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.Template:Sfn All Brauner spaces are stereotype and are in the stereotype duality relations with Fréchet spaces:Template:SfnTemplate:Sfn

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

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

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

See also

• Stereotype space
• Smith space

References

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