A Math book with an inspiring ethos?

Question

I was for some time curious about William Feller's probability tract (first volume); luckily, I could lay my hands on it recently and I find it of super qualities. It provides a complete exposition of elementary(no measures) probability. The book is rigorous "hard" math but doesn't escape from giving a solid intuitive feeling. The author discusses a topic, mentions an example, proposes different scenarios that gives back more math. His first chapter on "nature of probability" is essential. It gives a good feeling for what statistical probability means, and why/how it was defined as it is.

Question: I'm looking for other math books on fundamental mathematics(algebra, real analysis, etc...)- essential mathematics that is not very advanced(algebraic geometry for example) - of high qualities like Feller's probability text. Feller might not be used anymore, but its full of exercises that would make it a working textbook written by a master.

To be specific and not too general. I'm looking exactly for inspiring Feller style books in real analysis and abstract algebra. Rudin is good, but its not a master book. I don't know much about abstract algebra available textbooks/master expositions.

Answer 1

Here is a link to a free down-load of virtually verbatim lecture notes for a real analysis course taught by Fields Medal winner Vaughan Jones. They were my first introduction to real math - beautiful presentation, lots of motivation:

https://sites.google.com/site/math104sp2011/lecture-notes

Another nice book on real analysis is Pugh's "Real Math. Analysis." An unsung hero, again lots of motivation, excellent pictures for a real feel, and plenty of examples.

In addition to Paul Garrett's excellent notes, here is a link to great material by Keith Conrad:

http://www.math.uconn.edu/~kconrad/blurbs/

What I especially like here aside from the great presentation is the constant pointing out of anticipated misconceptions and many, many examples looking at the topic from many sides. They are relatively short and cover a wide range of primarily algebraic topics at many levels.

Answer 2

I don't know if this is exactly what you mean but the book visual group theory is a great way to develop intuition in abstract algebra.

Another great book is adventures in group theory where they use mathematical toys to give an insight of group theory

Finally, for a serious text I would recommend Paolo Aluffi

Answer 3

I highly recommend Artin's Algebra. In my opinion, it fits your criteria nicely; topic introduction, concrete example, and then thought provoking discussions that pique your interest. The exercises are fantastic.

If Artin is not advanced enough, then I second Jorge's recommendation for Paolo Aluffi.