I’d like to add my take on how to learn analysis, as a math student.
After seeing so many elementary analysis textbooks, I think that no analysis textbooks can surpass the following three, arranged in alphabetical order:
- Analysis by Amann & Escher,
- Mathematical Analysis by Zorich,
- Real Mathematical Analysis by Pugh.
Description of the first and second textbooks: These two books not only cover the typical analytical concepts but they also cover calculus rigorously, which is often ignored in other analysis textbooks. So you can directly start reading either of them without needing to know college-level calculus.
Moreover, they cover many advanced analytical concepts that are not usually covered in other analysis textbooks. As an example, there is a coverage of manifold theory in both textbooks.
However, these two textbooks are different in style. Analysis by Amann tries to introduce every concept in its generality. For example, there isn’t any presentation of multiple Riemann integrals. Rather, it covers a modern (and more complete) version of integration by Lebesgue, which is applied in the presentation of some modern aspects of Fourier analysis. In contrast, the textbook by Zorich usually presents concepts along classical lines as long as the classical presentation is considered to be sufficient for applicability in various branches of mathematics and physics. For example, it covers the theory of multiple Riemann integrals without mentioning that of Lebesgue integrals, and a classical presentation of Fourier analysis follows in time. This is not a limitation, as the reader is expected to learn such general topics in other courses.
It may also be useful to know that the textbook by Amann is completely mathematically minded, while the textbook by Zorich has in mind not only mathematically minded readers but also physics students. Therefore, it seems that the textbook by Amann is harder to grasp than the textbook by Zorich.
There is also a slight difference in content between these two textbooks. While many of the analytical topics are covered in both of them (sometimes in a different way, as mentioned above regarding the integration theory), there are topics that are covered in one without being covered in the other. For example, complex analysis is integrated in the textbook by Amann, while it is not covered in the textbook by Zorich. There is also a chapter on asymptotic expansions in the textbook by Zorich which is not covered in the textbook by Amann.
Finally, both books rigorously teach analysis in a ‘beautiful’ way. The esthetic aspect is sometimes ignored in many other mathematical and scientific books, which can discourage the student with their dryness. By the way, only the first chapter in the textbook by Amann is dry. But this (inevitable) dryness is justified in the subsequent chapters and therefore does not interfere with its beauty.
Description of the third textbook: This is another rigorous analysis textbook that covers many of the essential analytical concepts that an undergraduate student of mathematics needs to know. This textbook does not teach calculus, so a prior course in calculus will help the reader better appreciate new rigorous concepts. In particular, the other textbooks mentioned above are more comprehensive in coverage.
It emphasizes visualization when approaching mathematical concepts, so the reader will see plenty of useful pictures associated with the concepts, which will help with comprehending the material. Moreover, it has plenty of challenging problems which will help the reader prepare for important exams. It is this approach mixed with author’s insight and experience that has made this textbook such a “beautiful” read.
Almost all mathematical analysis textbooks cover the same topics as presented in the textbook by Pugh and most universities adopt one of them in their curricula. But the textbook by Pugh stands out as the most insightful and instructive among them.
Summary: Either of these three books will help the student not only learn but also enjoy analysis. A prior course in calculus is recommended if the reader decides to read the textbook by Pugh, while this is not needed with the other textbooks.