Fréchet space

A Fréchet space is considered in the mathematical branch of functional analysis. It is a topological vector space with special properties that characterize him as a generalization of the Banach space. Named is the space after the French mathematician Maurice René Fréchet.

The main representatives of Fréchet spaces are vector spaces of smooth functions. These rooms have no meaningful Banachraumtopologien: You can indeed equip with any other vector space topology, as defined topological vector spaces are not generally completely.

Definition

A Fréchet space is a Hausdorff shear, and complete locally convex topological vector space with a countable base of neighborhoods.

An equivalent property to the possession of a countable base of neighborhoods is the Metrisierbarkeit. A Fréchet space has no canonical metric.

Description of the topology of semi-norms

As with any locally convex topological vector space can also be the topology of a Fréchet space are described by a family of semi-norms. The existence of a countable base of neighborhoods ensures that only countably many seminorms generating the topology are necessary.

By means of this countable family of semi-norms, one can define a Fréchet metric in a Fréchet space. That is, the question of the Metrisierbarkeit can even be answered constructively.

Examples

Standard example of non- normalizable Fréchet spaces, the spaces of smooth functions on a compact manifold or on a compact subset of a finite dimensional real vector space. Your locally convex topology is a Fréchet topology in a canonical way.

The most important non- normalizable Fréchet spaces which are relevant in practice, are nuclear spaces. This includes most spaces that arise in the theory of distributions, the spaces of holomorphic functions on an open set or sequence spaces such as the space of rapidly falling number sequences. They have, for example, the Montel property: that is, every bounded set is relatively compact.

Properties

In complete metrizable vector spaces such as Banach spaces and Fréchet spaces, the phrase on the open mapping applies.

Other meanings

A topological space that satisfies the separation axiom T1 is sometimes also called " Fréchet space." To avoid confusion, but the name T is used ₁ - space for such rooms usually.

Swell

• Walter Rudin: Functional Analysis. McGraw- Hill, New York 1991. ISBN 0070542368th

• Locally convex space
• Functional Analysis