# Brauner space

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 Brauner,[1] who first started to study them. All Brauner spaces are stereotype and are in the stereotype duality relations with Fréchet spaces:[2][3]

• for any Fréchet space $X$ its stereotype dual space[4] $X^\star$ 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 biunded 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.[3]

## Notes

1. ^
2. ^
3. ^ a b
4. ^ The stereotype dual space to a locally convex space $X$ is the space $X^\star$ of all linear continuous functionals $f:X\to\mathbb{C}$ endowed with the topology of uniform convergence on totally bounded sets in $X$.

## References

• Schaefer, Helmuth H. (1966). Topological vector spaces. New York: The MacMillan Company. ISBN 0-387-98726-6.
• Robertson, A.P.; Robertson, W.J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics 53. Cambridge University Press.
• Brauner, K. (1973). "Duals of Frechet spaces and a generalization of the Banach-Dieudonne theorem". Duke Math. Jour. 40 (4): 845–855. doi:10.1215/S0012-7094-73-04078-7.