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

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