International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B, ch. 1.3, p. 30

A distribution T on Ω is a linear form over , i.e. a map which associates linearly a complex number to any , and which is continuous for the topology of that space. In the terminology of Section 1.3.2.2.6.2, T is an element of , the topological dual of .
Continuity over is equivalent to continuity over for all compact K contained in Ω, and hence to the condition that for any sequence in such that
then the sequence of complex numbers converges to 0 in .
If the continuity of a distribution T requires (ii) for only, T may be defined over and thus ; T is said to be a distribution of finite order m. In particular, for is the space of continuous functions with compact support, and a distribution is a (Radon) measure as used in the theory of integration. Thus measures are particular cases of distributions.
Generally speaking, the larger a space of test functions, the smaller its topological dual: This clearly results from the observation that if the ϕ's are allowed to be less regular, then less wildness can be accommodated in T if the continuity of the map with respect to ϕ is to be preserved.