The set of all real->real functions is still a vector space.
This vector space also has a basis (even if it is not as useful): there is a (uncountably infinite) subset of real->real functions such that every function can be expressed as a linear combination of a finite number of these basis functions, in exactly one way.
There isn't a clean way to write down this basis, though, as you need to use Zorn's lemma or equivalent to construct it.
This vector space also has a basis (even if it is not as useful): there is a (uncountably infinite) subset of real->real functions such that every function can be expressed as a linear combination of a finite number of these basis functions, in exactly one way.
There isn't a clean way to write down this basis, though, as you need to use Zorn's lemma or equivalent to construct it.