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.