[JadeNB]

> And vector spaces have bases. One basis for the vector space of functions is the collection of sin and cos functions. Thus we can see that finding the Fourier Transform is just finding how much of each basis vector we need to make the function.

You have indicated quite explicitly that you're being informal (by using non-technical terms like 'non-pathological functions'), but I think it's worth making the small observation here that sine and cosine functions do not form a basis (implicitly: "a Hamel basis") for, say, the space of continuous functions on R/Z in the usual sense of linear algebra, since not every function can be written as a finite linear combination of them.

Rather, they are a topological basis, in the sense that every function can be written as an infinite linear combination of them. Why is it worth making this point, which probably seemed too obvious to say? The reason is that, unlike finite sums, which are unambiguous algebraic constructs, infinite sums require topology to compute; and being a topological basis in, say, the L^2 topology is quite different to being a topological basis in, say, the topology of pointwise convergence. The study of the different kinds of summability of Fourier series is the subject of some very, very deep work.

[ColinWright]

All true - the explanation should be littered with "here be dragons ..." - but for the sake of grasping the concepts without drowning in detail, I thought it was worth leaving out such tricky (but important!) Issues.

But you're right, there is interesting and deep material here.