Top 4 Mathematical Analysis Books

In this video I will show you 4 mathematical analysis books. These are books you can use to learn real analysis on your own via self-study. You can also use these books to supplement any course you are taking on Advanced Calculus or Real Analysis at the undergraduate level. Knowing how to write proofs is very important before jumping into analysis, but these are always fun to look at even if you aren't an expert at proof writing. I hope you enjoy this video.

Here are the books.
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[devencapritta]

It’s so interesting looking at more complex math books. I’m barely beginning my mathematical journey with basic algebra and I’m super excited to get into higher level maths in the coming years.

[cybervigilante]

Who else was waiting for the easiest? 😜

[javiermd]

I would only recommend Baby Rudin as an intro to Real Analysis to someone that was my worst enemy. I think there are better (and more modern) alternatives. For example, Tao’s both volumes, Charles Pugh’s Real Mathematical analysis, or if you want to study graduate level real analysis, Sheldon Axler recently published a book on Measure, Integration and Real Analysis pitched at that level. All of them are way more pedagogical than Baby Rudin, they’re written in Anti-Bourbaki style, ie, they’re written to be understood first-hand.

[code_explorations]

The Jay Cummings long-form textbook on Real Analysis would have been a great inclusion.

[willyh.r.]

My favorite goes always to Analysis book. In french system math major, if you choose "pure" math, you have to take Real Analysis (Analysys 4), Differential Geometry, and Abstract Algebra.
Calculus is mandatory in 2 first university years.
Your great math books remind me my university math battle and joy: doing math everyday ... many hours a day.

[lukeatlas]

Consider reviewing

2-Volume Mathematical Analysis, by Zorich
3-Volume Analysis, by Herbert Amann

And for the serious mathematicians:
9-Volume Treatise on Analysis by Jean Dieudonné

[AndersBjornTH]

A book I found very readable is, A First Course in Mathematical Analysis by David Brannan, (CUP 2006).

[tristanfoss]

Hey, Math Sorcerer, have you ever heard of/considered doing a video on the Mathematics for Self-Study books by J.E. Thompson? Record has it that they're what Richard Feynman used to learn math and they were part of the reason for his nearly perfect math grades.

[nickknight]

I'm currently (after a gap) reviewing Real Analysis, primarily using Ethan Bloch's text. I have some issues with it, especially compared to Pugh's very friendly and breezy presentation: Bloch takes slow and steady to an extreme, and has a notion of simplicity which entails using some very inefficient proof methods; it's also 1D only. On the other hand, it has a clarity and a breadth and depth of perspective that I haven't seen in other texts, especially with regard to building up the real numbers, but also throughout the text. It also has a particularly strong didactic quality arising from its target audiences, which are both undergraduates but also secondary school teachers looking to solidify their foundations. 

Another good looking text is the introduction by Bruckner and Thomson - I haven't studied from it, but it's free and gives the impression of a very thorough and thoughtful presentation. They also have a graduate text which looks fabulous, but I'm not ready for it yet.

[alexanderkowalewski]

I graduated with my bachelor's degree in mathematics on Saturday which I'm pretty excited about. I plan to attend graduate school for mathematics during the fall semester of 2023.

[sunglee]

Mathematical analysis by apostol is really clear. I was frustrated by other books but this one I understood everything easily.

[feraudyh]

Here in France we have some fantastic mathematics books, most of which have never been translated.
It helps to know that there are several streams of mathematical education:
1) A two year intensive course as a preparation for the engineering schools ("Grandes Ecoles"). Once you get into one of these schools the schools have their own curriculum. In fact they throw so much at the students it is almost impossible to follow everything.
2) The more "leisurely" "licence" which is a 3 year degree (like a bachelor's). There is a lot of stuff to learn. In particular they teach differential forms in 2nd year.
I'd like to give you a feel for it by a scan and OCR of the table of contents of one of the best series of books for the Licence : By J.P. Marco, L. Lazzarini, H Boualem, R Brouzet, B Elsner, L Kaczmarek and D. Pennequin. I can do that in 2023 when I have access to a scanner and OCR software.
The books have very detailed proofs of most theorems: very little handwaving.
The French system has a particular style: emphasis on rigour and formality and they expect you to master one level before moving onto the rest. The abstract aspect may be going a bit far at times, in that visualisation is discouraged.
If you sweat through all of this, you will be able to prove theorems with a high degree of rigour.
After the licence there is the maitrise where more advanced subjects are dealt with, and I get the impression that you can often start to enter into the word of research, at least for some subjects, but not for stuff like Grothendieck.

[vaheakli]

Have you ever covered Khinchin's "Eight lectures on mathematical analysis"?
I can't remember that (I suppose that because it's very hard to find in paper), so I decided to write about, I think it is really worth the mention.
I recognized it at the 1:28 moment as the very opposite to Rudin. Whole book is dedicated to the simplest possible explanation of essentials and motivation behind the proofs and the way calculus is organized taken as a whole.

In case if anyone reading this haven't heard of it, but already interested in this, the preface says about the book better than me:

"We frequently encounter a situation in which an engineer, teacher, or economist who has at some time studied higher mathematics in a “simplified” course begins to feel the need for a broader and, what is more important, a more solid foundation for his mathematical knowledge. This need, whether it arises out of specific research by the specialist in his own scientific field or comes as an inevitable consequence of the general widening of his scientific and cultural horizons, must of course be satisfied. It might be supposed that the specialist might easily satisfy his need; he could merely take any comprehensive text on mathematical analysis and study it systematically, making use of the rudimentary knowledge he has already acquired. However, experience shows that this method, which seems so natural, almost never leads to the desired goal but instead often brings disillusionment and a consequent paralysis of any further effort. For such a student usually has only limited time at his disposal and therefore cannot undertake to work systematically through a full-length textbook. On the other hand (and this is probably the most important factor) he does not yet have a firm grounding in mathematics, and therefore he cannot, without outside help, pick out the essentials. He will be compelled instead to devote his attention to irrelevant details, and in these he will finally get lost, unable to see the forest for the trees."

"I renounced from the very beginning any idea of presenting even a single topic in full detail: instead, I limited myself to a vivid and concrete presentation of the essential points and spoke more of goals and perspectives, of problems and methods, of the connections of the fundamental notions of analysis with each other and with their applications, than of individual theorems and their proofs."

"This book has the same goal as the course of lectures I have just described, and tries to realize it by the same means. The reader should therefore be warned from the very beginning that he will not find here a complete presentation of a university course in analysis, or even of individual topics selected from such a course. I have set myself the task only of giving a general sketch of the basic ideas, concepts, and methods of mathematical analysis. But I have tried to make this sketch *as simple and as easy to retain as possible*, to make it something that can be read and assimilated by anyone familiar with even the crudest exposition of the subject, and one which, once assimilated, should enable the student to study the details of any part of the subject independently and effectively.
At the same time, I hope that this book may also be of real benefit to many students in the mathematics departments of universities. Neither a text nor a lecturer, limited as they both are by the exigencies of time and the program, can pay enough attention to the discussion of fundamental questions; both are compelled to concentrate on the exposition of all the details of the material they cover. And yet everyone knows how useful it is sometimes to turn one’s eyes away from the trees and look at the forest. I would like to believe that this book will help to reveal that broader view to more than one future mathematician who is studying analysis for the first time."

🥰

[fanalysis]

That second book is exactly what I've been looking for!

[robertpapp]

This is fantastic I especially like the rudin book

[billbez]

Really enjoyed this video. Have you reviewed Advanced Calculus by Avner Friedman. I'd be very interested in your opinion. Thanks

[soumyaj]

Mathematical Analysis by Tom M Apostol is a worthy inclusion. Its not too tough but it definitely prepares one for Analytic Number theory.

[WeilderofMathematics]

The 2nd edition of Abbott's book is a bit thicker with some added content. As far as solution availability, Springer has solutions to the exercises for professors. That's all I know about solution availability.

[sesburg]

Great video! I have been trying to brush up on real analysis that I learned ~ 2 years ago before I start studying complex analysis next semester. I have a paper copy of “Baby Rudin” that I have been reading through, and although I love the rigor and structure of the book it is definitely a hard read. After reading through a chapter, thinking I understood it, I still have no idea how to tackle the exercises. Maybe it’s just me, but I am struggling a bit. I really feel what you said at 1:26. Though I was able to find a compilation of solutions to the book online. I will definitely look into Abbott’s book 8:20.

[Anonymous-qw]

I used Apostol Mathematical Analysis and Dellillo Advanced Calculus with Applications back in the 1980s. I hoped you would look at those.