Test-function spaces

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, pp. 29-30 | 1 | 2 |

Section 1.3.2.3.3. Test-function spaces

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. Test-function spaces

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With this rationale in mind, the following function spaces will be defined for any open subset Ω of $[{\bb R}^{n}]$ (which may be the whole of $[{\bb R}^{n}]$ ):

(a) $[{\scr E}(\Omega)]$ is the space of complex-valued functions over Ω which are indefinitely differentiable;
(b) $[{\scr D}(\Omega)]$ is the subspace of $[{\scr E}(\Omega)]$ consisting of functions with (unspecified) compact support contained in $[{\bb R}^{n}]$ ;
(c) $[{\scr D}_{K} (\Omega)]$ is the subspace of $[{\scr D}(\Omega)]$ consisting of functions whose (compact) support is contained within a fixed compact subset K of Ω.

When Ω is unambiguously defined by the context, we will simply write $[{\scr E},{\scr D},{\scr D}_{K}]$ .

It sometimes suffices to require the existence of continuous derivatives only up to finite order m inclusive. The corresponding spaces are then denoted $[{\scr E}^{(m)},{\scr D}^{(m)},{\scr D}_{K}^{(m)}]$ with the convention that if [m = 0] , only continuity is required.

The topologies on these spaces constitute the most important ingredients of distribution theory, and will be outlined in some detail.

1.3.2.3.3.1. 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 ɛ, and compact subset K of Ω. Since a countable family of compact subsets K suffices to cover Ω, and since restricted values of ɛ 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.

1.3.2.3.3.2. 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.

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.

1.3.2.3.3.4. Topologies on $[{\scr E}^{(m)}, {\scr D}_{k}^{(m)},{\scr D}^{(m)}]$

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These are defined similarly, but only involve conditions on derivatives up to order m.

References

International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 29-30