Name: The Real Analysis Lifesaver
ISBN: 9780691172934

What Hacker News says

VovaPutinDec 2019

> . I agree with your broader point that the structure and presentation of math can substantively change, even if the fundamental content doesn't at the undergraduate and graduate levels.

That was my main point. I wasn't intending to recommend intro analysis book. But now that you mentioned it,

for introduction to real analysis, there are even better textbooks out there. For bare-bones hand-holding stuff, there are "How to Think About Analysis" by Lara Alcock and "Real Analysis" by Jay Cummings. The former spends a lot of pages explaining the concepts of convergence and boundlessness and the latter, in addition to what Alcock does, introduces a bit of topology on the real line. Speaking of topology necessary for analysis, Rafi Grinberg's "Real Analysis Lifesaver" kicks ass. It tackles topology and convergence in the Euclidean Spaces (a leg up on the above-mentioned books). These extremely easy to read books (read in the given order) would easily prepare a dedicated student for tacking modern analysis.

If a student wants to see worked out example after worked out example (in extreme detail), Indian authors have you covered. Such STEM books seem to be a tradition there.

My favorite elementary analysis books by an Indian author are "Real Analysis" and "Sequences and Infinite Series" by N.P. Bali (No idea what the initials stand for; all I know is this author is Indian).

A book that serves up elementary analysis in bite sized portions and gives you only what's unavoidable (not simply necessary) is "Intro to Analysis" by Mikusinskis. It has no fat and no excess whatsoever. It aims to save you time as much as possible and wherever possible. Such style is usually reserved for grad level books.

Out of the three books you recommend, the ones by Tao and Pugh require an instructor nearby. If I recall correctly, Tao starts out by carefully laying down the basics of naturals in a way that's sure to leave most any math novice (especially those interested in analysis) bored and lost in the weeds. Pugh, on the other hand, is roughly at the level of, say, baby Rudin (if a bit less formal and austere) which makes it a VERY tall order for a beginner. At this point, the student might as well read Axler's book as it has more material, is more advanced and Axler is a phenomenal writer.

If a student is taught to fear and avoid anything that is "outside the scope of this class", "graduate level" etc, there's "Lebesgue Integral" by Johnston and "Friendly Approach to Functional Analysis" by Sasane. An average high school student can read these books :)

I own all these books.

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prostitutkaNov 2017

My answer to your question is math. Learn to read and write proofs. Any intro to proofs will do: those employed in discrete math, the ones in analysis, the diagram chasing ones, whatever...Working with math proofs will definitely straighten out your thinking and whip your mind into shape.

Some suggestions to get you started:

Book of Proof by Richard Hammack: https://www.people.vcu.edu/~rhammack/BookOfProof/

Discrete Math by Susanna Epp: https://www.amazon.com/Discrete-Mathematics-Applications-Sus...

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al: https://www.amazon.com/Mathematical-Proofs-Transition-Advanc...

How to Think About Analysis by Lara Alcock: https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0...

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers: https://www.amazon.com/Learning-Reason-Introduction-Logic-Re...

Mathematics: A Discrete Introduction by Edward Scheinerman: https://www.amazon.com/Mathematics-Discrete-Introduction-Edw...

The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Rafi Grinberg: https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Pr...

Linear Algebra: Step by Step by Kuldeep Singh: https://www.amazon.com/Linear-Algebra-Step-Kuldeep-Singh/dp/...

Abstract Algebra: A Student-Friendly Approach by the Dos Reis: https://www.amazon.com/Abstract-Algebra-Student-Friendly-Lau...

That's probably plenty for a start.

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thousandpoundsAug 2017

I am in the same boat. I get the feeling that most Calculus books are just a compilation of tips and tricks. So I am suggesting you invest time into learning real analysis proper. Right now I am learning from [1]. It follows Rudin closely and as opposed to many other analysis books meant to "better explain" stuff, it goes deep into the trenches and actually tackles the subject.

[1] https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Pr...

I think time invested into studying real analysis pays off because then you can later study measure theory, functional analysis and more advanced probability to deal with curse of dimensionality and whatnot.

edit: I started studying the book linked above starting from chapter 4 since the first 3 chapters are familiar from discrete math. Then did chapter 5, skimmed chapters 6(little linear algebra), 7, 8 (most "transition to higher math" books contain this stuff) and am currently in chapter 9.

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mixtureMar 2020

Disclaimer: please, excuse the typos, grammatical and stylistical mistakes below as I am pretty freaking drunk at the moment.

Friendly reminder:

while the material in the linked books might appear trivial to many, the style might be discouraging to those who've never seen "actual math" before. If you're in the latter group, try out the books below (some are not free, but they're still helpful [libge*n might be helpful]):

1. BOOK OF PROOF by Richard Hammack. This FREE book is your passport into mathematics proper. It teaches you how to read math books. After this book you'll stop fearing the word "proof" and start demanding it wherever you go.

Link: https://www.people.vcu.edu/~rhammack/BookOfProof/

2. The book by Don Shimamoto is super trivial if you know linear algebra. The best books in the genre (imo) are (a) Linear Algebra: Step by Step by Kuldeep Singh whose secret is that he uses all the exercises from other LA books as his 100% worked out examples (which seems to be a tradition for Indian authors; I have tons of examples from other branches of math). Indian authors teach you how to drive a car by putting you at the steering wheel instead of lecturing you about it. BTW, if you know math at the level of, say, 7th grader, then you're perfectly suited to study from Singh's book. If that isn't clear, do you know how to add and multiply fractions? If yes, then you're perfectly ready.(b) A modern Intro to Linal by Henry Ricardo. This book is great in its ability to explain certain introductory concepts. For example, the linked Shimamoto book starts out with (going by memory) "the elements of R^n are finite sequences of length n... bla bla bla". Well, Ricardo explains what that means. It also helps if you know what a "sequence" is. But we're getting there. Another author who does a good job of it is Sheldon Axler, I believe. He does have his own idiosyncrasies, but I digress. The only difficulty here is, probably, the idea of diferentiability of multivariable functions.

Links:

https://www.amazon.com/Linear-Algebra-Step-Kuldeep-Singh-ebo...

https://www.amazon.com/Modern-Introduction-Linear-Algebra/dp...

http://sites.msudenver.edu/wp-content/uploads/sites/385/2017...

3. Now to analysis of real kind as opposed to the complex one. If you've never heard anything about this, then you should hook up with Lara Alcock's "How To Think About Analysis". To the best of my recollection, half the book is about how to think about the term "sequences". She goes above and beyond in trying to explain you what that means. The rest of the book is so-so. Once you learn about sequences, you can move on to Jay Cumming's book. It's a bit more involved with proofs and shyt, but still extremely accessible. Laughably so. But note, he only stays grounded on the real line and does not venture into the Euclidean Spaces proper. However, there's another easygoing dude named Raffi Grinberg who'll take you through R^n by hand, no problem. If you master this book, then you're at the level of what's referred to as "Baby Rudin". Next, if you're feeling adventurous, there's a new book by Sheldon Axler that introduces you into the modern analysis which is FREE FOR NOW. GRAB IT WHILE YOU CAN CAUSE I SAW IT ON AMAZON.

Now if you're a business type who ain't got no time for all this foolishness, try "Intro to Analysis: From Number To Integral" by Mikusinski's. It starts with axioms for reals and takes you straight through to Lebesgue Integral in no time. Every section has very few exercises. Say, about 4. It also skips through the traditionalist Riemann integral. Because why not.

Links:

https://www.amazon.com/Think-About-Analysis-Lara-Alcock-eboo...

https://www.amazon.com/Real-Analysis-Long-Form-Mathematics-T...

https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Pr...

https://link.springer.com/content/pdf/10.1007%2F978-3-030-33...

https://www.amazon.com/Introduction-Analysis-Number-Integral...

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Now to a few examples of Indian authors' propensity to give you everything you need to know about the subject unabashedly. Check out:

-- "Sequences and Infinite Series" by N.P. Bali. It's 100% worked out proofs after proofs after proofs. If you read through this book, I imagine you could easily crush any "sequences and series" portion of any elementary real analysis exam in existence. The author also has a book on real analysis which is structured very similarly to the aforementioned book. No idea what N.P stands for, though.

Now to a book you've never heard of. "Data Structures and Algos" by Gav Pai. It easily blows Cormens, Sedgewicks and Skienas out the park by giving you, as you guessed it, all the 100% worked out examples you need to understand the topic.

"Fundamentals of Complex Analysis by Bhat" is another one. All its problems are immediately followed by solutions. Look, it understands you're not at 40K a year school with 10 profs to help you out with any of your questions. Books like this are written for students, not to impress the profs.

I just remembered some other Indian textbooks on abstract algebra, discrete math etc, but I cannot remember their names. They are also all about giving you all the examples you need to grok the concept at hand.

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