Topology on

Bricogne, G.

doi:10.1107/97809553602060000551

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

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

Section 1.3.2.3.3.3. Topology on $[{\scr D}(\Omega)]$

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

1.3.2.3.3.3. Topology on $[{\scr D}(\Omega)]$

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It is defined by the fundamental system of neighbourhoods of the origin consisting of sets of the form $[\eqalign{&V((m), (\varepsilon)) \cr &\qquad = \left\{\varphi \in {\scr D}(\Omega)| |{\bf p}| \leq m_{\nu} \Rightarrow \sup\limits_{\|{\bf x}\| \leq \nu} |D^{{\bf p}} \varphi ({\bf x})| \;\lt\; \varepsilon_{\nu} \hbox{ for all } \nu\right\},}]$ where (m) is an increasing sequence $[(m_{\nu})]$ of integers tending to $[+ \infty]$ and (ɛ) is a decreasing sequence $[(\varepsilon_{\nu})]$ of positive reals tending to 0, as $[\nu \rightarrow \infty]$ .

This topology is not metrizable, because the sets of sequences (m) and (ɛ) are essentially uncountable. It can, however, be shown to be the inductive limit of the topology of the subspaces $[{\scr D}_{K}]$ , in the following sense: V is a neighbourhood of the origin in $[{\scr D}]$ if and only if its intersection with $[{\scr D}_{K}]$ is a neighbourhood of the origin in $[{\scr D}_{K}]$ for any given compact K in Ω.

A sequence $[(\varphi_{\nu})]$ in $[{\scr D}]$ will thus be said to converge to 0 in $[{\scr D}]$ if all the $[\varphi_{\nu}]$ belong to some $[{\scr D}_{K}]$ (with K a compact subset of Ω independent of ν) and if $[(\varphi_{\nu})]$ converges to 0 in $[{\scr D}_{K}]$ .

As a result, a complex-valued functional T on $[{\scr D}]$ will be said to be continuous for the topology of $[{\scr D}]$ if and only if, for any given compact K in Ω, its restriction to $[{\scr D}_{K}]$ is continuous for the topology of $[{\scr D}_{K}]$ , i.e. maps convergent sequences in $[{\scr D}_{K}]$ to convergent sequences in $[{\bb C}]$ .

This property of $[{\scr D}]$ , i.e. having a non-metrizable topology which is the inductive limit of metrizable topologies in its subspaces $[{\scr D}_{K}]$ , conditions the whole structure of distribution theory and dictates that of many of its proofs.

References

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