The basic topics covered doesn't necessarily mean that the book will be easy; this was my first introduction to proofs, and to trying to create my own (which I often failed at doing). However, even if I knew most of the material already (like that a negative number multiplied by a negative number results in a positive number), I had never known the reasoning behind why this was true. The logical approach presented taught me a different, more effective way to learn mathematics. Lang even has an entire section (somewhere after chapter 3, I think) where he simply covers logic and some notation, to help the reader get a better grasp of the idea. On the question of whether Lang will prepare you adequately for Spivak, the answer is maybe. It's better than most books for this, because it does devote considerable attention to proofs. On the other hand, it doesn't develop a high level of computational skill in important areas. That's one reason why in my answer here Jorgenson, Jay; Lang, Serge (2005). Posn(R) and Eisenstein Series. Lecture Notes in Mathematics. Vol.1868. Berlin: Springer-Verlag. doi: 10.1007/b136063. ISBN 978-3-540-25787-5. MR 2166237. Lang, Serge (1987). Elliptic functions. Graduate Texts in Mathematics. Vol.112. With an appendix by J. Tate (Second edition of 1973 originaled.). New York: Springer-Verlag. doi: 10.1007/978-1-4612-4752-4. ISBN 0-387-96508-4. MR 0890960. [22]

Algebraic Number Theory | SpringerLink Algebraic Number Theory | SpringerLink

Lang, Serge (1985). SL 2(R). Graduate Texts in Mathematics. Vol.105 (Reprint of the 1975 originaled.). New York: Springer-Verlag. doi: 10.1007/978-1-4612-5142-2. ISBN 0-387-96198-4. MR 0803508. [21] So what is the general consensus on Serge Lang as an author? Are some of his books just better than others or is it just a matter of personal preference? If the latter, what kind of person would enjoy working with one of his books? Basically, is it a text someone should go for or not?

Lang, Serge (1994). Algebraic number theory. Graduate Texts in Mathematics. Vol.110 (Second edition of 1970 originaled.). New York: Springer-Verlag. doi: 10.1007/978-1-4612-0853-2. ISBN 0-387-94225-4. MR 1282723. [25] The first edition was itself the second edition of Algebraic Numbers (1964) Rosenlicht, M. (1959). "Review: Introduction to algebraic geometry. By Serge Lang" (PDF). Bull. Amer. Math. Soc. 65 (6): 341–342. doi: 10.1090/s0002-9904-1959-10361-x. Lang, Serge (1978). Elliptic curves: Diophantine analysis. Grundlehren der Mathematischen Wissenschaften. Vol.231. Berlin–New York: Springer-Verlag. doi: 10.1007/978-3-662-07010-9. ISBN 3-540-08489-4. MR 0518817. [18] Se você é alguém que busca aprender matemática com este livro, leia somente as explicações e busque exercícios em outros lugares. Se você já tem experiência com matemática, pratique os exercícios e guarde o livro como referência para quando precisar lembrar de algo. If you're exceptionally concerned about rigor, you can get the full story on the set-theoretical foundations of mathematics from a book like Introduction to Set Theory, by Jech and Hrbacek. This builds up from the axioms of set theory to a construction of the natural numbers, and later of the integers, rational numbers and real numbers. The problem is that while such a program is pre-Spivak in purely logical terms, it is post-Spivak in the demands it places on readers' mathematical maturity. For comparison, Spivak constructs the real numbers in the last part of his book, but he takes the rational numbers and their properties as intuitively known. My opinion is that few people would benefit from working through a set theory book before Spivak, but it may be helpful to have a general idea of what the steps are in providing a firm logical foundation for mathematics.

Basic Mathematics by Serge Lang | Goodreads

Lang, Serge (1995). Introduction to Diophantine approximations (Second edition of 1966 originaled.). New York: Springer-Verlag. doi: 10.1007/978-1-4612-4220-8. ISBN 0-387-94456-7. MR 1348400. Lang, Serge (1997). Undergraduate analysis. Undergraduate Texts in Mathematics (Seconded.). New York: Springer-Verlag. doi: 10.1007/978-1-4757-2698-5. ISBN 0-387-94841-4. MR 1476913. The first edition (1983) of this title was itself the second edition of Analysis I (1968) Lang, Serge (1999). Fundamentals of differential geometry. Graduate Texts in Mathematics. Vol.191. New York: Springer-Verlag. doi: 10.1007/978-1-4612-0541-8. ISBN 0-387-98593-X. MR 1666820. This book is the fourth edition, previously published under the different titles of Introduction to Differentiable Manifolds (1962), Differential Manifolds (1972), and Differential and Riemannian Manifolds (1995). Lang also published a distinct second edition (preserving the title of the 1962 original) so as to provide a companion volume to Fundamentals of Differential Geometry which covers a portion of the same material, but with the more elementary exposition confined to finite-dimensional manifolds: O'Connor, John J.; Robertson, Edmund F., "Serge Lang", MacTutor History of Mathematics Archive, University of St Andrews

Lang, Serge (1993). Real and functional analysis. Graduate Texts in Mathematics. Vol.142 (Thirded.). New York: Springer-Verlag. doi: 10.1007/978-1-4612-0897-6. ISBN 0-387-94001-4. MR 1216137. This book is the third edition, previously published under the different titles of Analysis II (1968) and Real Analysis (1983) Lang, Serge (1988). Introduction to Arakelov theory. New York: Springer-Verlag. doi: 10.1007/978-1-4612-1031-3. ISBN 0-387-96793-1. MR 0969124. [24] Lang, Serge (1986). Introduction to linear algebra. Undergraduate Texts in Mathematics (Second edition of 1970 originaled.). New York: Springer-Verlag. doi: 10.1007/978-1-4612-1070-2. ISBN 978-0-387-96205-4. S2CID 117514050. Serge Lang (18 May 1978), "The Professors: A Survey of a Survey", The New York Review of Books available online as reprinted in Challenges Magill, K. D. (1965-01-01). "Review of A Second Course in Calculus". The American Mathematical Monthly. 72 (9): 1048–1049. doi: 10.2307/2313382. JSTOR 2313382.