Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 28   | 1 | 2 |

Section Topological vector 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 Topological vector spaces

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The function spaces E of interest in Fourier analysis have an underlying vector space structure over the field [{\bb C}] of complex numbers. A topology on E is said to be compatible with a vector space structure on E if vector addition [i.e. the map [({\bf x},{ \bf y}) \,\longmapsto\, {\bf x} + {\bf y}]] and scalar multiplication [i.e. the map [(\lambda, {\bf x}) \,\longmapsto\, \lambda {\bf x}]] are both continuous; E is then called a topological vector space. Such a topology may be defined by specifying a `fundamental system S of neighbourhoods of [{\bf 0}]', which can then be translated by vector addition to construct neighbourhoods of other points [{\bf x} \neq {\bf 0}].

A norm ν on a vector space E is a non-negative real-valued function on [E \times E] such that[\displaylines{\quad (\hbox{i}')\,\,\quad\nu (\lambda {\bf x}) = |\lambda | \nu ({\bf x}) \phantom{|\lambda | v ({\bf x} =i} \hbox{for all } \lambda \in {\bb C} \hbox{ and } {\bf x} \in E\hbox{\semi}\hfill\cr \quad (\hbox{ii}')\,\quad\nu ({\bf x}) = 0 \phantom{|\lambda | v ({\bf x} = |\lambda | vxxx} \hbox{if and only if } {\bf x} = {\bf 0}\hbox{\semi}\hfill\cr \quad (\hbox{iii}')\quad \nu ({\bf x} + {\bf y}) \leq \nu ({\bf x}) + \nu ({\bf y}) \quad \hbox{for all } {\bf x},{\bf y} \in E.\hfill}]Subsets of E defined by conditions of the form [\nu ({\bf x}) \leq r] with [r\,\gt\, 0] form a fundamental system of neighbourhoods of 0. The corresponding topology makes E a normed space. This topology is metrizable, since it is equivalent to that derived from the translation-invariant distance [d({\bf x},{ \bf y}) = \nu ({\bf x} - {\bf y})]. Normed spaces which are complete, i.e. in which all Cauchy sequences converge, are called Banach spaces; they constitute the natural setting for the study of differential calculus.

A semi-norm σ on a vector space E is a positive real-valued function on [E \times E] which satisfies (i′) and (iii′) but not (ii′). Given a set Σ of semi-norms on E such that any pair (x, y) in [E \times E] is separated by at least one [\sigma \in \Sigma], let B be the set of those subsets [\Gamma_{\sigma{, \,} r}] of E defined by a condition of the form [\sigma ({\bf x}) \leq r] with [\sigma \in \Sigma] and [r \,\gt\, 0]; and let S be the set of finite intersections of elements of B. Then there exists a unique topology on E for which S is a fundamental system of neighbourhoods of 0. This topology is uniformizable since it is equivalent to that derived from the family of translation-invariant pseudo-distances [({\bf x},{ \bf y}) \,\longmapsto\] [ \sigma ({\bf x} - {\bf y})]. It is metrizable if and only if it can be constructed by the above procedure with Σ a countable set of semi-norms. If furthermore E is complete, E is called a Fréchet space.

If E is a topological vector space over [{\bb C}], its dual [E^{*}] is the set of all linear mappings from E to [{\bb C}] (which are also called linear forms, or linear functionals, over E). The subspace of [E^{*}] consisting of all linear forms which are continuous for the topology of E is called the topological dual of E and is denoted E′. If the topology on E is metrizable, then the continuity of a linear form [T \in E'] at [f \in E] can be ascertained by means of sequences, i.e. by checking that the sequence [[T(\,f_{j})]] of complex numbers converges to [T(\,f)] in [{\bb C}] whenever the sequence [(\,f_{j})] converges to f in E.

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