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]

See also

• Stereotype space
• Smith space

References

1. Brauner 1973.
2. Akbarov 2003, p. 220.
3. Akbarov 2009, p. 466.
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 ${\displaystyle X}$.
5. I.e. a Stein manifold which is at the same time a topological group.
6. Akbarov 2009, p. 525.

• 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.