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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 29-30   | 1 | 2 |

## Section 1.3.2.3.3. Test-function spaces

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

#### 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 (which may be the whole of ):

 (a) is the space of complex-valued functions over Ω which are indefinitely differentiable; (b) is the subspace of consisting of functions with (unspecified) compact support contained in ; (c) is the subspace of 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 .

It sometimes suffices to require the existence of continuous derivatives only up to finite order m inclusive. The corresponding spaces are then denoted with the convention that if , 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 | top | pdf |

It is defined by the family of semi-norms where p is a multi-index and K a compact subset of Ω. A fundamental system S of neighbourhoods of the origin in is given by subsets of of the form for all natural integers m, positive real , and compact subset K of Ω. Since a countable family of compact subsets K suffices to cover Ω, and since restricted values of of the form lead to the same topology, S is equivalent to a countable system of neighbourhoods and hence is metrizable.

Convergence in may thus be defined by means of sequences. A sequence in will be said to converge to 0 if for any given there exists such that whenever ; in other words, if the and all their derivatives converge to 0 uniformly on any given compact K in Ω.

#### 1.3.2.3.3.2. Topology on | top | pdf |

It is defined by the family of semi-norms where K is now fixed. The fundamental system S of neighbourhoods of the origin in is given by sets of the form It is equivalent to the countable subsystem of the , hence is metrizable.

Convergence in may thus be defined by means of sequences. A sequence in will be said to converge to 0 if for any given there exists such that whenever ; in other words, if the and all their derivatives converge to 0 uniformly in K.

#### 1.3.2.3.3.3. Topology on | top | pdf |

It is defined by the fundamental system of neighbourhoods of the origin consisting of sets of the form where (m) is an increasing sequence of integers tending to and ( ) is a decreasing sequence of positive reals tending to 0, as .

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

A sequence in will thus be said to converge to 0 in if all the belong to some (with K a compact subset of Ω independent of ν) and if converges to 0 in .

As a result, a complex-valued functional T on will be said to be continuous for the topology of if and only if, for any given compact K in Ω, its restriction to is continuous for the topology of , i.e. maps convergent sequences in to convergent sequences in .

This property of , i.e. having a nonmetrizable topology which is the inductive limit of metrizable topologies in its subspaces , conditions the whole structure of distribution theory and dictates that of many of its proofs.

#### 1.3.2.3.3.4. Topologies on | top | pdf |

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