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

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. Bricognea

aMRC 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 [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.

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 ([epsilon]) 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 ([epsilon]) 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.








































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