[azernik]

This was an example used in my linear algebra class as soon as they started introducing the vector spaces in an abstract sense.

I think this post may still be too wedded to the idea of linear spaces and vectors being arrays of objects - specifically in insisting on decomposing functions like sin and cos to Taylor Series. In fact, you can have a vector space where, in addition to polynomial terms, there are also dimensions for sin(x), tan(x), sin(x - pi), e^x, etc. The fact that you can't enumerate these dimensions, or even describe the set of them until given a set of vectors you're trying to describe, doesn't keep this from being a vector space.

[tzahola]

Hm. Interesting.

I always viewed real functions as infinite-dimensional vectors in the "canonical" basis, that is, shifted Dirac impulses. I guess it can be transformed into your representation with a change of basis with some handwaving.

[azernik]

What you're describing may not be a subspace of the space I'm describing. Mine is definitely a subspace of yours - e.g. you can project functions into sums of shifted Dirac impulses by representing each function dimension as a linear combination of the "Dirac vectors" for that function's values.

[tzahola]

Hmmm. I think it’s easy to fix that:

Suppose that there’s a function f, that can be written as an infinite sum (integral) of shifted Dirac impulses, but cannot be written in your representation as a sum of those “base functions”. Then simply add a new dimension to your representation that will correspond to f, so that f will be represented as 1 at this new dimension, and zero everywhere else. (In other words: add f to the base functions)

Repeat until you have covered every function.