Understanding Analysis (Undergraduate Texts in Mathematics) by Stephen Abbott

by Stephen Abbott

Genres: Mathematics

Unknown Binding

Reviews

[Anna (ani) 🕷 · Rating 4 out of 5]

Did not read the last chapter (and it was probably the most interesting one, rip) but I can confidently say I’ve gotten through enough of this to mark it as read. I don’t think Goodreads is going to count it as an actual book I read, which is unfortunate because I have certainly read this more than any other book this year.

I do think this is a great intro to analysis. I learned a lot! Did I want to bang my head against the wall sometimes? Yes. Do I think numbers are cool and strange and bizarre and beautiful? Also yes. The Cantor set goes kinda crazy ngl.

[tmdubs · Rating 5 out of 5]

made me cry

[Pat · Rating 5 out of 5]

my professor based his entire course on this book and i found that it indeed helped me understand analysis 😀 !! i appreciate that the author chose to include details for readability instead of omitting them for style, because i came in with a bit of a weaker background and less confidence. it made the book approachable, even though some of the exercises were tough (good lord 🤣). my class skipped the discussion and project sections, and i will probably update after i go over them. my favorite chapter was chapter 6, on sequences and series of functions, but i also probably understood that one the least 💀 so it goes to show that this book actually made analysis interesting!

[Necot · Rating 4 out of 5]

An usual introductory course of mathematical analysis aims at providing the results of infinitesimal calculus in a rigorous way, with the theory of differentiation and integration of functions of a real variable being the usual final goals.

In order to reach this goal, a lot of ground has to be covered and along the way numerous results need to be enunciated and proved.

Abbott choosed a different approach in which coverage of the material is sacrificed in order to treat some selected topics with more depth. This, together with numerous counter examples helps understanding the particularities of the subject and the reason why so many definitions and theorems are necessary.

For example, the author illustrates the theory of Riemann integration by starting with continuous functions, then gives examples of functions with discontinuities that are Riemann-integrable, asks if also functions with infinite points of discontinuity can be integrated, shows some examples and finally proves why this is possible. He then mentions Lebesgue integration theory and that some functions that are not Riemann-integrable are Lebesgue-integrable.
In Chapter 8 he finally shows an extensions of Riemann integration that has some more advantageous properties than Lebesgue's theory.

This method of illustrating the results of mathematical analysis does indeed help understand the details of the subject, which makes this a very good book for accompanying a course in real analysis.

[Thea · Rating 5 out of 5]

Best book I’ve been lucky enough to lay my hands on in the past like 5 years.
The book explains related topics without immediately overcomplicating, and helps guide students every step along the way towards understanding.

If you’re studying on your own this might get a bit easy after a little while, and some things may need additional reading on the side as the book doesn’t cover all necessary fundamentals within real analysis. If your goal is to self study what would correspond to a course in analysis I’d suggest reading up on things like the Cantor set, convergence tests, and more/compare to baby Rudin (”standard”/common course book) to see what you might’ve missed.

But! this book definitely will help you build some solid ground on the topic, and it will not abandon you if something gets confusing or complicated. Every exercise feels very possible to do on your own, and I never felt like I was being lead astray throughout the reading. If studied properly, you will leave this book with a well developed confidence and understanding of the topic. I almost want to recommend even non-math people read this because it’s that good. :)

I love love love this!

Well done Abbott!!

[Victor]

"It is the pathologies that give rise to the need for rigor. A satisfying resolution to the questions raised will require that we be absolutely precise about what
we mean as we manipulate these infinite objects."

[Laurel! · Rating 3 out of 5]

middling understanding of analysis after reading. I feel as if in an analysis textbook you can never have too much explanation. pedantic is expected for learning! perhaps this book erred in the side of too little. really cool subject though!