Which Mathematical Analysis I Book or Textbook Is The Best?

Question

I'm in search of a mathematical analysis text that covers at least the same material as Walter Rudin's Principles of ... but does so in much more detail, without relegating the important results to the exercises, contrary to what Rudin does. Which one is it, if any?

Do the mathematics students at places like the MIT, Harvard, or UC Berkeley, where Rudin is used, cover this textbook fully, solving each and every problem? If not, then how much of it is taught and in what detail? Is there any university where this book is covered fully in their analysis courses?

Can I access any video lectures based on Rudin?

Is there any TV channel dedicated to higher level mathematics?

Answer 1

My class is using Intro to Real by Bartle and Sherbert. My previous class (9 years ago) used Introductory Real Analysis by Dangello and Seyfried, which I prefert to my current text. Neither one covers everything in what I would consider "great detail".

Answer 2

I would recommend Bartle's "The Elements of Real Analysis".

Answer 3

Not sure, but "Introductory Real Analysis" by Kolmogorov & Fomin (translation by RA Silverman, publ Dover) is rigorous and extensive and not expensive.

Answer 4

Check the many lecture notes available on the net, e.g. William Chen's. The Trillia Group has textbooks available for free too. Check the AMS for suggestions. MIT has lots of stuff on OCW, and there is now Coursera.

Answer 5

Tao's Analysis I is my favorite. It is very reader friendly and eloquently written.

Answer 6

I went to Berkeley and the real anaylsis class used Elementary Analysis: The Theory of Calculus by Ross. It is a bit simpler than Rudin but much more readable. We did pretty much everything.