It's not tautological — to pick one of the properties, you can conceive of spaces of functions where there is no -f to a function f. For example, consider the s

It's not tautological — to pick one of the properties, you can conceive of spaces of functions where there is no -f to a function f. For example, consider the space of all positive-valued real functions. Consider otherwise functions which take colours and output letters or the alphabet. Letters of the alphabet can't be added or subtracted, so there's no vector space structure on that. On the flip side, vector spaces can be defined not just on real numbers, but complex numbers as well, or even other sets — specifically, any "field", that is, any set with addition, subtraction, multiplication, and division defined on it in a self-consistent way. There are even finite fields; vector spaces over three are relevant in cryptography. If these rules seem silly and abstract to you (shouldn't a function just be a function?!), well, that's mathematics! By elucidating the very specific conditions under which results hold, and abstracting away all the irrelevant details, you end up with results of incredible generality.