Thanks to my university I have access to all Springer math books. Because of this I have tried to separate some books from Springer to learn the following subjects well (when I say well I refer to a book that requires as a prerequisite only subjects in the undergraduate level, for example, real analysis and that delves deep in the subject that it approaches):
- Metric Spaces
- Multivariate Calculus
- Topology of Metric Spaces
I have seen that the books "Analysis I", "Analysis II" and "Analysis III", written by Herbert Amann, teaches all these subjects, so I decided to read them. However, I have found other books, the ones listed below, for example, which better teach some subjects addressed by the Herbert Amann books.
The books to which I referred:
- "Calculus on Normed Vector Spaces", written by Rodney Coleman
- "Multivariate Calculus and Geometry", written by Seán Dineen
- "Integration - A Functional Approach", written by Klaus Bichteler
- "Topologies and Uniformities", written by I.M. James.
All of these books only require the prerequisites I have:
- Real Analysis
- Topology of the real line
- Linear Algebra
- Abstract Algebra(rings)
Please tell me the best books from Springer Publisher you know that only require the prerequisites that I have, but that go deep.