International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 28

The function spaces E of interest in Fourier analysis have an underlying vector space structure over the field 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 ] and scalar multiplication [i.e. the map ] 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 ', which can then be translated by vector addition to construct neighbourhoods of other points .
A norm ν on a vector space E is a nonnegative realvalued function on such thatSubsets of E defined by conditions of the form with 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 translationinvariant distance . 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 seminorm σ on a vector space E is a positive realvalued function on which satisfies (i′) and (iii′) but not (ii′). Given a set Σ of seminorms on E such that any pair (x, y) in is separated by at least one , let B be the set of those subsets of E defined by a condition of the form with and ; 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 translationinvariant pseudodistances . It is metrizable if and only if it can be constructed by the above procedure with Σ a countable set of seminorms. If furthermore E is complete, E is called a Fréchet space.
If E is a topological vector space over , its dual is the set of all linear mappings from E to (which are also called linear forms, or linear functionals, over E). The subspace of 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 at can be ascertained by means of sequences, i.e. by checking that the sequence of complex numbers converges to in whenever the sequence converges to f in E.