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.
Let and denote the set of smooth functions on with compact support.
Definition 1. Let and be a sequence in We say converges to denoted if
- for all for a fixed compact
- uniformly on
By we mean
is a vector space, and the above notion of convergence allows us to define closed sets and hence turns into a topological vector space called the space of test functions of and we denote it
Definition 2. A distribution is a continuous linear functional on We will denote the set of distributions on by –noting that we are only considering continuous linear functionals.
We say a sequence of distributions converges to the distribution denoted if for all This gives us a topological vector space structure on as well.
Definition 3. Let be a distribution and We define the -th distributional (weak) derivative of by
where We may also write
We will omit if The gradient of is simply
Let us define the space as the space of functions such that
for all compact We define strong convergence on as convergence in and weak convergence on as weak (pointwise) convergence in for all compact
Let then clearly
Let and and define
This integral is finite since both functions are integrable (use integration by parts and establish bounds). Also if uniformly, then
where which gives us the continuity of So We also have
 Lieb, Elliott and Michael Loss. Analysis. 2nd Edition. Graduate Studies in Mathematics. Vol. 14. American Mathematical Society. 2001.