathematics, for anyone interested in that.
Edit: Links
There are also some nice books on calculus, such as Excursions in Calculus by Robert M. Young and New Horizons in Geometry by Mamikon A. Mnatsakanian and Tom M. Apostol (of Calculus and Analytic Number theory fame).
Please, simply disregard everything below if the info is old news to you.
------------
Algebraic geometry requires the knowledge of commutative algebra which requires the knowledge of some basic abstract algebra (consists of vector spaces, groups, rings, modules and the whole nine yards). There are many books written on abstract algebra like those of Dummit&Foote, Artin, Herstein, Aluffi, Lang, Jacobson, Hungerford, MacLane/Birkhoff etc. There are a million much more elementary intros out there, though. Some of them are:
Discovering Group Theory: A Transition to Advanced Mathematics by Barnard/Neil
A Friendly Introduction to Group Theory by Nash
Abstract Algebra: A Student-Friendly Approach by the Dos Reis
Numbers and Symmetry: An Introduction to Algebra by Johnston/Richman
Rings and Factorization by Sharpe
Linear Algebra: Step by Step by Singh
As far as DE go, you probably want to see them done rigorously first. I think the books you are looking for are titled something along the lines of "Analysis on Manifolds". There are famous books on the subject by Sternberg, Spivak, Munkres etc. If you don't know basic real analysis, these books will be brutal. Some elementary analysis and topology books are:
Understanding Analysis by Abbot
The Real Analysis Lifesaver by Grinberg
A Course in Real Analysis by Mcdonald/Weiss
Analysis by Its History by Hirer/Wanner
Introductory Topology: Exercises and Solutions by Mortad
Pick up mathematics. Now if you have never done math past the high school and are an "average person" you probably cringed.
Math (an "actual kind") is nothing like the kind of shit you've seen back in grade school. To break into this incredible world all you need is to know math at the level of, say, 6th grade.
Intro to Math:
These books only serve as samplers because they don't even begin to scratch the surface of math. After you familiarized yourself with the basics of writing proofs you can get started with intro to the largest subsets of math like:
Intro to Abstract Algebra:
There are tons more books on abstract/modern algebra. Just search them on Amazon. Some of the famous, but less accessible ones are
Intro to Real Analysis:
Again, there are tons of more famous and less accessible books on this subject. There are books by Rudin, Royden, Kolmogorov etc.
Ideally, after this you would follow it up with a nice course on rigorous multivariable calculus. Easiest and most approachable and totally doable one at this point is
At this point it's clear there are tons of more famous and less accessible books on this subject :) I won't list them because if you are at this point of math development you can definitely find them yourself :)
From here you can graduate to studying category theory, differential geometry, algebraic geometry, more advanced texts on combinatorics, graph theory, number theory, complex analysis, probability, topology, algorithms, functional analysis etc
Most listed books and more can be found on libgen if you can't afford to buy them. If you are stuck on homework, you'll find help on [MathStackexchange] (https://math.stackexchange.com/questions).
Good luck.
The Gagliargo-Nirenberg inequalities you mention originate here.
Some of Nirenberg's "greatest hits" at a glance: some of his early work concerned the Minkowski problem of finding surfaces with prescribed Gauss curvature, and the related Weyl problem of finding isometric embeddings of positive curvature metrics on the sphere. For a gentle introduction to this type of problem accessible (in principle) after a basic course in differential geometry and some analysis, see these notes by Khazdan. For a more advanced treatment, including a discussion of the Minkowski problem and generalizations see these notes by Guan. This line of research owes a lot to Nirenberg.
In this legendary paper (2700+ citations, for a math paper!) and another with the same co-authors (Agmon and Douglis), he investigated boundary Schauder and L^p estimates for solutions of general linear elliptic equations. You can look at Gilbarg-Trudinger, or Krylov's books (1, 2) for the basics of linear elliptic equations, including boundary estimates. Here is a course by Viaclovsky in case you don't want to buy the books. This last set is far more basic stuff than Agmon-Douglis-Nirenberg, though, but it should give you an idea of what its about.
Another extremely famous contribution of Nirenberg is his introduction with Kohn of the (Kohn-Nirenberg) calculus of pseudodifferential operators. Shortly thereafter, Hoermander began his monumental study of the subject, later summarized in his books I, II, III, IV. If you know nothing about pseudo-differential operators, I suggest starting with this book by Alinhac and Gérard.
Another gigantic result is the Newlander-Nirenberg theorem on integrability of almost complex structures. An almost-complex structure is a structure on the tangent space of a manifold which mimics the effect that rotation by i has on the tangent vectors. The Newlander-Nirenberg tells you that if a certain simple necessary condition holds, you can actually choose locally holomorphic coordinates for the manifold compatible which induce this a.c. structure. A proof that should be reasonably accessible, provided you understand what I just wrote and have some basic notions of several complex variables can be found here.
Nirenberg also studied the important problem of (local) solvability of (pseudo)-differential equations with Francois Treves. In this paper, he introduced the famous condition Psi, which was only recently proved by Dencker to be necessary and sufficient for local solvability. An exposition of the problem at a basic level can be found in this undergrad thesis from UW.
Another massively influential paper was this one, with Fritz John, where he introduces the space of BMO functions, and proved the Nirenberg-John lemma to the effect that any BMO function is exponentially integrable. Fefferman later identified BMO as the dual of the Hardy space Re H_1, and the BMO class plays a crucial role in the Calderon-Zygmund theory of singular integral operators. You can read about this in any decent book on harmonic analysis. I myself like Duoandicoetxea's Fourier Analysis. BMO functions are treated in chapter 6. For a more "old school" treatment using complex analysis, including a proof of Fefferman's theorem, check out Koosis' lovely Introduction to H^p spaces.
Another noted contribution was his "abstract Cauchy-Kowalevski" theorem, where he formulated the classical theorem in terms of an iteration in a scale of spaces, instead of the more direct treatment based on power series. This point of view has now become classical. Look at the proof in Treve's book Basic Linear Partial Differential Equations.
Next, his landmark paper with Gidas and Ni (2000+ citations) on symmetry of positive solutions to second order nonlinear elliptic PDE are absolute classics. The technique is now a basic part of the "elliptic toolbox".
His series of papers with Caffarelli, Spruck and Kohn (starting here) on fully nonlinear equations is also classic, and the basis for much of the later work. It's gotten sustained attention in part because optimal transport equations are of (real) Monge-Ampere type.
The theorem about partial regularity of NS you are referring to is this absolute classic with Cafarelli and Kohn. A simple recent proof, together with an accessible exposition of de Georgi's method, can be found here.
Let me finish by mentioning my personal favorite, one of the most cited papers in analysis of the 20th century, an absolute landmark of variational analysis, Brezis-Nirenberg 1983. A pedagogical exposition appears in Chapter III of Struwe's excellent book.
TLDR: Nirenberg is one of the most important analysts of the past 60 years.
edit: Thanks for the gold! Glad this was useful/interesting to someone, given how advanced and specialized the material is.
/u/another_user_name posted this list a while back. Actual aerospace textbooks are towards the bottom but you'll need a working knowledge of the prereqs first.
Non-core/Pre-reqs:
Mathematics:
Calculus.
1-4) Calculus, Stewart -- This is a very common book and I felt it was ok, but there's mixed opinions about it. Try to get a cheap, used copy.
1-4) Calculus, A New Horizon, Anton -- This is highly valued by many people, but I haven't read it.
1-4) Essential Calculus With Applications, Silverman -- Dover book.
More discussion in this reddit thread.
Linear Algebra
3) Linear Algebra and Its Applications,Lay -- I had this one in school. I think it was decent.
3) Linear Algebra, Shilov -- Dover book.
Differential Equations
4) An Introduction to Ordinary Differential Equations, Coddington -- Dover book, highly reviewed on Amazon.
G) Partial Differential Equations, Evans
G) Partial Differential Equations For Scientists and Engineers, Farlow
More discussion here.
Numerical Analysis
5) Numerical Analysis, Burden and Faires
Chemistry:
Physics:
2-4) Physics, Cutnel -- This was highly recommended, but I've not read it.
Programming:
Introductory Programming
Programming is becoming unavoidable as an engineering skill. I think Python is a strong introductory language that's got a lot of uses in industry.
Core Curriculum:
Introduction:
Aerodynamics:
Thermodynamics, Heat transfer and Propulsion:
Flight Mechanics, Stability and Control
5+) Flight Stability and Automatic Control, Nelson
5+)[Performance, Stability, Dynamics, and Control of Airplanes, Second Edition](http://www.amazon.com/Performance-Stability-Dynamics-Airplanes-Education/dp/1563475839/ref=sr_1_1?ie=UTF8&qid=1315534435&sr=8-1, Pamadi) -- I gather this is better than Nelson
Engineering Mechanics and Structures:
3-4) Engineering Mechanics: Statics and Dynamics, Hibbeler
6-8) Analysis and Design of Flight Vehicle Structures, Bruhn -- A good reference, never really used it as a text.
G) Introduction to the Mechanics of a Continuous Medium, Malvern
G) Fracture Mechanics, Anderson
G) Mechanics of Composite Materials, Jones
Electrical Engineering
Design and Optimization
Space Systems
https://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192
​
https://www.amazon.com/Mathematics-Form-Function-Saunders-MacLane/dp/1461293405/ref=sr_1_3?keywords=MacLane%2C+Saunders&qid=1555006726&s=books&sr=1-3 (unfortunately, very expensive)
Some books even use those for set membership. Like this one. Good book, but the ε's really annoy me.
As /u/DR6 said, he is incorrect. It is trivial to show that the numbers in question are exactly the 10-adics with a partial order.
If you want further reading, I recommend this book, which gives a development of the p-adics and their properties, as well as the m-adics for composite m (section 1.10). It is a good book.
Most of the book is about Q_p for p prime, but if you read section 1.9 you will see why this is so. Ostrowski's Theorem shows why the prime case is a natural object of study, and the composite case is not.
An Introduction to Manifolds by Tu is a very approachable book that will get you up to Stokes. Might as well get the full version of Stokes on manifolds not just in analysis. From here you can go on to books by Ramanan, Michor, or Sharpe.
A Guide to Distribution Theory and Fourier Transforms by Strichartz was my introduction to Fourier analysis in undergrad. Probably helps to have some prior Fourier experience in a complex analysis or PDE course.
Bartle's Elements of Integration and Legesgue Measure is great for measure theory. Pretty short too.
Intro to Functional Analysis by Kreysig is an amazing introduction to functional analysis. Don't know why you'd learn it from any other book. Afterwards you can go on to functional books by Brezis, Lax, or Helemskii.
First you have to decide what kind. The easiest to understand IMHO is Nelson's IST, and the best introduction to that is the first chapter of an unpublished book by Nelson.
Robinson-style NSA doesn't have logical/set theoretical foundations. It has model theoretic foundations. And there isn't just one Robinson-style NSA. Rather there are lots of different ones based on different models.
If you want to know more about different axiomatic NSA the book for that is Nonstandard Analysis, Axiomatically.
Although I have a bunch of books on Robinson-style NSA, I don't recommend any of them (for me the subject is very confusing compared to IST). Goldblatt is OK.
You need some grounding in foundational topics like Propositional Logic, Proofs, Sets and Functions for higher math. If you've seen some of that in your Discrete Math class, you can jump straight into Abstract Algebra, Rigorous Linear Algebra (if you know some LA) and even Real Analysis. If thats not the case, the most expository and clearly written book on the above topics I have ever seen is Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.
Some user friendly books on Real Analysis:
Some user friendly books on Linear/Abstract Algebra:
Topology(even high school students can manage the first two titles):
Some transitional books:
Plus many more- just scour your local library and the internet.
Good Luck, Dude/Dudette.
>When university starts, what can I do to ensure that I can compete and am just as good as the best mathematics students?
Read textbooks for mathematics students.
For example for Linear Algebra I heard that Axler's book is very good (I studied Linear Algebra in another language, so I can't really suggest anything from personal experience). For Calculus I personally suggest Spivak's book.
There are many books that I could suggest, but one of the greatest books I've ever read is The Art and Craft of Problem Solving.
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.
You can get a cheap international 3rd edition very easily. That would be better than looking for an errata for the previous editions.
It depends on where they are and what the purpose is. If you are trying to discourage them (and there might be valid reasons to do that), I'd say try measure theory.
Maybe use the Bartle book.
That would give them a taste for how abstract things can get and also drive home the point tiny books can require a lot o work.
On the other hand, if you want to do something that will help them, they An Introduction to Mathematical Reasoning.
It won't break the bank and, despite a few small typos, covers a lot material fairly gently.
You are missing Abstract Algebra that usually comes before or after Real Analysis. As for that 4chan post, Rudin's book will hand anyone their ass if they havent seen proofs and dont have a proper foundation (Logic/Proofs/Sets/Functions). Transition to Higher Math courses usually cover such matters. Covering Rudin in 4 months is a stretch. It has to be the toughest intro to Real Analysis. There are tons of easier going alternatives:
Real Mathematical Analysis by Charles Pugh
Understanding Analysis by Stephen Abbot
A Primer of Real Functions by Ralph Boas
Yet Another Introduction to Analysis
Elementary Analysis: The Theory of Calculus
Real Analysis: A Constructive Approach
Introduction to Topology and Modern Analysis by George F. Simmons
...and tons more.
My university took an analysis approach to calculus. Our classes were 95% proofs and 5% application. I believe this gave us a better understanding for what calculus really is and a good introduction to later pure math courses. I would recommend you look in to [Introduction to Real Analysis by Robert G. Bartle & Donald R. Sherbert] (http://www.amazon.ca/Introduction-Real-Analysis-Robert-Bartle/dp/0471433314). The first few pages teach you the basics of proofs (proof structuring, sets, axioms, theorems such as well ordering property and etc) and then it dives into calculus. Good luck!
The 1st Course In Math Analysis by Brannan
Analysis I by Terrence Tao
Yet Another Intro To Real Analysis by Bryant
Understanding Analysis Stephen Abbott
Elementary Analysis: The Theory of Calculus Kenneth A. Ross
Metric Spaces by Robert Reisel
A Problem Text in Advanced Calculus by John Erdman. PDF
Advanced Calculus by Shlomo Sternberg and Lynn Loomis.PDF
This is a compilation of what I gathered from reading on the internet about self-learning higher maths, I haven't come close to reading all this books or watching all this lectures, still I hope it helps you.
General Stuff:
The books here deal with large parts of mathematics and are good to guide you through it all, but I recommend supplementing them with other books.
Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.
Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.
Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.
Abstract Algebra: One of the most important, and in my opinion fun, subjects in mathematics. Deals with algebraic structures, which are roughly sets with operations and properties of this operations.
There are many other beautiful fields in math full of online resources, like Number Theory and Combinatorics, that I would like to put recommendations here, but it is quite late where I live and I learned those in weirder ways (through olympiad classes and problems), so I don't think I can help you with them, still you should do some research on this sub to get good recommendations on this topics and use the General books as guides.
If you're interested in Fourier series in general, I'd recommend a couple of different books. They all contain these results (some contain more constructive versions than others).
[Stein and Shakarchi's Fourier Analysis: An Introduction] (http://www.amazon.com/Fourier-Analysis-Introduction-Princeton-Lectures/dp/069111384X) is probably the most accessible book I can think of. It doesn't assume much analysis background, and it's a pretty easy read. It contains all the classical goodies you should see on Fourier analysis and Fourier series without having to use any measure theory. It also springboards into the 3rd volume in this series, which is on measure theory.
Sticking with the classical camp but adding in a bit of measure theory and functional analysis, there's Katznelson's An Introduction to Harmonic Analysis and the infamous Zygmund Trigonometric Series. Zygmund is an exceedingly comprehensive introduction to Fourier series at the beginning graduate level. And I do mean comprehensive. It was published in 1935, and it's a fair bet that it captured close to everything that was known about convergence results concerning Fourier series at that time.
The last way I'd go (and I wouldn't really look at it until you have some background in the above) is Javier Duoandikoetxea's Fourier Analysis. The book makes very free use of measure theory and functional analysis. It also assumes a pretty good working familiarity with the theory of distributions (which it introduces at rapid speed).
I used Burden and Faires for three courses in numerical methods. I really enjoyed the book and it comes with free code online in a variety of languages. It is a little pricey but if you search hard enough (cough cough, first link) you will find it.
p-adic Analysis Compared with Real is the main text I've used. If you haven't taken real analysis, then parts of the book may be lost on you, but there are definitely large sections that will still be accessible.
It has lots of exercises as well as "Answers, Hints, and Solutions for Selected Exercises" at the end.
George Bergman's companion exercises to Rudin's textbook for Chapters 1-7.
Roger Cooke's solutions manual for Rudin's analysis
A subreddit devoted to Baby Rudin with further resources in the sidebar.
Tom Apostol's textbook
I find that Rudin is to Analysis textbooks what C++ is to programming languages. A little difficult at first, but with so many auxiliary sources that it becomes one of the best texts to learn from in spite of this.
Check out Mathematics Form and Function by Saunders Mac Lane. Mac Lane is one of the fathers of category theory, which is essentially the theory of the "big picture" in mathematics. The book is easy to read and seems pretty accessible.
That exam looks particularly computational from my experience but it is still pretty standard. It is very algebraic, don't be fooled by all the actual numbers :P.
The point is to use the general language of abstract algebra so you can answer questions about number theory. So you'd be working with very particular examples of rings like
[; \mathbb{Z}[\sqrt{d}] ;]and you'd have to make actual computations but the theory is based on very heavy algebra.The 'geometric' part comes from from actual basic algebraic geometry as you can associate geometric objects to number theoreic questions and so going back and forward between algebra and geometry will shed a better light on how to solve things in number theory.
As of selfstudying Galois theory I wouldn't know, depends on you I guess since to me it is kind of a dry subject if you are not properly motivated by good examples. Check out this book if you can, they are very good motivated notes on the topic. If you get engaged then go for it and study it in the summer, if you find it a little bit confusing then maybe it's better to have a professor delivering the content properly.
Edit: The notes I mentioned are heavy on the 'geometric' side of the theory and I don't think they cover the abstract setting of the theory in algebraic terms, but that's 'easy' to get once you get the point of what the whole topic is about and these notes are good at doing that IMO. So you'd still need to read a book dedicated entirely to the general theory.
>My first goal is to understand the beauty that is calculus.
There are two "types" of Calculus. The one for engineers - the plug-and-chug type and the theory of Calculus called Real Analysis. If you want to see the actual beauty of the subject you might want to settle for the latter. It's rigorous and proof-based.
There are some great intros for RA:
Numbers and Functions: Steps to Analysis by Burn
A First Course in Mathematical Analysis by Brannan
Inside Calculus by Exner
Mathematical Analysis and Proof by Stirling
Yet Another Introduction to Analysis by Bryant
Mathematical Analysis: A Straightforward Approach by Binmore
Introduction to Calculus and Classical Analysis by Hijab
Analysis I by Tao
Real Analysis: A Constructive Approach by Bridger
Understanding Analysis by Abbot.
Seriously, there are just too many more of these great intros
But you need a good foundation. You need to learn the basics of math like logic, sets, relations, proofs etc.:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers
Discrete Mathematics with Applications by Epp
Mathematics: A Discrete Introduction by Scheinerman
If you want something a bit more in depth than Wikipedia, baby Rudin is the standard analysis textbook and can be had for about ten dollars on Amazon. It's pretty dense though.
https://www.amazon.com/Principles-Mathematical-Analysis-Walter-Rudin/dp/1259064786/ref=pd_sbs_14_t_0?_encoding=UTF8&psc=1&refRID=2JHRTDF15JZR0HJF56GH
I also have a lot of love for the Bartle real analysis text. It's light on prerequisites other than calculus and the text spends time explaining the logical reasoning. A lot of upper-level math texts are simply a collection of theorems. The good ones present a coherent narrative that give context as each theorem is presented.
I can't speak to Gouvea, but in my undergrad p-adic course we used Katok which wasn't to bad. As long as your analysis and topology are reasonably well practiced than it should make a good companion text.
If you're interested, my professor for that class (Keith Conrad) wrote up some great handouts on various topics in p-adic analysis which can be found here under the Algebraic Number Theory section. I'd really recommend them.
Like justrasputin says, there usually is quite a lot of work to be done before you start to really see the beauty everyone refers to. I'd like to suggest a few book about mathematics, written by mathematicians that explicitly try to capture the beauty -
By Marcus Du Sautoy (A group theorist at oxford)
By G.H. Hardy,
Also, a good collection of seminal works -
God Created the Integers
And a nice starter -
What is Mathematics
Good luck and don't give up!
Terrence Tao gave some very nice examples in his book Analysis I on the point of rigor and mathematical analysis as a rigorous ground for calculus in general. He gave several examples for which, when we are in fact to apply mathematical theorems that are so common among calculus circles without rigor, would result in errors.
For example, L'Hôpital's rule was applied to x/(1+x) for x → 0 on page 10 of Analysis I, which gave a result of 1, which is clearly an incorrect answer. To quote: "-all that is going on here is that L'Hôpital rule is only applicable when both f(x) and g(x) go to zero as x → x_0, a condition which was violated in the above example. "
Quite a popular example would be the notion of differentials and the potential errors that may arise from an unrigorous handling of such mathematical objects.
In sufficiently rigorous mathematics books, mathematical theorems are derived always with the key assumptions (or conditions) stated for which the theorem holds true, from the existence of the derivative for the intermediate value theorem of differential calculus (keeping in mind that there exists functions that are everywhere uniformly continuous and nowhere differentiable, a classic example being Weierstrass's trigonometric series) to the condition of, for example, that any arbitrary curve must be periodic and sectionally smooth in any closed interval that does not contain a point of discontinuity in order for the Fourier series representing said curve to converge uniformly to said curve, e.g. Introduction to Calculus and Analysis I, page 604, section 8.6, a. etc..
Therefore, an unrigorous training may be just fine if we were to aspire to become technicians and stay always within the boundaries for which we were trained for, however, when attempting to apply mathematics to systems that are much more complex (e.g. dynamic systems in modern science being the frontier etc.), that are to be modeled, it seems quite clear to me that a rigorous training in analysis particularly could be very useful and, no less, results in much more efficient research and a lot less hair pulling.
Noting fully that many models of systems in reality are approximations at best, that we must satisfy ourselves with various assumptions, various assumptions for which perhaps we may try to find closest partners in the entirety of all mathematical theorems (analysis being the most popular, of course). If we aren't even intimate with the mathematical theorems, however, there may be problems.
.
Supplement:
Depending on the derivations and/or proofs, the set of mathematical conditions may be different. Sometimes, alternative derivations/proofs may furnish "relaxed" or less restrictive conditions.
> In particular, I am struggling with concepts of topology and mappings between metric spaces. I just cannot visualize what is going on in an intuitive way.
I think Apostol's text should work for you. The first few chapters use a lot of point-set topology language with a fair number of examples & proofs...that should hopefully address the issues you mentioned.
I agree with /u/canadabrah's suggestions.
I'll just add that McGill is also relatively strong in econometrics and applied econ. Queens also has a strong international-macro group.
If you're interested in international macro or labor economics, I would definitely add Western to your list of top Canadian PhD programs.
As canadabrah mentioned, real analysis is unlikely to be a high priority for Canadian MA programs, but it wouldn't hurt to take a course during your MA, or to start early through independent reading. I would advise going straight to a reputable textbook with exercises rather than relying on youtube videos. Here are a couple of recommendations to start:
http://math.harvard.edu/~ctm/home/text/books/royden-fitzpatrick/royden-fitzpatrick.pdf
https://www.amazon.ca/Real-Analysis-3rd-Halsey-Royden/dp/0024041513
I used Intro to Real Analysis by Bartle and Sherbert for my first analysis course. I just checked, and it has quite a few examples for most sections. The book was sufficient without lecture, as far as learning proofs and applying theorem goes. It was also relatively easy to read; there are a lot of analysis books which are hard to read due to terseness. You can probably find it for cheaper too. Good luck.
I know it's been a few hours, but here's Wolfram Alpha's article on Nonstandard Analysis which uses the terminology "genuine infinitesimal", which it defines as less than 1/n for all natural n but greater than 0. I'm guessing they were defining these so-called "genuine" infinitesimals in the article you posted.
If you want to learn a bit about Nonstandard Analysis and Hyperreals, I recommend the text "Lectures on the Hyperreals: An Introduction to Nonstandard Analysis" by Goldblatt.
Just pick up an intro to proofs text and work through it. Play around with sets/relations proofs, read up on various axiomatic constructions and see how you like that.
I haven't read it, but the people in the know say that Analysis I by Terry Tao is a very gentle intro.
My prof recommended me Bartle
Called The Art and Craft of Problem Solving, http://www.amazon.ca/Art-Craft-Problem-Solving/dp/0471135712
It focuses not only on really cool Olympiad problems but also on the psychology on dealing with problems. Must read.
My man Arnold be doing that for long time:
teaching 14 yr olds Abel's theorem:
https://www.amazon.com/Abels-Theorem-Problems-Solutions-Professor/dp/9048166098
introduction to calculus and analysis (3 book set) by courant and john:
volume 1
volume 2 book 1
volume 2 book 2
One book which comes to mind, although doesn't cover all of your topics, is "Introductory Mathematics: Algebra and Analysis" which I and many others found a very readable introduction as a beginning undergraduate. http://www.amazon.com/dp/3540761780/
There was a sister publication for applied topics called "Introductory Mathematics: Applications and Methods" but I haven't read it and don't know if it achieves the same easy-going and conversational style. http://www.amazon.com/dp/3540761799/
I loved topology and had no problem with it. For me it was a fourth year statistics course, I forget the details but I dropped it halfway through, did not need the credits.
I prefer http://www.amazon.com/Real-Analysis-3rd-Halsey-Royden/dp/0024041513 to Rudin's books.
Goldblatt can.
Relating to enjoying math, try reading some problem solving books. They stimulate logic and critical thinking, and show a "real" side of math that adds a lot of insight to any math course. I recommend The Art and Craft of Problem Solving. It truly opened up my eyes to the art and elegance of math. Just force yourself to really try the problems on your own before looking at the solutions, it makes a world of a difference.
Here is a great book, since the Putnam focuses much more on creative problem solving than specific knowledge:
http://www.amazon.ca/Art-Craft-Problem-Solving/dp/0471135712
This is slightly off topic, but if anyone is like me, and finds the style of rudin a bit too terse and magical, I'd recommend Apostol as a nice book in the same spirit, but (IMO) better written:
https://www.amazon.com/Mathematical-Analysis-Second-Tom-Apostol/dp/0201002884
It also has great, classic style cover art, and hardcover copies are well bound. Mine's goin on 15 years and is in good shape.
Has anyone read Mathematics Form and Function by Saunders Maclane?
Kreyszig is the best first book on functional analysis IMO. For measure theory I liked Royden, specifically the 3rd edition.
I hear that Rudin's book is pretty dense, so initially, I won't be using it, though I'm not entirely familiar with Spivak/Rudin beyond the comments on Amazon/Reddit.
Instead, I'm reading from Ross and [Bartle] (https://www.amazon.ca/Introduction-Real-Analysis-Robert-Bartle/dp/0471433314) right now, which I hear are good books for people starting out in Analysis. As I progress through the series, I might start teaching from Rudin and a variety of other sources.