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.

[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."