Fréchet topology

The topological structure (topology) of an -space (a space of type ; cf. also Fréchet space), i.e. a completely metrizable topological vector space. The term was introduced by S. Banach in honour of M. Fréchet. Many authors, however, demand additionally local convexity. A complete topological vector space is an -space if and only if it has a countable basis of neighbourhoods of the origin. The topology of an -space can be given by means of an -norm, i.e. a function satisfying:

i) and if and only if ;

ii) for all ;

iii) for each scalar , , and for each , . This means that the (complete) topology of can be given by means of a distance of the form . The completion of any metrizable topological vector space (cf. Completion) is an -space and, consequently, the topology of any metric vector space can be given by means of a translation-invariant distance. Without loss of generality it can be assumed that depends only upon and that the function is non-decreasing for each . If one relaxes condition i) so that can hold for a non-zero , one obtains an -semi-norm. The topology of an arbitrary topological vector space can be given by means of a family of -semi-norms; consequently, every complete topological vector space is an inverse (projective) limit of a directed family of -spaces.

Important classes of -spaces.

Locally convex -spaces.

Such spaces are also called spaces of type (some authors call them just Fréchet spaces, but see Fréchet space). The topology of such a space can be given by means of an increasing sequence of (homogeneous) semi-norms

(a1)

so that a sequence of elements of tends to if and only if for . An -norm giving this topology can be written as

If is a continuous linear operator from a -space to a -space , then for each there are a and a such that , , for all (it is important here that the systems of semi-norms giving, respectively, the topologies of and satisfy (a1)). The dual space of a -space (the space of all continuous linear functionals provided with the topology of uniform convergence on bounded sets) is said to be an -space; it is non-metrizable (unless is a Banach space). Any space of type is an inverse (projective) limit of a sequence of Banach spaces.

Complete locally bounded spaces.

A topological vector space is said to be locally bounded if it has a bounded neighbourhood of the origin (then it has a basis of such neighbourhoods consisting of bounded sets). The topology of such a space is metrizable and can be given by means of a -homogeneous norm, , i.e. an -norm satisfying instead of iii) the more restrictive condition

iiia) for all scalars and all .

For this reason, locally bounded spaces are sometimes called -normed spaces. The class of all Banach spaces is exactly the intersection of the class of locally convex -spaces and the class of complete locally bounded spaces. The dual space of a locally bounded space can be trivial, i.e. consist only of a zero functional.

Locally pseudo-convex -spaces.

They are like -spaces, but with -homogeneous semi-norms instead of homogeneous semi-norms (the exponent may depend upon the semi-norm). This class contains the class of locally convex -spaces and the class of complete locally bounded spaces.

Examples of -spaces.

The space of all Lebesgue-measurable functions on the unit interval with the topology of convergence in measure (asymptotic convergence) is a space of type . Its topology can be given by the -norm

This space is not locally pseudo-convex.

The space of all infinitely differentiable functions on the unit interval with the topology of unform convergence of functions together with all derivatives is a -space. Its topology can be given by semi-norms

The space of all entire functions of one complex variable with the topology of uniform convergence on compact subsets of the complex plane is a -space. Its topology can be given by the semi-norms

The space on the unit interval, , is a complete locally bounded space with trivial dual. Its topology can be given by (its discrete analogue, the space of all sequences summable with the -th power, has a non-trivial dual).

The space with the semi-norms , where , is a locally pseudo-convex space of type which is not locally bounded.

General facts about -spaces.

A linear operator between spaces is continuous if and only if it maps bounded sets onto bounded sets.

Let be a family of continuous linear operators from an -space to an -space space . If the set is bounded in for each fixed , then is equicontinuous (the Mazur–Orlicz theorem; it is a theorem of Banach–Steinhaus type, cf. also Banach–Steinhaus theorem).

If and are -spaces and is a sequence of continuous linear operators from to such that for each the limit exists, then is a continuous linear operator from to .

The image of an open set under a continuous linear operator between -spaces is open (the open mapping theorem).

The graph of a linear operator between -spaces is closed if and only if is continuous (the closed graph theorem).

If a one-to-one continuous linear operator maps an -space onto an -space, then the inverse operator is continuous (the inverse operator theorem).

A separately continuous bilinear mapping between -spaces is jointly continuous (cf. also Continuous function).

References

[a1]	S. Banach, "Théorie des operations lineaires" , Warszawa (1932)
[a2]	N. Bourbaki, "Espaces vectorielles topologiques" , Paris (1981) pp. Chapt. 1–5
[a3]	N. Dunford, J.T. Schwartz, "Linear operators" , I. General theory , Wiley, reprint (1988)
[a4]	A. Grothendieck, "Topological vector spaces" , New York (1973)
[a5]	H. Jarchow, "Locally convex spaces" , Teubner (1981)
[a6]	G. Köthe, "Topological vector spaces" , I–II , New York (1969–1979)
[a7]	S. Rolewicz, "Metric linear spaces" , PWN & Reidel (1972)
[a8]	H.H. Schaefer, "Topological vector spaces" , Springer (1971)
[a9]	L. Waelbroeck, "Topological vector spaces and algebras" , Lecture Notes in Mathematics , 230 , Springer (1971)
[a10]	A. Wilansky, "Modern methods in topological vector spaces" , New York (1978)