Cardinality of a vector space?

Another question that has been bugging me for a while:

Consider a vector space of all piecewise continuous functions (more specifically, those that satisfy the Dirichlet conditions) defined on an interval of the reals, for example f:(-pi, pi) => R.

This space has (I think!) a number of dimensions equal to the cardinality of the continuum, alef[1].

However, consider the Fourier series approximation:

g(x) = a[0] + sum(n = 1, 2, ...)(a[n]*cos(nx) + b[n]*sin(nx))

There are a countable number of parameters for g, namely: a[0], a[1], ..., and b[1], b[2], ...; in other words, the space of all such g's has aleph[0] dimensions.

How is it possible that for every such f, there is a corresponding choice of the a's and b's?

Treated a slightly different way:  There are alef[2] such f's, and there are only alef[1]^alef[0] = alef[1].

I asked one of my prof's, and he pointed out that although the set of rationals is smaller than the set of reals, any real number can be approximated arbitrarily close by the rationals.  Using that analogy, he argued that any f(x) can be approximated arbitrarily close by a g(x).  However, there is a ton of hand-waving here, and I'm not sure whether I'm convinced.

Another resolution to this issue would be if there are only countably many dimensions for the vector space of all f's satisfying the Dirichlet conditions.  However, I think that, at least in the case where the domain for f is the whole number line, there are uncountably many dimensions; please correct me if that is my error.

While this space has uncountable vector space dimension, it has countable dimension as a Hilbert space or Banach space (depending on what norm you are using). The two are different concepts, involving finite sums and infinite sums respectively. Here is a good explanation of the distinction.

Thanks.  I hadn't realized there was so much of a difference between finite and infinite linear combinations.  To much fiddling around in R^n makes one start to make false assumptions about other structures

Hilbert space has an infinite number of dimensions of course. Physicists often use this, so feel free to ask people outside of your math department!

Indeed. A big inspiration for the rapid development of functional analysis in the early and mid-20th century was the needs of quantum mechanics. The briefest history of functional analysis I have ever seen shows how the theory developed simultaneously with the development of quantum mechanics, with both taking off in the mid-1920s.

(looking again, the content of the article I linked from post #2 is good, but the editing is terrible! But there are many more articles about the different types of basis).

For the difference between finite sums and infinite sums, your professor's example is a good one. With finite sums of rationals, you get just more rationals. With infinite sums of rationals (more precisely, convergent series) you get all the real numbers. Quite a difference!