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. K.Brauner (1973).
2. S.S.Akbarov (2003).
3. S.S.Akbarov (2009).
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.

• 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. (subscription required (help)).