How to Think About Analysis

by Lara Alcock

Analysis (sometimes called Real Analysis or Advanced Calculus) is a core subject in most undergraduate mathematics degrees. It is elegant, clever and rewarding to learn, but it is hard. Even the best students find it challenging, and those who are unprepared often find it incomprehensible at first. This book aims to ensure that no student need be unprepared. It is not like other Analysis books. It is not a textbook containing standard content. Rather, it is designed to be read before arriving at university and/or before starting an Analysis course, or as a companion text once a course is begun. It provides a friendly and readable introduction to the subject by building on the students existing understanding of six key topics: sequences, series, continuity, differentiability, integrability and the real numbers. It explains how mathematicians develop and use sophisticated formal versions of these ideas, and provides a detailed introduction to the central definitions, theorems and proofs, pointing out typical areas of difficulty and confusion and explaining how to overcome these. The book also provides study advice focused on the skills that students need if they are to build on this introduction and learn successfully in their own Analysis courses: it explains how to understand definitions, theorems and proofs by relating them to examples and diagrams, how to think productively about proofs, and how theories are taught in lectures and books on advanced mathematics. It also offers practical guidance on strategies for effective study planning. The advice throughout is research-based and is presented in an engaging style that will be accessible to students who are new to advanced abstract mathematics.

Genres: Mathematics, Nonfiction, Science, Academic, Reference

274 pages, Kindle Edition

First published January 1, 2014

Reviews

[Valéria]

Why am I, an Astrophysics student, taking pure mathematics modules? Well dear reader there is only one answer, academic inflicted masochism.

[Burhan Malik · Rating 4 out of 5]

Great book for students of mathematics who aren't accustomed to mathematics past the high school level and the rigour of analysis, this book is NOT an analysis book but it will provide you with the needed tools to be able to start reading more advanced mathematics. Alcock is a professor of mathematics whose research is geared towards teaching and uses that research to make pure math more accessible to everyone.

[Alexander Snitsarenko · Rating 5 out of 5]

I have read her two main books, including this one. Her math and writing style are perfect. This book provides some practical insights on how to study advanced math, in particular the last few pages would be very helpful:). For me, even though it is quite banal, her advice on always writing my own denifition list in any math course was a turning point in substantially improving a material understanding. Apart from this, this book is a math book with many proofs quite deeply explained. Topics of sequences, series, continuity, and reals are covered well. In sum, there are very few such math books out there that, on the one hand, are a real math book demanding a good math level, but at the same time, are written by an author that wants to make every theorem or math fact in a book intuitively understandable as well. Sometimes she succeedes

[Rodrigo d'Orey · Rating 5 out of 5]

This is fantastic little book that will help you get rid of that instinctive fear of Real Analysis (those epsilons, deltas and countless mathematical symbols). The book is short but very engaging with very good diagrams and explanations. Analysis is hard but it is also elegant, clever and very rewarding. This book goes a long way to easy your path in its study ("pointing out typical areas of difficulty and confusion and explaining how to overcome these").

Topics covered: sequences, series, continuity, differentiability, integrability and the real numbers.

[Marc Watt · Rating 5 out of 5]

I have plenty of time on the train each morn and aft and I have been using this time to revisit analysis. It doesn't elicit the terror it did as an undergrad in fact I really enjoyed reading this and going through the proofs she cites. Fun Stuff... And before you know it. You're at work with your brain humming.

[Syed Nouman Hasany · Rating 5 out of 5]

Lovely read. Motivated me enough that I might, someday, go down the path of formally pursuing Analysis. In general also it's a great book to improve mathematical literacy. Gives a neat birds eye view of Analysis, makes you comfortable into reading mathematical statements and proofs, and also presents excellent tips on how to structure your study for this as well as other mathematical domains.

[Mohammad Tanviruzzaman · Rating 5 out of 5]

I never felt easy with the limiting process involved in Calculus. To remedy that I wanted to teach myself Analysis. However I got intimidated by the sheer amount of logical symbols in a typical passage in a typical Analysis book.

I found this book to be a good friend. I could follow it well. It gave me ammunition and a mindset to tackle further Analysis.

[Klaire Hoang · Rating 5 out of 5]

8.2025 - I have to reread this masterpiece


Best book on Real Analysis. Seriously!!! Universities need to adopt this book as their textbook, please. At least replace recommending Introduction to Real Analysis by Bartle, Robert G. to this

[Ivan Koma · Rating 5 out of 5]

I really get it (and i'm not math student at all). This book can be example for all other math study books. Easy examples, easy explanations and best introduction into theme (even some humor), this things it's a pillars of good book for me
If you think it's hard:
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∀a, b ∈ R, a + b = b + a;
∃ 0 ∈ R s.t. ∀a ∈ R, a + 0 = a = 0 + a.
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or
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f : X → R is bounded above on X if and only if
∃ M ∈ R s.t. ∀x ∈ X, f(x) ≤ M.
-
You just don't readed second chapter of this book, where you can have all definitions of RA things

[Margaret Cathcart · Rating 4 out of 5]

I read this to prepare for grad level analysis and refresh my knowledge but I am HEAVILY wishing I had read this last year to prepare for my undergraduate course! The author is super knowledgeable and gives tips on how to transition into the class. I really like all of the diagrams and the conversational style ( something that's often missing from Analysis materials ). She also talks about why students might misunderstand or misinterpret concepts and why! I will definitely recommend this book to any undergrads anxious about taking analysis.

[Angelino Desmet · Rating 4 out of 5]

I only read the first part for practical reasons (for now), and I'm blown away. This book is a must-read for anyone embarking upon a challenging study. Even beyond mathematics this book is relevant because it teaches you how to tackle the course material and inoculates you from your brain's self-sabotaging tendencies. Looking forward to part two.

[Henry · Rating 4 out of 5]

I have feared analysis since studying it at first-year undergraduate level when I didn't really get it at all. This book makes it seem easy, and even enjoyable. A great way to prepare to study this topic.

[David]

Gentle introduction to analysis with valuable advice for almost any reader.

[Joshua · Rating 4 out of 5]

Good intro but needs more depth or examples of how to write proofs.

[Malcolm Gardner · Rating 5 out of 5]

Very readable and motivating

[Charlie Moll]

A useful resource

[Rhys Tilford · Rating 5 out of 5]

As with all nightmares this one had an ending.

[Ole · Rating 5 out of 5]

I wish I had this book when I studied Analysis in university!
Lara explains concepts and ideas so well, it's a pleasure read and follow along.

[Simona · Rating 4 out of 5]

Maybe I got too excited after my Analysis 3 course, I overestimated the level of this book it is actually basic first year university material. I skimmed it and I think it written was very clearly and nicely.

[Elizabeth · Rating 4 out of 5]

This book was extremely helpful to me while I was taking a Graduate level Analysis. It served as an extra resource I could go to in order to better understand the material. I only gave four stars because I found the book lacking in some of the concepts I struggled with like the theorems involving open, closed, compact, etc. It is possible I struggle with them because they were thrown in before the midterm, but it still would have been helpful to have information in the book about them. I do understand that not everything can be covered in books. At the same time, my course did not cover sequences and this book had an entire chapter on them. I was most intrigued about the sequences chapter and tried to read it anyway, but I ended up just skimming it because I was already finished with my class and wanted to move on to other reading. I hope to delve into the sequences chapter further at some points. When I do, I will update this review to reflect that. An overall very good book to give some good foundation to my Analysis grad class.

[Timothy Corbett-Clark · Rating 5 out of 5]

Quite superb. Compulsory reading for any nascent maths undergraduates.