Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

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

Section Support of a distribution

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 Support of a distribution

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A distribution [T \in {\scr D}\,'(\Omega)] is said to vanish on an open subset ω of Ω if it vanishes on all functions in [{\scr D}(\omega)], i.e. if [\langle T, \varphi \rangle = 0] whenever [\varphi \in {\scr D}(\omega)].

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 [T = T_{f}] for [f \in L_{\rm loc}^{1} (\Omega)], then Supp [T = \hbox{Supp } f], so that the two notions coincide. Clearly, if Supp T and Supp ϕ are disjoint subsets of Ω, then [\langle T, \varphi \rangle = 0].

It can be shown that any distribution [T \in {\scr D}\,'] with compact support may be extended from [{\scr D}] to [{\scr E}] while remaining continuous, so that [T \in {\scr E}\,']; and that conversely, if [S \in {\scr E}\,'], then its restriction T to [{\scr D}] is a distribution with compact support. Thus, the topological dual [{\scr E}\,'] of [{\scr E}] consists of those distributions in [{\scr D}\,'] 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 [{\scr E}] rather than [{\scr D}].

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