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

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

## Section 1.3.2.3.3.1. Topology on

G. Bricognea

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#### 1.3.2.3.3.1. Topology on

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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.