# Brauner space

Jump to navigation
Jump to search

In functional analysis and related areas of mathematics **Brauner space** is a complete compactly generated locally convex space having a sequence of compact sets such that every other compact set is contained in some .

Brauner spaces are named after Kalman Brauner,^{[1]} who first started to study them. All Brauner spaces are stereotype and are in the stereotype duality relations with Fréchet spaces:^{[2]}^{[3]}

## Examples

- Let be a -compact locally compact topological space, and the space of all functions on (with values in or ), endowed with the usual topology of uniform convergence on compact sets in . The dual space of measures with compact support in with the topology of uniform convergence on compact sets in is a Brauner space.

- Let be a smooth manifold, and the space of smooth functions on (with values in or ), endowed with the usual topology of uniform convergence with each derivative on compact sets in . The dual space of distributions with compact support in with the topology of uniform convergence on bounded sets in is a Brauner space.

- Let be a Stein manifold and the space of holomorphic functions on with the usual topology of uniform convergence on compact sets in . The dual space of analytic functionals on with the topology of uniform convergence on biunded sets in is a Brauner space.

- Let be a compactly generated Stein group. The space of holomorphic functions of exponential type on is a Brauner space with respect to a natural topology.
^{[3]}

## Notes

- ↑ Template:Harvtxt.
- ↑ Template:Harvtxt.
- ↑
^{3.0}^{3.1}Template:Harvtxt. Cite error: Invalid`<ref>`

tag; name "Akbarov-2" defined multiple times with different content - ↑ The
*stereotype dual*space to a locally convex space is the space of all linear continuous functionals endowed with the topology of uniform convergence on totally bounded sets in .

## References

- {{#invoke:citation/CS1|citation

|CitationClass=book }}

- {{#invoke:citation/CS1|citation

|CitationClass=book }}

- {{#invoke:Citation/CS1|citation

|CitationClass=journal }}

- {{#invoke:Citation/CS1|citation

|CitationClass=journal }}

- {{#invoke:Citation/CS1|citation

|CitationClass=journal }}