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Author : A. The Chicago undergraduate mathematics bibliography is a nice annotated list of books. Ditto for a few more things I've seen here. It would be great for those that could, but we're not going to ditch half our students. Needham, Visual Complex Analysis.
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I read this while in high school, and it's simply beautiful. I recommend this book as a supplement to any first course in complex analysis a different book should probably be used for the main textbook since Needham's is very pretty, very engaging, but not very rigorous. The Princeton Lectures in Analysis by Stein and Shakarchi are great introductions to Fourier, complex, and real analysis in that order! Real and Complex Analysis by W. Rudin is a beatiful and extremely well written book which presents the fundamentals of real and complex analysis highlighting the interactions between different results and ideas.
There is a nice geometrical philosophy and plenty of motivation. I didn't see any suggested books from the great Russian school of mathematics, here is a brief list of superb, well written, example oriented books:. I have also found a Spanish translation of a book written by S. Banach about "Differential and integral calculus".
It is very good as an undergraduate book. The spanish translation for those who want to search is Calculo diferencial e integral. Books that undergraduate should not touch in my humble opinion are books written in bourbaki's style. In spite of being part of "Graduate Texts in Mathematics" series and unlike Rudin's Real and Complex Analysis see a comment above , this is a book at the undergraduate level. It only presupposes undergraduate algebra as in Herstein Topics in Algebra or M.
Artin's Algebra , undergraduate analysis like in Rudin's Principles of Mathematical Analysis and basic number theory. In fact it recalls or proves many of the necessary results in each of those fields. As an undergraduate, I loved Shafarevich's book Basic notions of algebra. This is not a textbook, but gives small beautiful tastes of a broad choice of topics in algebra, emphasizing connections with other fields. I found it very stimulating, in the sense that every example or overview of some topic in this book made me want to learn more details about it. In fact I became interested in algebraic geometry because of this book.
Dummit and Foote's Abstract Algebra is an excellent book for learning group theory, ring theory, and module theory. There's also a section on basic algebraic geometry and homological algebra. Topics in Algebra by I. A new edition will be coming out this year. It consists of 38 in my edition chapters that give often largely self-contained introductions to various areas of the field. Although it doesn't go nearly as in depth as, say, Stanley's "Enumerative Combinatorics" or a text focused solely on graph theory, I found it excellent for giving a broad overview and indicating to me where I wanted to explore deeper.
My one caveat would be that some chapters require background in either linear algebra or basic group theory, though those are easily skippable due to the structure of the book. Not so much a textbook as a collection of essays in particular, it doesn't have exercises , but all of the essays are instructive and enlightening. Here is a list of titles.
Introductory Real Analysis (Dover Books on Mathematics)
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Active today. Viewed 70k times. I think its better as a community wiki though. Do I need to do something to make it a community wiki or is it already like that? But moderators have the power to convert a post to community wiki, which is what David has done here actually, three moderators independently thought it should be converted to CW As it's been popping back to the front page fairly frequently, we've decided to close it. Harrison Brown. The third edition doesn't differ that much from the second one, though.
David E Speyer. Gerald Edgar. That said, when I first read this book I loved it so much that it made me want to be a mathematician.
Introductory Real Analysis
It's both rigorous AND intuitive, in a way that both qualities complement one another. I have the hardcover and it looks like the one you link to. Apparently Artin is working on a new edition. But you can't help but love the infectious passion with which Artin weaves his craft in front of the students. He loves algebra and he's trying to prosyletize his students to it. A book with a similar geometric bent,level and also by a master that students will probably find easier going is E. But Artin's book is very good and it's good news for all of us that Artin is revising it.
Dmitri Pavlov. It's a matter of taste, of course. Milnor does much less material that Guillemin and Pollack, but reading it was an amazing experience for me. Guillemin and Pollack is a very good book, but I never got nearly as much from it. This book is awful as an undergraduate text! It's a great reference for someone who already knows the material, but the proofs skip many "simple" steps, and the author makes no attempt to explain the concepts from a intuitive point of view!
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Our professor assigned this book for the undergraduate course in Topology at SUNY Stony Brook, and at the time it was of absolutely no help to me whatsoever. Michael Lugo. Even if the list itself is somewhat obsolete, one gets a good feel of relative strengths and weaknesses of "canonical texts" ca Clemens Koppensteiner. Max M.