What is the significance of the Frechet room

Fréchet room

A Fréchet room is considered in the mathematical sub-area of ​​functional analysis. It is a topological vector space with special properties that characterize it as a generalization of the Banach space. The room is named after the French mathematician Maurice René Fréchet.

The main representatives of Fréchet spaces are vector spaces of smooth functions. Although these rooms can be equipped with different standards, they are not complete with regard to any standard, i.e. no Banach rooms. But one can define a topology on them, so that many theorems that are valid in Banach spaces keep their validity.

definition

Description of the topology using semi-standards

As with any locally convex topological vector space, the topology of a Fréchet space can also be described by a family of semi-norms. The existence of a countable zero environment base guarantees that only a countable number of semi-norms are necessary to generate the topology.

Using this countable family of semi-norms, one can define a Fréchet metric in a Fréchet space. This means that the question of metrisability can even be answered constructively.

Examples

Every Banach room is a Fréchet room.

Standard examples for non-normalizable Fréchet spaces are the spaces of smooth functions on a compact manifold or on a compact subset of a finite-dimensional real vector space. Its locally convex topology is canonically a Fréchet topology.

The most important non-normalizable Fréchet spaces that are relevant in practice are nuclear spaces. This includes most of the spaces that appear 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. You have e.g. B. the Montel property, i. H. any constrained set is relatively compact.

properties

In completely metrizable vector spaces such as Banach spaces or Fréchet spaces, the theorem about the open mapping applies.

Other meanings

A topological space that follows the axiom of separation T1 is sometimes also called the “Fréchet room”. In order to avoid confusion, the name T₁ room is usually used for such rooms.

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