# Brauner space

In functional analysis and related areas of mathematics 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 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 ${\displaystyle X}$ 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.

## Examples

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

## References

• Schaefer, Helmuth H. (1966). Topological vector spaces. New York: The MacMillan Company. ISBN 0-387-98726-6.