Brauner space

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

Brauner spaces are named after Kalman George Brauner, who began their study. All Brauner spaces are stereotype and are in the stereotype duality relations with Fréchet spaces:
 * for any Fréchet space $$X$$ its stereotype dual space $$X^\star$$ is a Brauner space,
 * and vice versa, for any Brauner space $$X$$ its stereotype dual space $$X^\star$$ is a Fréchet space.

Special cases of Brauner spaces are Smith spaces.

Examples

 * Let $$M$$ be a $$\sigma$$-compact locally compact topological space, and $${\mathcal C}(M)$$ the Fréchet space of all continuous functions on $$M$$ (with values in $${\mathbb R}$$ or $${\mathbb C}$$), endowed with the usual topology of uniform convergence on compact sets in $$M$$. The dual space $${\mathcal C}^\star(M)$$ of Radon measures with compact support on $$M$$ with the topology of uniform convergence on compact sets in $${\mathcal C}(M)$$ is a Brauner space.
 * Let $$M$$ be a smooth manifold, and $${\mathcal E}(M)$$ the Fréchet space of all smooth functions on $$M$$ (with values in $${\mathbb R}$$ or $${\mathbb C}$$), endowed with the usual topology of uniform convergence with each derivative on compact sets in $$M$$. The dual space $${\mathcal E}^\star(M)$$ of distributions with compact support in $$M$$ with the topology of uniform convergence on bounded sets in $${\mathcal E}(M)$$ is a Brauner space.
 * Let $$M$$ be a Stein manifold and $${\mathcal O}(M)$$ the Fréchet space of all holomorphic functions on $$M$$ with the usual topology of uniform convergence on compact sets in $$M$$. The dual space $${\mathcal O}^\star(M)$$ of analytic functionals on $$M$$ with the topology of uniform convergence on bounded sets in $${\mathcal O}(M)$$ is a Brauner space.

In the special case when $$M=G$$ possesses a structure of a topological group the spaces $${\mathcal C}^\star(G)$$, $${\mathcal E}^\star(G)$$, $${\mathcal O}^\star(G)$$ become natural examples of stereotype group algebras.
 * Let $$M\subseteq{\mathbb C}^n$$ be a complex affine algebraic variety. The space $${\mathcal P}(M)={\mathbb C}[x_1,...,x_n]/\{f\in {\mathbb C}[x_1,...,x_n]:\ f\big|_M=0\}$$ of polynomials (or regular functions) on $$M$$, being endowed with the strongest locally convex topology, becomes a Brauner space. Its stereotype dual space $${\mathcal P}^\star(M)$$ (of currents on $$M$$) is a Fréchet space. In the special case when $$M=G$$ is an affine algebraic group, $${\mathcal P}^\star(G)$$ becomes an example of a stereotype group algebra.
 * Let $$G$$ be a compactly generated Stein group. The space $${\mathcal O}_{\exp}(G)$$ of all holomorphic functions of exponential type on $$G$$ is a Brauner space with respect to a natural topology.