In almost all the literature that I have seen, one of the examples of vector space is as follows:
Set of all real-valued functions $f(x)$ defined on the real line
what confuses me here is that the word "linear functions" should be used instead of just "functions" because I think we also have non-linear functions and they (non-linear functions)do not follow the rule of addition and multiplication as:
$ (f + g)(x)=f(x) + g(x) $ and $(kf)(x) = k f(x) $
thus, non-linear functions cannot make a vector space.
am I right or wrong?