I'm a physics major (undergrad) who wants to learn real and complex analysis, but don't have the time to do the courses in my programme. Can anyone recommend a good text book for learning the subjects on your own?
Arthur Mattuck's Introduction to Analysis is great and perfect for self-study (I read it on my own). It combines the formality of a regular textbook with informal notes to the reader explaining the stuff that trips up most students. He's an awesome lecturer and his style carries through to the book.
I'm currently reading Tristan Needham's Visual Complex Analysis and it's just outstanding. I've seen some reviews say "This is great, but use it as a secondary book, not as a primary one"; I think that mostly means that you might not have the rigor to ace your Complex Analysis final if you get all your complex analysis from this book. But it's totally self-contained, very clear, and beautiful. I bet there's no better way to really get a feeling for what the subject is all about.
It is hard, but a great book is "foundations of modern analysis" by jean dieudonne. mainly for reals. it covers metric spaces, banach and hilbert spaces, real calculus in finite and infinite dimensions, complex analysis, and soime differential equations and sturm liouville theory. this is a great book for reference if not for self study. for years i noticed almost every tricky question i tried to find a proof of occurs as an exercise in this book with hints. there is even a proof of the jordan curve theorem. but no lebesgue integration.
rudin is notoriously difficult to learn from but is the favorite of professional analysts. i do not know if any of them learned from it, but they all seem to like to teach from it.
mattuck is a terrific teacher, and i think his book is a lot more elementary than rudin.
i think i have never seen a bad complex analysis book. my favorite is by cartan.
i agree that dieuodenne is great . aso try lang analysis i and Ii easy to readand comphrensive goldberg book method of real abnalysis bartle and sherbat intro to real analysis bartke (elements of real analysis are all excellent.
but dieuodene lang and cartan all suffer ias they use regulated functions. they failed to acknowledge henstock integral.