# Brauner space

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]

## Notes

1. Akbarov 2003, p. 220.
2. Akbarov 2009, p. 466.
3. The stereotype dual space to a locally convex space $\displaystyle{ X }$ is the space $\displaystyle{ X^\star }$ of all linear continuous functionals $\displaystyle{ f:X\to\mathbb{C} }$ endowed with the topology of uniform convergence on totally bounded sets in $\displaystyle{ X }$.
4. I.e. a Stein manifold which is at the same time a topological group.
5. Akbarov 2009, p. 525.

## 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.