Topology in function spaces

Bricogne, G.

doi:10.1107/97809553602060000551

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

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 28 | 1 | 2 |

Section 1.3.2.2.6. Topology in function spaces

G. Bricogne^a

^a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.2.2.6. Topology in function spaces

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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 infinite-dimensional, 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.

1.3.2.2.6.1. General topology

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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 $[E \times E]$ to the non-negative reals which satisfies: $[\matrix{(\hbox{i})\hfill & d(x, y) = d(y, x)\hfill &\forall x, y \in E\hfill &\hbox{(symmetry);}\hfill\cr\cr (\hbox{ii})\hfill &d(x, y) = 0 \hfill &\hbox{iff } x = y\hfill &\hbox{(separation);}\hfill\cr\cr (\hbox{iii})\hfill & d(x, z) \leq d(x, y) + d(y, z)\hfill &\forall x, y, z \in E\hfill &\hbox{(triangular}\hfill\cr& & &\hbox{inequality).}\hfill}]$ 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 $[{\scr O}(E)]$ of open subsets of E: two distance functions leading to the same $[{\scr O}(E)]$ lead to identical topological properties. An axiomatic reformulation of topological notions is thus possible: a topology in E is a collection $[{\scr O}(E)]$ 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 non-metrizable 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 pseudo-distances $[(d_{\alpha})_{\alpha \in A}]$ , where each $[d_{\alpha}]$ satisfies (i) and (iii) but not (ii). Such spaces are called uniformizable. If for every pair $[(x, y) \in E \times E]$ there exists $[\alpha \in A]$ such that $[d_{\alpha} (x, y) \neq 0]$ , then the separation property can be recovered. If furthermore a countable subfamily of the $[d_{\alpha}]$ 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.

1.3.2.2.6.2. 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 $[\displaylines{\quad (\hbox{i}')\;\;\quad\nu (\lambda {\bf x}) = |\lambda | \nu ({\bf x}) \phantom{|\lambda | v ({\bf x} =i} \hbox{for all } \lambda \in {\bb C} \hbox{ and } {\bf x} \in E\hbox{;}\hfill\cr \quad (\hbox{ii}')\;\quad\nu ({\bf x}) = 0 \phantom{|\lambda | v ({\bf x} = |\lambda | vxxx}\; \hbox{if and only if } {\bf x} = {\bf 0}\hbox{;}\hfill\cr \quad (\hbox{iii}')\quad \nu ({\bf x} + {\bf y}) \leq \nu ({\bf x}) + \nu ({\bf y}) \quad \hbox{for all } {\bf x},{\bf y} \in E.\hfill}]$ 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.

References

Dieudonné, J. (1969). Foundations of modern analysis. New York and London: Academic Press.Google Scholar

Dieudonné, J. (1970). Treatise on analysis, Vol. II. New York and London: Academic Press.Google Scholar

Friedman, A. (1970). Foundations of modern analysis. New York: Holt, Rinehart & Winston. [Reprinted by Dover, New York, 1982.]Google Scholar

Trèves, F. (1967). Topological vector spaces, distributions, and kernels. New York and London: Academic Press.Google Scholar

Yosida, K. (1965). Functional analysis. Berlin: Springer-Verlag.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 28