I am not sure I fully understand your question, but I’ll try to answer the question I understood. Very generally, every field of math I know concerns itself wit

I am not sure I fully understand your question, but I’ll try to answer the question I understood. Very generally, every field of math I know concerns itself with mappings and what they do (e.g. matrixes are mappings, as are metrics and norms). The differences between the different areas of math (oversimplified) is that they concern themselves with different kinds of mappings. The area I specialized in was partial differential equations, here the mappings of interest are solutions to partial differential equations (usually generalized functions). One of the important questions to ask is of these functions are continuous, differentiable, or any other of the many variations. All of these properties are defined over how a function maps an input to an output. BTW: There’s usually an infinite number of solutions to a differential equation, and IIRC (it’s been a while) those solutions too form a vector space.