Let and be the collection of functions (or if preferred) such that

It turns out are Banach spaces (see [2] 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

Thus

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

[1] Lieb, Elliot and Michael Loss. *Analysis*. 2nd Edition. Graduate Studies in Mathematics. Vol. 14. American Mathematical Society. 2001.

[2] Evans, Lawrence. *Partial Differential Equations*. Graduate Studies in Mathematics. Vol. 19. American Mathematical Society. 1998.