Hilbert spaces of distributions

[Jan Medlock <medlock@amath.washington.edu>]

To answer Tim's question today as to the existence of Hilbert spaces
of distributions, the answer such a thing can be constructed as
follows:

Let D={\phi : \phi=limit \phi_n where \phi_n are infinitly smooth and
    form a Cauchy sequence} (I'm not sure how restrictive this is---it
    certainly includes *some* distributions which are not functions).

Then define the inner product as
(\phi, \psi) = limit (\phi_n, \psi_n).

This situation is exactly as Eric said: some other function space (in
this case, the space of infinitely smooth functions) is dense this
distribution space.

There's a more in Griffel's Applied Functional Analysis book.  I have
the AMath library's copy and it's probably going with me over the
weekend.  Let me know if you'd like to have a look.

Jan