Let me briefly talk about some points that need to be paid attention to in self-study:
1, abstract algebra (modern algebra) does not need other basic knowledge (linear algebra or higher algebra knowledge is better), and mainly studies the properties of groups, rings and fields. As long as you have a little concept, you can understand what symbols represent, their operations and properties. For a simple example, A is an element in a group and can represent a number (real number and complex number, etc.). ), matrix (with certain properties, such as diagonal, invertible, n-order and so on. ), a map, or even a set (group, ring, etc. ).
2. Learning functional analysis requires several courses (mathematical analysis, advanced algebra, real variable function). This course is a bit difficult for students who are not majoring in mathematics. Don't want to go into details, just say a few points: make clear the norm in normed linear space, the elements in linear space and the properties of normed linear space. This course is not very easy to learn, but it is very powerful. You should be mentally prepared!
3. Topology (briefly talk about point set topology). The compulsory course of point set topology is mathematical analysis, and the most important thing is to have the foundation of set theory. Point set topology mainly studies the invariant properties of topological space, including connectivity, countability axiom, separation axiom, compactness and so on. Of course, we should make clear what a topological space is, and what its nature and structure are! Long-winded: topology is equally powerful, but it is also difficult to learn!
Ps: The aforementioned mathematical analysis is a basic course for mathematics majors. If it is calculus or advanced mathematics and other courses, it is equally difficult to learn these courses. Remember!