Distributions

Chapter 2Space of Test Functions

In order to be able to build in the next chapter the space ′(Ω; E) of distributions, i.e. of continuous linear mappings from (Ω) into E, we construct here the space (Ω) of test functions.

We endow it with the semi-norms indexed by p ∈ ⁺(Ω), and give the following properties.

• — It is a Neumann space (Theorem 2.8) which is sequentially separable (Theorem 2.16) and in which every bounded sequence has a convergent subsequence (Theorem 2.14).

• — We give characterizations of its convergent sequences, including a new one (Theorem 2.13 (c)) which will serve later (Theorem 4.14) to show that ′(Ω; E-weak) = ′(Ω; E).

• — We establish an inequality (Theorem 2.22), also new, that allows us to “control” a series of norms of the by one semi-norm of the (Ω), which gives ...