Fréchet space

The idea of a nuclear operator can be adapted to Fréchet spaces.

The strong dual of a nuclear Fréchet space is nuclear.

(The term Fréchet space also has an entirely different meaning in functional analysis.

A closed bounded subset of a nuclear Fréchet space is compact.

S(R) is a Fréchet space over the complex numbers.

The set of C functions over D also forms a Fréchet space.

A converse does hold when the domain is pseudometrisable, a case which includes Fréchet spaces.

More generally, every Fréchet space is locally convex.

If furthermore the space is complete, the space is called a Fréchet space.

In particular, any Fréchet space is bornological.

Such a limit of Fréchet spaces is known as an LF space.

Convergence in this Fréchet space is equivalent to element-wise convergence.

Note that this space is infinite-dimensional, and is commonly taken to be a Fréchet space.

However, it is not metrisable, and so it is not a Fréchet space.

More generally, any Nuclear Fréchet space has the Heine-Borel property.

However, seminorms are useful in that they enable us to construct Fréchet spaces, as follows:

Not all vector spaces with complete translation-invariant metrics are Fréchet spaces.

On the other hand, some infinite-dimensional Fréchet spaces do have the Heine-Borel property.

A locally convex F-space is a Fréchet space.

Frequently the Fréchet spaces that arise in practical applications of the derivative enjoy an additional property: they are tame.

Spaces of infinitely differentiable functions are typical examples of Fréchet spaces.

LF-spaces are limits of Fréchet spaces.

This makes each DK a Fréchet space.

Roughly speaking, a tame Fréchet space is one which is almost a Banach space.

A Fréchet space can be defined as a complete locally convex space with a separated countable family of seminorms.