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 ${𝒞}(M)$ 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 ${{𝒞}}^{\star }(M)$ of Radon measures with compact support on $M$ with the topology of uniform convergence on compact sets in ${𝒞}(M)$ is a Brauner space.
• Let $M$ be a smooth manifold, and ${ℰ}(M)$ 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 ${{ℰ}}^{\star }(M)$ of distributions with compact support in $M$ with the topology of uniform convergence on bounded sets in ${ℰ}(M)$ is a Brauner space.
• Let $M$ be a Stein manifold and ${𝒪}(M)$ the Fréchet space of all holomorphic functions on $M$ with the usual topology of uniform convergence on compact sets in $M$. The dual space ${{𝒪}}^{\star }(M)$ of analytic functionals on $M$ with the topology of uniform convergence on bounded sets in ${𝒪}(M)$ is a Brauner space.

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

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

See also

• Stereotype space
• Smith space

References

• Brauner, K. (1973). "Duals of Fréchet spaces and a generalization of the Banach-Dieudonné theorem". Duke Mathematical Journal 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. doi:10.1007/s10958-009-9646-1.