There is only one definition of "vector space" (up to isomorphism anyway), and that's what the author uses. You'll note that he doesn't talk about bases at all, the assumption of a basis is entirely in your mind. The entire point of the article is that the ℝ→ℝ function space is a vector space. A vector space is not required to have a basis, but assuming the axiom of choice, every vector space does have (at least) one, including that of ℝ→ℝ functions.
The choice of basis is very important for the applications. It doesnt matter that function could be vector spaces in theory without constructing a base, the article is about "hey look at functions they're as easy to operate in as this thing you already know with 3D vectors"
