Distributions

Chapter 3Space of Distributions

The goal of this chapter is to construct the space ′(Ω; E) of distributions on an open subset Ω of ^d with values in a Neumann space E. It is the space of continuous linear mappings from (Ω) into E.

We endow it (Definition 3.1) with the semi-norms indexed by φ ∈ (Ω) and ν ∈ _E, namely with the topology of simple (pointwise) convergence on (Ω). We then give various characterizations of distributions (Theorem 3.4) and we identify every continuous function with a distribution (Theorem 3.8).

In § 3.4, we show that if E is not a Neumann space, i.e. if it is not sequentially complete, this construction does not give the expected properties, since not every continuous function is (identifiable with) a mapping of ((Ω); E), mappings that should not then be called “distributions”. ...