Topology on

Bricogne, G.

doi:10.1107/97809553602060000551

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 30 | 1 | 2 |

Section 1.3.2.3.3.2. Topology on $[{\scr D}_{k} (\Omega)]$

G. Bricogne^a

^a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

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

| top | pdf |

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.

References

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 30