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, pp. 2728

Geometric intuition, which often makes `obvious' the topological properties of the real line and of ordinary space, cannot be relied upon in the study of function spaces: the latter are infinitedimensional, and several inequivalent notions of convergence may exist. A careful analysis of topological concepts and of their interrelationship is thus a necessary prerequisite to the study of these spaces. The reader may consult Dieudonné (1969, 1970), Friedman (1970), Trèves (1967) and Yosida (1965) for detailed expositions.
Most topological notions are first encountered in the setting of metric spaces. A metric space E is a set equipped with a distance function d from to the nonnegative reals which satisfies:By means of d, the following notions can be defined: open balls, neighbourhoods; open and closed sets, interior and closure; convergence of sequences, continuity of mappings; Cauchy sequences and completeness; compactness; connectedness. They suffice for the investigation of a great number of questions in analysis and geometry (see e.g. Dieudonné, 1969).
Many of these notions turn out to depend only on the properties of the collection of open subsets of E: two distance functions leading to the same lead to identical topological properties. An axiomatic reformulation of topological notions is thus possible: a topology in E is a collection of subsets of E which satisfy suitable axioms and are deemed open irrespective of the way they are obtained. From the practical standpoint, however, a topology which can be obtained from a distance function (called a metrizable topology) has the very useful property that the notions of closure, limit and continuity may be defined by means of sequences. For nonmetrizable topologies, these notions are much more difficult to handle, requiring the use of `filters' instead of sequences.
In some spaces E, a topology may be most naturally defined by a family of pseudodistances , where each satisfies (i) and (iii) but not (ii). Such spaces are called uniformizable. If for every pair there exists such that , then the separation property can be recovered. If furthermore a countable subfamily of the suffices to define the topology of E, the latter can be shown to be metrizable, so that limiting processes in E may be studied by means of sequences.
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.
References
Dieudonné, J. (1969). Foundations of Modern Analysis. New York, London: Academic Press.Dieudonné, J. (1970). Treatise on Analysis, Vol. II. New York, London: Academic Press.
Friedman, A. (1970). Foundations of Modern Analysis. New York: Holt, Rinehart & Winston. [Reprinted by Dover, New York, 1982.]
Trèves, F. (1967). Topological Vector Spaces, Distributions, and Kernels. New York, London: Academic Press.
Yosida, K. (1965). Functional Analysis. Berlin: SpringerVerlag.