Brauner space

In functional analysis and related areas of mathematics a 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, who began their study. All Brauner spaces are stereotype and are in the stereotype duality relations with Fréchet spaces:
Special cases of Brauner spaces are Smith spaces.

Examples

Let be a -compact locally compact topological space, and the Fréchet space of all continuous functions on, endowed with the usual topology of uniform convergence on compact sets in. The dual space of Radon measures with compact support on with the topology of uniform convergence on compact sets in is a Brauner space.
Let be a smooth manifold, and the Fréchet space of all smooth functions on, 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 Fréchet space of all 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 bounded sets in is a Brauner space.

In the special case when possesses a structure of a topological group the spaces,, become natural examples of stereotype group algebras.

Let be a complex affine algebraic variety. The space of polynomials on, being endowed with the strongest locally convex topology, becomes a Brauner space. Its stereotype dual space is a Fréchet space. In the special case when is an affine algebraic group, becomes an example of a stereotype group algebra.
Let be a compactly generated Stein group. The space of all holomorphic functions of exponential type on is a Brauner space with respect to a natural topology.