I will be in the next month finishing up Spivak´s Calculus and I was wondering what would be a good continuation.
Background: I will probably start my university studies this fall and thought that I might be able to take on one more book before i start. I have been self studying mathematics for a about three years soon. I dropped out of high school at approximatly the same time as I started taking mathematics seriously and have since been very interested in mathematics as a whole. My studies can be summarized roughly as
- The highschool material i did not complete while in school
- How to prove it - Daniell Velleman
linear algebra and its applications(matrices, determinatns, vector spaces, euclidean space.)- Lay
Calculus - Spivak
- A bit of introductory abstract algebra in the form of Fraleigh
Now i have been particulary pleased with the material of Spivak and I have found that the exercises in that book felt like they where proper exercises as compared to linear algebra and its applications , where , atleast for me, they seemed a bit too routine and not very interesting in general. I feel comfortable doing most proofs and have not struggled (too hard) on the material in spivak. I particulary enjoyed the later chapters where sums, uniform convergence and all that good stuff was introduced.
Should I continue my study of analysis? and if so what books do your reccomend?
Should I aim to take up some other branch (algebra, number theory, linear algebra, geometry)? and if so what books give a "simillar" approach as spivak to analysis?