# Brauner space

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In functional analysis and related areas of mathematics 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.

## Examples

• Let $M$ be a $\sigma$ -compact locally compact topological space, and ${\mathcal {C}}(M)$ the space of all 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 measures with compact support in $M$ 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 space of 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 space of 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.
• Let $G$ be a compactly generated Stein group. The space ${\mathcal {O}}_{\exp }(G)$ of holomorphic functions of exponential type on $G$ is a Brauner space with respect to a natural topology.