Let and be the collection of functions (or if preferred) such that
It turns out are Banach spaces (see  for proof), called Sobolev spaces, under the above norm. In particular The spaces are Hilbert with inner product
We can also define the local Sobolev spaces as the subset of such that for We have
and where under dictionary ordering on provided has a finite Lebesgue measure. Thus in particular
is a -module by a simple boundedness argument following from: if and then is finite in Thus Correspondingly this gives us action on (which denotes the distributions on for us, although the action works on the dual space as well) defined by
Theorem 1. Let be bounded in and Then there exists a sequence of functions such that
We can also define (and similarly ) as the set of functions such that the following norm is finite
I’m wondering if we could consider attempting to define (and similarly ) as the collection of functions such that
is finite. This would require all derivatives to eventually get to in such a way that the sum converges. It would thus include polynomials if is bounded and hence be nonempty. In particular it would obey for all And for all
 Lieb, Elliot and Michael Loss. Analysis. 2nd Edition. Graduate Studies in Mathematics. Vol. 14. American Mathematical Society. 2001.
 Evans, Lawrence. Partial Differential Equations. Graduate Studies in Mathematics. Vol. 19. American Mathematical Society. 1998.