Brauner space explained

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^[1] 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 { l 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 { l C} ^{\star(M)} of Radon measures with compact support on M with the topology of uniform convergence on compact sets in { l C}(M) is a Brauner space.

• Let

M be a smooth manifold, and { l 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 { l E} ^{\star(M)} of distributions with compact support in M with the topology of uniform convergence on bounded sets in { l E}(M) is a Brauner space.

• Let

M be a Stein manifold and { l 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 { l O} ^{\star(M)} of analytic functionals on M with the topology of uniform convergence on bounded sets in { l O}(M) is a Brauner space.

In the special case when

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

• Let

M\subseteq{ C} ^{n} be a complex affine algebraic variety. The space { l P}(M) = { C}[x _{1,...,x} _{n]/\{f\in} { C}[x _{1,...,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 { l P} ^{\star(M)} (of currents on M ) is a Fréchet space. In the special case when M = G is an affine algebraic group, { l P} ^{\star(G)} becomes an example of a stereotype group algebra.

• Let

G be a compactly generated Stein group.^[2] The space { l 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.. Duals of Fréchet spaces and a generalization of the Banach-Dieudonné theorem. Duke Mathematical Journal. 1973. 40. 4. 845–855. 10.1215/S0012-7094-73-04078-7.
• Akbarov. S.S.. Pontryagin duality in the theory of topological vector spaces and in topological algebra. Journal of Mathematical Sciences. 2003. 113. 2. 179–349. 10.1023/A:1020929201133. 115297067. free.
• Akbarov. S.S.. Holomorphic functions of exponential type and duality for Stein groups with algebraic connected component of identity. Journal of Mathematical Sciences. 2009. 162. 4. 459–586. 0806.3205. 10.1007/s10958-009-9646-1. 115153766.

Notes and References

1. The stereotype dual space to a locally convex space X is the space X ^{\star} of all linear continuous functionals f:X\to C endowed with the topology of uniform convergence on totally bounded sets in X .
2. I.e. a Stein manifold which is at the same time a topological group.