# Brauner space

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 George Brauner, who began their study^{[1]}. All Brauner spaces are stereotype and are in the stereotype duality relations with Fréchet spaces:^{[2]}^{[3]}

- for any Fréchet space its stereotype dual space
^{[4]}is a Brauner space, - and vice versa, for any Brauner space its stereotype dual space is a Fréchet space.

- for any Fréchet space its stereotype dual space

## Contents

## Examples[edit]

- 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.
^{[5]}

## See also[edit]

## Notes[edit]

**^**Brauner 1973.**^**Akbarov 2003, p. 220.**^**Akbarov 2009, p. 466.**^**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 .**^**Akbarov 2009, p. 525.

## References[edit]

- Brauner, K. (1973). "Duals of Frechet spaces and a generalization of the Banach-Dieudonne theorem".
*Duke Math. Jour*.**40**(4): 845–855. doi:10.1215/S0012-7094-73-04078-7. - Akbarov, S.S. (2003). "Pontryagin duality in the theory of topological vector spaces and in topological algebra".
*Journal of Mathematical Sciences*.**113**(2): 179–349. doi:10.1023/A:1020929201133. - Akbarov, S.S. (2009). "Holomorphic functions of exponential type and duality for Stein groups with algebraic connected component of identity".
*Journal of Mathematical Sciences*.**162**(4): 459–586. arXiv:0806.3205 . doi:10.1007/s10958-009-9646-1. (Subscription required (help)).

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