Brauner space

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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]

  • 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[edit]

  • 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[edit]

  1. ^ K.Brauner (1973).
  2. ^ S.S.Akbarov (2003).
  3. ^ a b S.S.Akbarov (2009).
  4. ^ 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[edit]

  • Schaefer, Helmuth H. (1966). Topological vector spaces. New York: The MacMillan Company. ISBN 0-387-98726-6.