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 ${\mathcal {C}}(M)$ 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 ${\mathcal {C}}^{\star }(M)$ of Radon measures with compact support on $M$ 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 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 ${\mathcal {E}}^{\star }(M)$ 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 Fréchet space of all 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 bounded sets in ${\mathcal {O}}(M)$ is a Brauner space.

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

• Let $M\subseteq {\mathbb {C} }^{n}$ be a complex affine algebraic variety. The space ${\mathcal {P}}(M)={\mathbb {C} }[x_{1},...,x_{n}]/\{f\in {\mathbb {C} }[x_{1},...,x_{n}]:\ f{\big |}_{M}=0\}$ of polynomials (or regular functions) on $M$, being endowed with the strongest locally convex topology, becomes a Brauner space. Its stereotype dual space ${\mathcal {P}}^{\star }(M)$ (of currents on $M$) is a Fréchet space. In the special case when $M=G$ is an affine algebraic group, ${\mathcal {P}}^{\star }(G)$ becomes an example of a stereotype group algebra.
• Let $G$ be a compactly generated Stein group. The space ${\mathcal {O}}_{\exp }(G)$ 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. S2CID 115297067.
• 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. arXiv:0806.3205. doi:10.1007/s10958-009-9646-1. S2CID 115153766.