International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 30   | 1 | 2 |

## Section 1.3.2.3.4. Definition of distributions

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

#### 1.3.2.3.4. Definition of distributions

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A distribution T on Ω is a linear form over , i.e. a mapwhich 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

 (i) Supp is contained in some compact K independent of ν, (ii) the sequences converge uniformly to 0 on K for all multi-indices p;

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.