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

| top | pdf |

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-normswhere 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 formfor 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-normswhere K is now fixed. The fundamental system S of neighbourhoods of the origin in is given by sets of the formIt 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 formwhere (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.