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

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 ${\displaystyle X}$ its stereotype dual space^[4] ${\displaystyle X^{\star ))$ is a Brauner space,
• and vice versa, for any Brauner space ${\displaystyle X}$ its stereotype dual space ${\displaystyle X^{\star ))$ is a Fréchet space.

Special cases of Brauner spaces are Smith spaces.

Examples

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

In the special case when ${\displaystyle M=G}$ possesses a structure of a topological group the spaces ${\displaystyle {\mathcal {C))^{\star }(G)}$, ${\displaystyle {\mathcal {E))^{\star }(G)}$, ${\displaystyle {\mathcal {O))^{\star }(G)}$ become natural examples of stereotype group algebras.

• Let ${\displaystyle M\subseteq {\mathbb {C} }^{n))$ be a complex affine algebraic variety. The space ${\displaystyle {\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 ${\displaystyle M}$, being endowed with the strongest locally convex topology, becomes a Brauner space. Its stereotype dual space ${\displaystyle {\mathcal {P))^{\star }(M)}$ (of currents on ${\displaystyle M}$) is a Fréchet space. In the special case when ${\displaystyle M=G}$ is an affine algebraic group, ${\displaystyle {\mathcal {P))^{\star }(G)}$ becomes an example of a stereotype group algebra.
• Let ${\displaystyle G}$ be a compactly generated Stein group.^[5] The space ${\displaystyle {\mathcal {O))_{\exp }(G)}$ of all holomorphic functions of exponential type on ${\displaystyle G}$ is a Brauner space with respect to a natural topology.^[6]