A variant of Jim Propp's original question, as I understood it, is: "Let X=\R/\Z. How can a person give an intuitive explanation of the Hilbert space isomorphism L^2(X) \to \ell^2(X) when the frequency space \ell^2 seems so much smaller than the physical space L^2?" I don't think that there is an easy answer, but coming to understand the issues is rewarding. This is an isomorphism of Hilbert spaces, not of sets, so a person needs an intuitive feel for the topologies arising from L^2-norm and the \ell^2-norm, neither of which is the (continuous or discrete) C^0-norm or any C^k-norm. L^2-functions don't have pointwise values, and this is a very hard sell. L^2(X) is naturally the completion of C^\infty(X) under the L^2-norm, but this is also a hard sell because people seem to find the more ad hoc construction of L^2(X) as {square-integrable Lebesgues}/{equality a.e.} more tangible. Bessel/Parseval statements about Fourier series representation don't say what a person hopes because they assume a basis, and the existence of the basis (or the fact that the "obvious" basis really is one) is the real issue. The old-fashioned proof-of-basis via the "Fejer kernel" is (imho) unexplanatory. A clear argument interweaves the Banach spaces C^k(X) with Sobolev spaces, which are Hilbert even though the C^k-spaces aren't, and then argues that the spectral theorem basis of differentiation-eigenfunctions in L^2 (L^2 functions can be differentiated despite not having pointwise values, using a Friedrichs extension of differentiation on C^\infty, even though that operator is "unbounded" [discontinuous])... a clear argument interweaves and then argues that the spectral theorem basis of differentiation-eigenfunctions must lie in the littlest space C^\infty, and in that space we know what they are: the oscillations. I doubt that any of this is helpful to Jim. On the other hand, these ideas scale to many spaces X, such as Euclidean space (where the basis of Laplacian eigenfunctions isn't the oscillations, which aren't L^2, but rather comprises the Gaussian times harmonic polynomials), spheres, and quotients of complex upper half spaces, where we are talking about automorphic forms and nobody knows a basis of cuspforms but we know that there is one. Jerry Shurman