Brauner space

In functional analysis and related areas of mathematics a Brauner space is a complete compactly generated locally convex space [math]\displaystyle{ X }[/math] having a sequence of compact sets [math]\displaystyle{ K_n }[/math] such that every other compact set [math]\displaystyle{ T\subseteq X }[/math] is contained in some [math]\displaystyle{ K_n }[/math]. 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 [math]\displaystyle{ X }[/math] its stereotype dual space [math]\displaystyle{ X^\star }[/math] is a Brauner space,
• and vice versa, for any Brauner space [math]\displaystyle{ X }[/math] its stereotype dual space [math]\displaystyle{ X^\star }[/math] is a Fréchet space.

Special cases of Brauner spaces are Smith spaces.

Examples

• Let [math]\displaystyle{ M }[/math] be a [math]\displaystyle{ \sigma }[/math]-compact locally compact topological space, and [math]\displaystyle{ {\mathcal C}(M) }[/math] the Fréchet space of all continuous functions on [math]\displaystyle{ M }[/math] (with values in [math]\displaystyle{ {\mathbb R} }[/math] or [math]\displaystyle{ {\mathbb C} }[/math]), endowed with the usual topology of uniform convergence on compact sets in [math]\displaystyle{ M }[/math]. The dual space [math]\displaystyle{ {\mathcal C}^\star(M) }[/math] of Radon measures with compact support on [math]\displaystyle{ M }[/math] with the topology of uniform convergence on compact sets in [math]\displaystyle{ {\mathcal C}(M) }[/math] is a Brauner space.
• Let [math]\displaystyle{ M }[/math] be a smooth manifold, and [math]\displaystyle{ {\mathcal E}(M) }[/math] the Fréchet space of all smooth functions on [math]\displaystyle{ M }[/math] (with values in [math]\displaystyle{ {\mathbb R} }[/math] or [math]\displaystyle{ {\mathbb C} }[/math]), endowed with the usual topology of uniform convergence with each derivative on compact sets in [math]\displaystyle{ M }[/math]. The dual space [math]\displaystyle{ {\mathcal E}^\star(M) }[/math] of distributions with compact support in [math]\displaystyle{ M }[/math] with the topology of uniform convergence on bounded sets in [math]\displaystyle{ {\mathcal E}(M) }[/math] is a Brauner space.
• Let [math]\displaystyle{ M }[/math] be a Stein manifold and [math]\displaystyle{ {\mathcal O}(M) }[/math] the Fréchet space of all holomorphic functions on [math]\displaystyle{ M }[/math] with the usual topology of uniform convergence on compact sets in [math]\displaystyle{ M }[/math]. The dual space [math]\displaystyle{ {\mathcal O}^\star(M) }[/math] of analytic functionals on [math]\displaystyle{ M }[/math] with the topology of uniform convergence on bounded sets in [math]\displaystyle{ {\mathcal O}(M) }[/math] is a Brauner space.

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

• Let [math]\displaystyle{ M\subseteq{\mathbb C}^n }[/math] be a complex affine algebraic variety. The space [math]\displaystyle{ {\mathcal P}(M)={\mathbb C}[x_1,...,x_n]/\{f\in {\mathbb C}[x_1,...,x_n]:\ f\big|_M=0\} }[/math] of polynomials (or regular functions) on [math]\displaystyle{ M }[/math], being endowed with the strongest locally convex topology, becomes a Brauner space. Its stereotype dual space [math]\displaystyle{ {\mathcal P}^\star(M) }[/math] (of currents on [math]\displaystyle{ M }[/math]) is a Fréchet space. In the special case when [math]\displaystyle{ M=G }[/math] is an affine algebraic group, [math]\displaystyle{ {\mathcal P}^\star(G) }[/math] becomes an example of a stereotype group algebra.
• Let [math]\displaystyle{ G }[/math] be a compactly generated Stein group. The space [math]\displaystyle{ {\mathcal O}_{\exp}(G) }[/math] of all holomorphic functions of exponential type on [math]\displaystyle{ G }[/math] 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. 
• 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.