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 E}(\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 E}(\Omega)]

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It is defined by the family of semi-norms[\varphi \in {\scr E}(\Omega) \,\longmapsto\, \sigma_{{\bf p}, \, K} (\varphi) = \sup\limits_{{\bf x} \in K} |D^{{\bf p}} \varphi ({\bf x})|,]where p is a multi-index and K a compact subset of Ω. A fundamental system S of neighbourhoods of the origin in [{\scr E}(\Omega)] is given by subsets of [{\scr E}(\Omega)] of the form[V (m, \varepsilon, K) = \{\varphi \in {\scr E}(\Omega)| |{\bf p}| \leq m \Rightarrow \sigma_{{\bf p}, K} (\varphi) \,\lt\, \varepsilon\}]for all natural integers m, positive real [epsilon], and compact subset K of Ω. Since a countable family of compact subsets K suffices to cover Ω, and since restricted values of [epsilon] of the form [\varepsilon = 1/N] lead to the same topology, S is equivalent to a countable system of neighbourhoods and hence [{\scr E}(\Omega)] is metrizable.

Convergence in [{\scr E}] may thus be defined by means of sequences. A sequence [(\varphi_{\nu})] in [{\scr E}] will be said to converge to 0 if for any given [V (m, \varepsilon, K)] there exists [\nu_{0}] such that [\varphi_{\nu} \in V (m, \varepsilon, K)] 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 on any given compact K in Ω.

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