Reddit reviews Understanding Analysis (Undergraduate Texts in Mathematics)
We found 15 Reddit comments about Understanding Analysis (Undergraduate Texts in Mathematics). Here are the top ones, ranked by their Reddit score.
Springer
For real analysis I really enjoyed Understanding Analysis for how clear the material was presented for a first course. For abstract algebra I found A book of abstract algebra to be very concise and easy to read for a first course. Those two textbooks were a lifesaver for me since I had a hard time with those two courses using the notes and textbook for the class. We were taught out of rudin and dummit and foote as mainly a reference book and had to rely on notes primarily but those two texts were incredibly helpful to understand the material.
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If any undergrads are struggling with those two courses I would highly recommend you check out those two textbooks. They are by far the easiest introduction to those two fields I have found. I also like that you can find solutions to all the exercises so it makes them very valuable for self study also. Both books also have a reasonable amount of excises so that you can in theory do nearly every problem in the book which is also nice compared to standard texts with way too many exercises to realistically go through.
Try picking up a book. I recommend this one. You can also use Rudin but it will be more difficult.
If you are using notes and online research, it may be that the exercises you've been working on are coming from many different areas and aren't really focused on one topic in particular. This may be the reason that every problem seems to require a new trick.
While it's certainly not the best or broadest advice, I've always found that, whenever a problem starts to get excessively complicated, the mean value theorem always seems to be the why-didn't-I-think-of-that trick that solves it.
Please, simply disregard everything below if the info is old news to you.
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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.
Understanding Analysis by Stephen Abbott https://www.amazon.com/dp/1493927116/
Topics in Algebra by I.N. Herstein https://www.goodreads.com/book/show/1264762.Topics_in_Algebra
The Feynman lectures on physics http://www.feynmanlectures.caltech.edu/
I've got nothing for Economics, but the above would be my personal recommendations for self-study and just general reading.
I recommend thumbing through an introductory real analysis textbook like Abbot - and perhaps speaking to a professor - before declaring a second major. Mathematics beyond sophomore level are a lot different, even at the applied level.
FWIW, I quit a PChem PhD program to pursue applied math, it definitely gives you a lot more flexibility, but it's not for everyone.
The Nature of Computation
(I don't care for people who say this is computer science, not real math. It's math. And it's the greatest textbook ever written at that.)
Concrete Mathematics
Understanding Analysis
An Introduction to Statistical Learning
Numerical Linear Algebra
Introduction to Probability
I'll be using some LaTeX markup here; see the sidebar for free browser plugins to translate my code into something readable.
Uniform continuity is a stronger condition than "mere" continuity (on a particular domain or subset of a domain), and the difference between uniform continuity and mere continuity can be subtle. In order to build up your intuition, it's worth going through a few examples. (Most of these can be generalized to arbitrary metric spaces—except some of the examples involving derivatives, of course—but I'll use real-valued functions for now to help make things more concrete.)
[; f \colon K \to \mathbb{R} ;]is a continuous function on[; K, ;]where[; K ;]is compact, then[; f ;]is uniformly continuous on[; K. ;]That is, if
[; f ;]is "minimally nice" (i.e., continuous) on a "nice" domain (i.e., a compact one), then[; f ;]is uniformly continuous on that domain, meaning it's even nicer than initially thought.[; S \subseteq \mathbb{R} ;]is a function with bounded derivative, then[; f \colon S \to \mathbb{R} ;]is uniformly continuous.This is a good heuristic to detect when a given function might not be uniformly continuous. For example, the function
[; f \colon x \mapsto x^2 ;]is not uniformly continuous on the domain[; S := \mathbb{R}, ;]and part of the obstacle to uniform continuity on[; \mathbb{R} ;]is that[; |f'(x)| ;]is not bounded on[; \mathbb{R}. ;]Warning: The converse to #2 above is false, though. Consider the function
[; f(x) := \sqrt{x} ;]on the interval[; [0,1]. ;]Then[; \lim_{x \to 0^{+}} f'(x) = +\infty, ;]but since[; [0,1] ;]is compact and[; x \mapsto \sqrt{x} ;]is continuous,[; f ;]is uniformly continuous on[; [0,1] ;]by #1 above.[; f \colon S \to \mathbb{R} ;]is a uniformly continuous on[; S. ;]Then if[; (s_n) ;]is a Cauchy sequence in[; S, ;][; (f(s_n)) ;]is a Cauchy sequence in[; \mathbb{R}. ;]That is, uniformly continuous functions preserve "Cauchyness".Note: This condition fails for functions which are merely continuous. Consider the function
[; f \colon (0,1) \to \mathbb{R}, ;][; f(x) := 1/x. ;]Then the sequence[; (1/n) ;]is a Cauchy sequence in[; (0,1), ;]but the image sequence[; (f(1/n)) = (n) ;]is not a Cauchy sequence.[; f \colon S \to \mathbb{R} ;]is uniformly continuous on[; S, ;]then[; f ;]can be extended to a unique continuous function[; F \colon \overline{S} \to \mathbb{R} ;]such that[; F|_{S} = f. ;]That is, we can take a given function[; f ;]which is uniformly continuous on[; S ;]and extend it in a unique way to continuous function on[; \overline{S}, ;]the closure of[; S. ;]The proof of this basically depends on #3 above: since Cauchyness is preserved, and since
[; \mathbb{R} ;]is complete, we can construct "candidates" for elements[; F(s), ;]where[; s \in \overline{S} \setminus S. ;]Once we show that our values for[; F(s) ;]don't depend upon the specific Cauchy sequence in the domain, all that remains is to show that the extension function is continuous on the closure[; \overline{S} ;]of[; S. ;]Note: This gives a method of detecting nonuniform continuity. For example, if you have the function
[; f \colon (0,1) \to \mathbb{R}, ;][; f(x) = 1/x, ;]then there's no way to extend that to a continuous function[; F \colon [0,1] \to \mathbb{R}. ;]The problem is that since[; \lim_{x \to 0^{+}} f(x) = +\infty, ;]any finite choice for[; F(0) ;]will introduce a discontinuity into any hypothetical extension function[; F \colon [0,1] \to \mathbb{R}. ;]"A function
[; f \colon A \to \mathbf{R} ;]fails to be uniformly continuous on[; A ;]if and only if there exists a particular[; \epsilon_0 > 0 ;]and two sequences[; (x_n) ;]and[; (y_n) ;]in[; A ;]satisfying[; |x_n - y_n| \to 0 ;]but[; |f(x_n) - f(y_n)| \geq \epsilon_0. ;]"I'm sure there's much more I could add to this list, but this seems to be a good starting point. The bottom line is that while continuous functions are "nice", functions which are continuous on a particular domain are even nicer where they happen to be uniformly continuous.
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As for motivating the definition of uniform continuity, there might be some degree of reverse-engineering going on here. For example, having a function such that the image of a Cauchy sequence is also Cauchy might be a useful property. The criterion for this turns out to be that of uniform continuity (on the relevant domain or subdomain).
Anyway, I hope the above helps clarify your sense of intuition about the differences between uniform continuity and mere continuity. Good luck!
In addition to Baby Rudin, I really liked this book when I first start learning analysis.
Calculus by James Stewart is the best introductory Calculus book that I used in college - I definitely recommend it. It will get you through both single-variable calculus, as well as most of multi-variable calculus that you will need for for master's level probability and statistical theory. In particular, if you plan to use the book, you should focus on chapters 1-7 (for single variable calculus), chapter 11 (infinite sequences and series) and chapters 14 and 15 (partial derivatives and multiple integrals). These chapter numbers are based on the 7th edition.
If you have previously taken calculus, you might consider looking at Khan Academy for an overview instead.
If you have not previously taken linear algebra, or it has been awhile, you will definitely need to work through a linear algebra textbook (don't have any particular recommendations here) or visit Khan academy.
Finally, a book such as Stephen Abbott's Understanding Analysis is not necessary for master's level statistics, but could be helpful for getting into the mindset of calculus-based proofs.
I'm not sure what level of math you have previously completed, and what level of rigor the MS in Statistics program is, but you will likely need be very familiar with single- and multi-variable calculus as well as linear algebra to be successful in probability and statistical theory. It's certainly possible, just pointing out that there could be a lot of work! If you have any other questions, I'm happy to answer them.
Here are some great books that I believe you may find helpful :)
and last but definitely not least:
Later on:
This is just my perspective, but . . .
I think there are two separate concerns here: 1) the "process" of mathematics, or mathematical thinking; and 2) specific mathematical systems which are fundamental and help frame much of the world of mathematics.
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Abstract algebra is one of those specific mathematical systems, and is very important to understand in order to really understand things like analysis (e.g. the real numbers are a field), linear algebra (e.g. vector spaces), topology (e.g. the fundamental group), etc.
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I'd recommend these books, which are for the most part short and easy to read, on mathematical thinking:
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How to Solve It, Polya ( https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X ) covers basic strategies for problem solving in mathematics
Mathematics and Plausible Reasoning Vol 1 & 2, Polya ( https://www.amazon.com/Mathematics-Plausible-Reasoning-Induction-Analogy/dp/0691025096 ) does a great job of teaching you how to find/frame good mathematical conjectures that you can then attempt to prove or disprove.
Mathematical Proof, Chartrand ( https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094 ) does a good job of teaching how to prove mathematical conjectures.
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As for really understanding the foundations of modern mathematics, I would start with Concepts of Modern Mathematics by Ian Steward ( https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Books/dp/0486284247 ) . It will help conceptually relate the major branches of modern mathematics and build the motivation and intuition of the ideas behind these branches.
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Abstract algebra and analysis are very fundamental to mathematics. There are books on each that I found gave a good conceptual introduction as well as still provided rigor (sometimes at the expense of full coverage of the topics). They are:
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A Book of Abstract Algebra, Pinter ( https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178 )
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Understanding Analysis, Abbott ( https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ).
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If you read through these books in the order listed here, it might provide you with that level of understanding of mathematics you talked about.
Your question is pretty vague because studying "mathematics" could mean a lot of things. And yes, your observation is correct: "There are a lot of Mathematical problems which are extremely difficult". In fact, that's true for a lot of people as well. So I suggest that you choose a certain field and delve into that.
For proof based subjects, the most basic to start with is Real Analysis. I recommend Stephen Abbott's Understanding Analysis as it is a pretty well-explained book.
This free pdf book should help you: Proof, Logic, and Conjecture - The Mathematician's Toolbox
It's really well written (I like it better than Velleman's How to Prove It.) After this you should go through something easier than Rudin, like Spivak Calculus. Then you can try a real analysis book, but try using Abbott or Pugh instead; I hear those books are much better than Rudin.
I found 'Understanding Analysis' by Stephen Abbott ( https://www.amazon.co.uk/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ) to be super helpful/enlightening post Real Analysis insofar that it helped me build an intuition and understanding for some of the key ideas. Earlier today someone highly recommended this book as well: 'A Story of Real Analysis'
http://textbooks.opensuny.org/how-we-got-from-there-to-here-a-story-of-real-analysis/ (download link on the right). I had a quick glance through it and it seems pretty good.