This list "is All You Need"
I graduated in Mathematics a couple of years ago but I know feel that I am forgetting quite a few things so I would like to go over most of the mathematical syllabus on my own.
I would like to build a physical library (meaning I want to buy books, not read them online). I am looking to fill the areas shown below. Importantly, I would like to have books that are clear, fairly modern (I struggle to read 100-year-old manuscripts but if there is no alternative, I'll go for an old book) and allow me to cover the areas below without many gaps.
Can you please suggest books following a similar style to Sheldon Axler's "Linear Algebra Done Right" and "Measure, Integration & Real Analysis"? By similar style I mean that they take the reader by hand and reinforce topics by using different colors, repetition of concepts and, importantly, building a lot of intuition with diagrams, figures or examples. They also shouldn't be huge 1000-page-long bricks.
Here are the areas I am trying to cover. Striken over text means I have already bought them and happy with them. Texts in parenthesis are books I heard are good but possibly not quite similar to Sheldon Axler's style.
| Area | Found | Book |
|---|---|---|
| Calculus | No | C - Spivak or VCLADF - Hubbard |
| Yes | LADR - Axler | |
| Analysis | Maybe | A1- Tao or UA - Abbot |
| Metric Spaces | Maybe | A2 - Tao or PMA - Rudin |
| Differential Geometry | No | |
| Functional Analysis | Maybe | FAFA - Sasane or IFAA - Kreyszig or FA - Stein |
| Yes | MIRA - Axler | |
| Probability | Maybe | PI - Grimmett or PT - Klenke or PM - Billingsley |
| ODEs | No | DEDSLA - Hirsch |
| Dynamical Systems | Maybe | NDC - Strogatz |
| Stochastic Differential Equations | No | |
| Optimization | No | |
| Differential Calculus | Maybe | DCNS - Cartan or VCLADF - Hubbard |
| Statistics | Maybe | NSLT - Vapnik or SLT - Vapnik |
For calculus (multivariable calculus and differential forms) I highly recommend Hubbard and Hubbard. It is very clear and written with great care. For dynamical systems, Nonlinear Dynamics and Chaos by Strogatz is a great book that you will like a whole lot.
For differential geometry, I recommend Needham's new book Visual Differential Geometry and Forms, which will be published July 2021.