Understanding Analysis
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- Type：Epub+TxT+PDF+Mobi
- Create Date：2021-05-10 10:54:02
- Update Date：2025-09-06
- Status：finish
- Author：Stephen Abbott
- ISBN：1493927116
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Summary
This book outlines an elementary， one-semester course that exposes students to both the process of rigor， and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable。 The aim of a course in real analysis should be to challenge and improve mathematical intuition rather than to verify it。 The philosophy of this book is to focus attention on questions which give analysis its inherent fascination。
This new edition is extensively revised and updated with a refocused layout。 In addition to the inclusion of extra exercises， the quality and focus of the exercises in this book has improved， which will help motivate the reader。 New features include a discussion of infinite products， and expanded sections on metric spaces， the Baire category theorem， multi-variable functions， and the Gamma function。
Reviews from the first edition:
"This is a dangerous book。 Understanding Analysis is so well-written and the development of the theory so well-motivated that exposing students to it could well lead them to expect such excellence in all their textbooks。 。。。 Understanding Analysis is perfectly titled; if your students read it that s what s going to happen。 This terrific book will become the text of choice for the single-variable introductory analysis course; take a look at it next time you re preparing that class。"
-Steve Kennedy， The Mathematical Association of America， 2001
"Each chapter begins with a discussion section and ends with an epilogue。 The discussion serves to motivate the content of the chapter while the epilogue points tantalisingly to more advanced topics。 。。。 I wish I had written this book！ The development of the subject follows the tried-and-true path， but the presentation is engaging and challenging。 Abbott focuses attention immediately on the topics which make analysis fascinating 。。。 and makes them accessible to an inexperienced audience。"
-Scott Sciffer， The Australian Mathematical Society Gazette， 29:3， 2002"
This new edition is extensively revised and updated with a refocused layout。 In addition to the inclusion of extra exercises， the quality and focus of the exercises in this book has improved， which will help motivate the reader。 New features include a discussion of infinite products， and expanded sections on metric spaces， the Baire category theorem， multi-variable functions， and the Gamma function。
Reviews from the first edition:
"This is a dangerous book。 Understanding Analysis is so well-written and the development of the theory so well-motivated that exposing students to it could well lead them to expect such excellence in all their textbooks。 。。。 Understanding Analysis is perfectly titled; if your students read it that s what s going to happen。 This terrific book will become the text of choice for the single-variable introductory analysis course; take a look at it next time you re preparing that class。"
-Steve Kennedy， The Mathematical Association of America， 2001
"Each chapter begins with a discussion section and ends with an epilogue。 The discussion serves to motivate the content of the chapter while the epilogue points tantalisingly to more advanced topics。 。。。 I wish I had written this book！ The development of the subject follows the tried-and-true path， but the presentation is engaging and challenging。 Abbott focuses attention immediately on the topics which make analysis fascinating 。。。 and makes them accessible to an inexperienced audience。"
-Scott Sciffer， The Australian Mathematical Society Gazette， 29:3， 2002"
Reviews
Kevin Doran
，
Brilliant。 Covers the essence of concepts that are easy to miss in other texts。Update 1: it's really amazing。 I highly recommend reading it even if you have read other analysis texts。 I feel like a lot of other books make you work hard but don't reveal the fundamental insights at play。 If you have done the hard work from other texts， please have an amazing experience of skimming though this book。 Brilliant。 Covers the essence of concepts that are easy to miss in other texts。Update 1: it's really amazing。 I highly recommend reading it even if you have read other analysis texts。 I feel like a lot of other books make you work hard but don't reveal the fundamental insights at play。 If you have done the hard work from other texts， please have an amazing experience of skimming though this book。 。。。more
Rafał Grochala
，
Quite clear， approachable， nice to follow。 Filled in a lot of gaps after Tao's Analysis。 Quite clear， approachable， nice to follow。 Filled in a lot of gaps after Tao's Analysis。 。。。more
Thor
，
I wish he covered the transition into higher analysis better。 If you have a decent point set topology background this book mostly covers just that。 Finding a book that transitions， more strongly， into Lp Space and Fourier Series would be great。 So， for example， you can't easily read this book and then jump into Rudin's Real and Complex Analysis。Otherwise， this is a 5 star book and very readable。 I wish he covered the transition into higher analysis better。 If you have a decent point set topology background this book mostly covers just that。 Finding a book that transitions， more strongly， into Lp Space and Fourier Series would be great。 So， for example， you can't easily read this book and then jump into Rudin's Real and Complex Analysis。Otherwise， this is a 5 star book and very readable。 。。。more
James Povilonis
，
Easily the best math textbook I've ever used。 One major frustration I have with university math texts that challenge their readers to think deeper is that the student's work can often feel useless or misguided。 The writing can be unfocused or too presumptuous of what the the student can do。 Yet， a balance must be struck against the other extreme: so simple and repetitive that a student begins to think of the concepts as a collection of algorithms to complete。This book achieves a perfect balance Easily the best math textbook I've ever used。 One major frustration I have with university math texts that challenge their readers to think deeper is that the student's work can often feel useless or misguided。 The writing can be unfocused or too presumptuous of what the the student can do。 Yet， a balance must be struck against the other extreme: so simple and repetitive that a student begins to think of the concepts as a collection of algorithms to complete。This book achieves a perfect balance of asking the reader to think deeper and persistently， gradually guiding them along。 This book should be a gold standard of how to begin each topic （and the entire book） with simple concepts and problems， and grow each with complexity and difficulty until the student reaches such great heights of mathematical thinking and capability。 Yet all of this happens without the reader dreading its difficulty。 Material is presented so clearly， intuitively， and lightly。 I highly recommend this book to anyone who wants the clearest picture of introductory analysis or an understanding of why calculus works so well。 。。。more
Mahin
，
For clarification: Putting it down right now， not because it is bad。 I have plan to come towards it later。
Owen Jepps
，
I like this book very much as an introduction to analysis， because it motivates the concepts much more strongly than most books from the traditional canon。 I use this book for a reading course in analysis: the chapters are well structured with a hook to motivate the content， and a nice summary at the end。 I agree with another reviewer's comment that the treatment of the real numbers is a little confusing due to the order， and one student noted that with so many parts of proofs left as exercises I like this book very much as an introduction to analysis， because it motivates the concepts much more strongly than most books from the traditional canon。 I use this book for a reading course in analysis: the chapters are well structured with a hook to motivate the content， and a nice summary at the end。 I agree with another reviewer's comment that the treatment of the real numbers is a little confusing due to the order， and one student noted that with so many parts of proofs left as exercises （a great idea in principle）， readers have to suspend disbelief if they are unable to fill in those blanks。 But overall a great book。， motivating some important concepts in modern analysis 。。。more
Darian
，
Great introductory book in analysis。 Presents the material in a straightforward manner accessible to those with introductory calculus， and the many stories， backgrounds， and explorations the author offers help ground understanding。 Just an all-around great introductory math text for building conceptual understanding of analysis as a first look。
Max Wang
，
Beautifully written。
Davi Barreira
，
This is the best book I found for learning analysis the first time， specially for self-study！ The exercises evolve in difficulty， which gives you confidence that you are actually learning the subject （because you are able to do the easy exercises as you read the chapter）。 Also， the examples and the writing are just great！ It felt like a novel！ I just wish I could find other textbooks with the same “spirit”。
Robbie Carlton
，
This is the best presentation on Analysis I've read。He does a really great job of motivating the ideas， and gives really interesting and unusual examples， not just the normal basic examples。Lot's of exercises for each chapter ranging from simple to pretty involved proofs （no solutions）。Above all that it's just well written and a pleasure to read。 This is the best presentation on Analysis I've read。He does a really great job of motivating the ideas， and gives really interesting and unusual examples， not just the normal basic examples。Lot's of exercises for each chapter ranging from simple to pretty involved proofs （no solutions）。Above all that it's just well written and a pleasure to read。 。。。more
Tim Wetzel
，
Honestly the thing that pisses me off most about math books is when they're disorganized and written for the author， rather than for the students。 This one has all the definitions， theorems， proofs， and problems arranged in a really digestible way。 I really struggled with the concepts in this class （fuck series， am I right） but the book made it possible to teach myself a lot of the harder concepts， which is a tough feat especially for a proofs class。 This is one of the notoriously hard classes a Honestly the thing that pisses me off most about math books is when they're disorganized and written for the author， rather than for the students。 This one has all the definitions， theorems， proofs， and problems arranged in a really digestible way。 I really struggled with the concepts in this class （fuck series， am I right） but the book made it possible to teach myself a lot of the harder concepts， which is a tough feat especially for a proofs class。 This is one of the notoriously hard classes at Cal Poly， but this book made it a lot more palatable。 。。。more
Randall
，
It's a fine book， but the second edition is much better in that it has several additional sections which cover the Lebesgue integral and some other very fine topics that the first edition doesn't cover at all。 Not that there isn't plenty for a first-semester course in analysis in the 1st but the 2nd is just much better for that very reason。 It's a fine book， but the second edition is much better in that it has several additional sections which cover the Lebesgue integral and some other very fine topics that the first edition doesn't cover at all。 Not that there isn't plenty for a first-semester course in analysis in the 1st but the 2nd is just much better for that very reason。 。。。more
Ayush Bhat
，
Simple and great book to get started with real analysis。
Joie
，
skimmed， imo， good for getting an idea of what analysis is and its general "flavor"， however for me it was not rigorous for self studying because descriptions focused on being intuitive rather than precise。 I will be using the tao book along with baby rudin instead :| skimmed， imo， good for getting an idea of what analysis is and its general "flavor"， however for me it was not rigorous for self studying because descriptions focused on being intuitive rather than precise。 I will be using the tao book along with baby rudin instead :| 。。。more
Erickson
，
Very nice introduction to analysis with good background and motivation of why we are interested in various theorems or problems。 Even the chapter on Fourier series was excellent。 Naturally， this book doesn't go too far so more difficult analysis must search elsewhere。 Very nice introduction to analysis with good background and motivation of why we are interested in various theorems or problems。 Even the chapter on Fourier series was excellent。 Naturally， this book doesn't go too far so more difficult analysis must search elsewhere。 。。。more
Richard
，
This is a fantastic introduction to real analysis。 All the concepts are explained and motivated very well。 It is great for self study， and requires only as much as the calculus sequence and a previous exposure to proofs。 This is a relatively easy book， though some of the exercises are really good。 Do them all if you can。 My only gripes with this book is that it could at least attempt to be more ambitious in its complexity。 For example， it describes the properties of the real numbers in the first This is a fantastic introduction to real analysis。 All the concepts are explained and motivated very well。 It is great for self study， and requires only as much as the calculus sequence and a previous exposure to proofs。 This is a relatively easy book， though some of the exercises are really good。 Do them all if you can。 My only gripes with this book is that it could at least attempt to be more ambitious in its complexity。 For example， it describes the properties of the real numbers in the first few chapters really only implicitly， and doesn't mention fields。 I was confused by this， as fields aren't all that difficult to understand as a general structure at this level （especially if one has， say， seen vector spaces in a linear algebra course）。 Ultimately， I would recommend this book to someone who is struggling with analysis， though you could certainly argue that this book isn't really necessary if you are studying from a more advanced book， with a stronger mathematical background。 That is really my ultimate recommendation， as I think that students coming into real analysis should already have a fair amount of mathematical maturity。 If that isn't you， then go for this one。 。。。more
Kiên Trần
，
concise， crystal clear， complete core of calculus/real analysis。
Trip
，
THis book is ok
Arzu Erdem
，
Easy way to understand analysis
Nina
，
This book was intense pain and utter pleasure for a class I worked extremely hard on - ultimately my favorite class。 A great reference for me as I learned the proofs behind why Calculus works。
Reference
google book https://www.google.com/search?tbm=bks&q=Understanding Analysis
hathitrust digital library https://babel.hathitrust.org/cgi/ls?q1=Understanding Analysis&field1=ocr&a=srchls&ft=ft&lmt=ft