# Fréchet room

A Fréchet space 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 . These spaces can indeed with different standards provide, but any standard with respect fully , so no Banach spaces. But one can define a topology on them, so that many theorems that are valid in Banach spaces keep their validity.

## definition

A Fréchet space is a Hausdorffian , locally convex and complete topological vector space with a countable zero neighborhood basis .

An equivalent property to having a countable zero neighborhood base is metrizability . A Fréchet space has no canonical metric .

## 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 . That means, the question of the 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.e. H. any constrained set is relatively compact .

## properties

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

## Other meanings

A topological space that fulfills the axiom of separation T 1 is sometimes also called a “Fréchet space”. In order to avoid confusion, the name T₁ room is usually used for such rooms .