# Montel room

The mathematical term **Montel space** describes a special class of locally convex spaces . They carry its name from the Montel's theorem from the theory of functions . Many locally convex spaces from the theory of distributions are Montel spaces.

## definition

A locally convex space is called a **Montel space** if it is quasitonneliert and the closure of each bounded set is compact .

## Examples

- A normalized space is a Montel space if and only if it is finite-dimensional.
- If is a domain and is the space of the holomorphic functions on G with the semi-norms , where the compact subsets of G runs through, then by Montel's theorem each in bounded set has a compact closure. Since the Fréchet room is also quasi-tiled , it turns out to be a Montel room.
- Be open and the space of any number of differentiable functions with the semi-norms , that's a Montel space. It was for the multiindex used notation.
- Be open and the subspace of any number of differentiable functions with a compact carrier in . For compact, let the space of the functions with carrier in K with the subspace topology induced by . Then there is a finest locally convex topology that makes all embeddings continuous. with this topology is the
*space of the**test functions*and is an example of a non- metrizable Montel space. - Be the space of all functions for which all suprema are finite. The multi-index notation was used again. The room with the semi-norms is called the room of rapidly falling functions and is a Montel room.
- Complete, quasi-tiled Schwartz rooms are Montel rooms.
- Every locally convex space with the
*finest locally convex topology*, that is to say with the topology generated by all absolutely convex , absorbing sets as a zero neighborhood basis , is a Montel space.

## Properties of Montel rooms

- Montel rooms are reflexive and therefore barreled .
- Montel spaces are
*quasi-complete*; H. every bounded Cauchy network converges. There are incomplete Montel rooms. - Direct products (with the product topology ) and direct sums (with the final topology ) of Montel spaces are again Montel spaces.
- In general, neither closed subspaces nor quotients of Montel spaces are again Montel spaces.
- If E is a Montel space, so is the strong dual space E '. So in particular, those in the distribution theory spaces occur , and Montel spaces.

## swell

- Klaus Floret, Joseph Wloka :
*Introduction to the theory of locally convex spaces*(=*Lecture Notes in Mathematics.*Vol. 56, ISSN 0075-8434 ). Springer, Berlin et al. 1968, doi : 10.1007 / BFb0098549 . - HH Schaefer: Topological Vector Spaces, Springer, 1971 ISBN 0-387-98726-6
- H. Jarchow: Locally Convex Spaces, Teubner, Stuttgart 1981 ISBN 3-519-02224-9
- R. Meise, D. Vogt: Introduction to Functional Analysis, Vieweg, 1992 ISBN 3-528-07262-8