Books for a second lecture I'm looking here for references for a second lecture. When you start learning mathematics, there are books that you have to read because there is every basic thing that you should know about this topic, because the course is extremely clear, the examples are well chosen to apply the definitions, and there are basic problems to evaluate how well you understand the theorems. 
Here I'm looking for different kinds of books : The books that you read when you know the theory, that choose a different point of view (not recommended for a beginner). The exercises and problems are here to test the limit of the theory, to expand the theory, or to develop counterexamples that you didn't want to see at first. For example, some people may say that the 4-volume course of Analysis by R. Godement is one of those books.
Do you have some other examples of these kinds of books? (about what a math undergraduate  should know : Algebra, Linear Algebra, Integration, Real Analysis, Probability Theory, Arithmetic, Complex Analysis, Geometry...) 
Until now, I only have suggestions for Algebra, Differential Geometry and Riemann Surfaces (this domain may be to specific, I'm not sure that all undergrads are aware of Riemann Surfaces). 
 A: For algebra I would recommend Eisenbud's  Commutative Algebra: it contains many interesting examples and non trivial applications (as the full title suggests).
I would have also a look at Jacobson's Basic Algebra , second volume: it is very well written (there is also an introduction to homological algebra).
For geometry (differential geometry) I would suggest Kobayashi & Nomizu's great  Foundations of differential geometry. It is technically difficult, but it contains what it has to be known :-).
For complex analysis, I think it strongly depends on applications which are of interest to the reader (even more than for other branches of Mathematics). For Riemann surfaces & complex geoemtry I recommend Farkas & Kra's Riemann Surfaces and Miranda's Algebraic Curves & Riemann Surfaces, as well. 
A: As for Real Analysis, Vladimir Zorich's "Mathematical Analysis" (I and II) is very hard-core in the sense that it is totally proof-based and very comprehensive, and it is very suitable for those people who have once studied Real Analysis and now need a second study. Zorich's Analysis is also famous for both its broad coverage of the application of mathematical principles to physics, and the difficulty of the review problems. It can benifit you greatly if you put great efforts into thinking and solving them independently.
As for Functional Analysis, I highly recommend "Real and Functional Analysis" written by V. I. Bogachev and O. G. Smolyanov. This book is also totally proof-based and the authors prove every theorem or lemma as detailed as they can. Also, it is the most comprehensive and hard-core book on Functional Analysis in the sense it even covers some chapters that will usually not be covered. Another thing is that the review problems after every chapter are very challenging. So I think this book is also suitable for you.
