Brauner space - Alchetron, The Free Social Encyclopedia

In functional analysis and related areas of mathematics 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 ⊆ X is contained in some K n .

Brauner spaces are named after Kalman Brauner, who first started to study them. All Brauner spaces are stereotype and are in the stereotype duality relations with Fréchet spaces:

for any Fréchet space X its stereotype dual space X ⋆ is a Brauner space,

and vice versa, for any Brauner space X its stereotype dual space X ⋆ is a Fréchet space.

Examples

Let M be a σ -compact locally compact topological space, and C ( M ) the space of all 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 ⋆ ( M ) of measures with compact support in 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 space of 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 ⋆ ( 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 space of holomorphic functions on M with the usual topology of uniform convergence on compact sets in M . The dual space O ⋆ ( M ) of analytic functionals on M with the topology of uniform convergence on biunded sets in O ( M ) is a Brauner space.

Let G be a compactly generated Stein group. The space O exp ( G ) of holomorphic functions of exponential type on G is a Brauner space with respect to a natural topology.

References

Brauner space Wikipedia

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