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 1.3.2.3.7. Support of a distribution

G. Bricognea

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#### 1.3.2.3.7. Support of a distribution

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A distribution is said to vanish on an open subset ω of Ω if it vanishes on all functions in , i.e. if whenever .

The support of a distribution T, denoted Supp T, is then defined as the complement of the set-theoretic union of those open subsets ω on which T vanishes; or equivalently as the smallest closed subset of Ω outside which T vanishes.

When for , then Supp , so that the two notions coincide. Clearly, if Supp T and Supp ϕ are disjoint subsets of Ω, then .

It can be shown that any distribution with compact support may be extended from to while remaining continuous, so that ; and that conversely, if , then its restriction T to is a distribution with compact support. Thus, the topological dual of consists of those distributions in which have compact support. This is intuitively clear since, if the condition of having compact support is fulfilled by T, it needs no longer be required of ϕ, which may then roam through rather than .