Brauner space
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In functional analysis and related areas of mathematics Brauner space is a complete compactly generated locally convex space having a sequence of compact sets such that every other compact set is contained in some .
Brauner spaces are named after Kalman Brauner,[1] who first started to study them. All Brauner spaces are stereotype and are in the stereotype duality relations with Fréchet spaces:[2][3]
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- for any Fréchet space its stereotype dual space[4] is a Brauner space,
- and vice versa, for any Brauner space its stereotype dual space is a Fréchet space.
Examples
- Let be a -compact locally compact topological space, and the space of all functions on (with values in or ), endowed with the usual topology of uniform convergence on compact sets in . The dual space of measures with compact support in with the topology of uniform convergence on compact sets in is a Brauner space.
- Let be a smooth manifold, and the space of smooth functions on (with values in or ), endowed with the usual topology of uniform convergence with each derivative on compact sets in . The dual space of distributions with compact support in with the topology of uniform convergence on bounded sets in is a Brauner space.
- Let be a Stein manifold and the space of holomorphic functions on with the usual topology of uniform convergence on compact sets in . The dual space of analytic functionals on with the topology of uniform convergence on biunded sets in is a Brauner space.
- Let be a compactly generated Stein group. The space of holomorphic functions of exponential type on is a Brauner space with respect to a natural topology.[3]
Notes
- ↑ K.Brauner (1973).
- ↑ S.S.Akbarov (2003).
- ↑ 3.0 3.1 S.S.Akbarov (2009).
- ↑ The stereotype dual space to a locally convex space is the space of all linear continuous functionals endowed with the topology of uniform convergence on totally bounded sets in .
References
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