I've been learning real analysis from this book:
Elementary Analysis, K.A. Ross
I really liked the style of this book. It is quite old, and sometimes very difficult, but I guess I liked the way it lays out analysis very precise and rigorously. Although, I'm a first-year math student and I don't have much to compare it with.
We just finished this book at my university, and now we are studying: "Analysis from $ℝ$ to $ℝ^n$: multivariable functions" as it is called. The "book" we use is written by the university, and to be fair, I just don't understand it. I think that if I had attended all the lectures then it would have worked out otherwise. But now I'm quite sure that some definitions/theorems/tricks to solve the questions are just missing in the text. I get completely frustrated by this.
Would anybody know a good sequel to Elementary Analysis by Ross?
These are the names of the chapters of the book we use:
1. Normed vector spaces and limits
1.1 Normed vector spaces
1.2 Limits of functions and continuity
1.3 Operator norm
2. Differentiation of functions from $ℝ^n$ to $ℝ^m$
3. The chain rule
4. Geometric interpretation of derivatives
5. Mean value theorem: Higher partial derivatives
6. Taylor series in use
8. Higher total derivatives: Taylor series