Английская Википедия:Brauner space
In functional analysis and related areas of mathematics a Brauner space is a complete compactly generated locally convex space <math>X</math> having a sequence of compact sets <math>K_n</math> such that every other compact set <math>T\subseteq X</math> is contained in some <math>K_n</math>.
Brauner spaces are named after Kalman George Brauner, who began their study.Шаблон:Sfn All Brauner spaces are stereotype and are in the stereotype duality relations with Fréchet spaces:Шаблон:SfnШаблон:Sfn
- for any Fréchet space <math>X</math> its stereotype dual space[1] <math>X^\star</math> is a Brauner space,
- and vice versa, for any Brauner space <math>X</math> its stereotype dual space <math>X^\star</math> is a Fréchet space.
Special cases of Brauner spaces are Smith spaces.
Examples
- Let <math>M</math> be a <math>\sigma</math>-compact locally compact topological space, and <math>{\mathcal C}(M)</math> the Fréchet space of all continuous functions on <math>M</math> (with values in <math>{\mathbb R}</math> or <math>{\mathbb C}</math>), endowed with the usual topology of uniform convergence on compact sets in <math>M</math>. The dual space <math>{\mathcal C}^\star(M)</math> of Radon measures with compact support on <math>M</math> with the topology of uniform convergence on compact sets in <math>{\mathcal C}(M)</math> is a Brauner space.
- Let <math>M</math> be a smooth manifold, and <math>{\mathcal E}(M)</math> the Fréchet space of all smooth functions on <math>M</math> (with values in <math>{\mathbb R}</math> or <math>{\mathbb C}</math>), endowed with the usual topology of uniform convergence with each derivative on compact sets in <math>M</math>. The dual space <math>{\mathcal E}^\star(M)</math> of distributions with compact support in <math>M</math> with the topology of uniform convergence on bounded sets in <math>{\mathcal E}(M)</math> is a Brauner space.
- Let <math>M</math> be a Stein manifold and <math>{\mathcal O}(M)</math> the Fréchet space of all holomorphic functions on <math>M</math> with the usual topology of uniform convergence on compact sets in <math>M</math>. The dual space <math>{\mathcal O}^\star(M)</math> of analytic functionals on <math>M</math> with the topology of uniform convergence on bounded sets in <math>{\mathcal O}(M)</math> is a Brauner space.
In the special case when <math>M=G</math> possesses a structure of a topological group the spaces <math>{\mathcal C}^\star(G)</math>, <math>{\mathcal E}^\star(G)</math>, <math>{\mathcal O}^\star(G)</math> become natural examples of stereotype group algebras.
- Let <math>M\subseteq{\mathbb C}^n</math> be a complex affine algebraic variety. The space <math>{\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>M</math>, being endowed with the strongest locally convex topology, becomes a Brauner space. Its stereotype dual space <math>{\mathcal P}^\star(M)</math> (of currents on <math>M</math>) is a Fréchet space. In the special case when <math>M=G</math> is an affine algebraic group, <math>{\mathcal P}^\star(G)</math> becomes an example of a stereotype group algebra.
- Let <math>G</math> be a compactly generated Stein group.[2] The space <math>{\mathcal O}_{\exp}(G)</math> of all holomorphic functions of exponential type on <math>G</math> is a Brauner space with respect to a natural topology.Шаблон:Sfn
See also
Notes
References
Шаблон:Functional analysis Шаблон:Topological vector spaces
- ↑ The stereotype dual space to a locally convex space <math>X</math> is the space <math>X^\star</math> of all linear continuous functionals <math>f:X\to\mathbb{C}</math> endowed with the topology of uniform convergence on totally bounded sets in <math>X</math>.
- ↑ I.e. a Stein manifold which is at the same time a topological group.
