Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 29   | 1 | 2 |

Section Topology on [{\scr D}_{k} (\Omega)]

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 Topology on [{\scr D}_{k} (\Omega)]

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It is defined by the family of semi-norms[\varphi \in {\scr D}_{K} (\Omega) \,\longmapsto\, \sigma_{\bf p} (\varphi) = \sup\limits_{{\bf x} \in K} |D^{{\bf p}} \varphi ({\bf x})|,]where K is now fixed. The fundamental system S of neighbourhoods of the origin in [{\scr D}_{K}] is given by sets of the form[V (m, \varepsilon) = \{\varphi \in {\scr D}_{K} (\Omega)| |{\bf p}| \leq m \Rightarrow \sigma_{\bf p} (\varphi) \,\lt\, \varepsilon\}.]It is equivalent to the countable subsystem of the [V (m, 1/N)], hence [{\scr D}_{K} (\Omega)] is metrizable.

Convergence in [{\scr D}_{K}] may thus be defined by means of sequences. A sequence [(\varphi_{\nu})] in [{\scr D}_{K}] will be said to converge to 0 if for any given [V(m, \varepsilon)] there exists [\nu_{0}] such that [\varphi_{\nu} \in V(m, \varepsilon)] whenever [\nu \,\gt \,\nu_{0}]; in other words, if the [\varphi_{\nu}] and all their derivatives [D^{\bf p} \varphi_{\nu}] converge to 0 uniformly in K.

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