But the vector space of (all) real functions is a completely different beast from the space of computable functions on finite-precision numbers. If you restrict the equality of these functions to their extension,
defined as f = g iff forall x\in R: f(x)=g(x),
then that vector space appears to be not only finite dimensional, but in fact finite.
Otherwise you probably get a countably infinite dimensional vector space indexed by lambda terms (or whatever formalism you prefer.)
But nothing like the space which contains vectors like
A vector is a list of numbers, at its most basic. You can add a lot of extra functionality to it, but at its core, its just a list.
Functions from the reals to the reals are an example of a vector space with elements which can not be represented as a list of numbers.
It still can be, just not on infinite precision as nothing can with fp.
But the vector space of (all) real functions is a completely different beast from the space of computable functions on finite-precision numbers. If you restrict the equality of these functions to their extension,
defined as f = g iff forall x\in R: f(x)=g(x),
then that vector space appears to be not only finite dimensional, but in fact finite. Otherwise you probably get a countably infinite dimensional vector space indexed by lambda terms (or whatever formalism you prefer.) But nothing like the space which contains vectors like
F_{x_0}(x) := (1 if x = x_0; 0 otherwise)
where x_0 is uncomputable.