Brauner space

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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\subseteq X is contained in some K_n.

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 X its stereotype dual space[4] X^\star is a Brauner space,
  • and vice versa, for any Brauner space X its stereotype dual space X^\star is a Fréchet space.

Examples[edit]

  • Let M be a \sigma-compact locally compact topological space, and {\mathcal C}(M) the space of all functions on M (with values in {\mathbb R} or {\mathbb C}), endowed with the usual topology of uniform convergence on compact sets in M. The dual space {\mathcal C}^\star(M) of measures with compact support in M with the topology of uniform convergence on compact sets in {\mathcal C}(M) is a Brauner space.
  • Let M be a smooth manifold, and {\mathcal E}(M) the space of smooth functions on M (with values in {\mathbb R} or {\mathbb C}), endowed with the usual topology of uniform convergence with each derivative on compact sets in M. The dual space {\mathcal E}^\star(M) of distributions with compact support in M with the topology of uniform convergence on bounded sets in {\mathcal E}(M) is a Brauner space.
  • Let M be a Stein manifold and {\mathcal O}(M) the space of holomorphic functions on M with the usual topology of uniform convergence on compact sets in M. The dual space {\mathcal O}^\star(M) of analytic functionals on M with the topology of uniform convergence on biunded sets in {\mathcal O}(M) is a Brauner space.
  • Let G be a compactly generated Stein group. The space {\mathcal O}_{\exp}(G) of holomorphic functions of exponential type on G 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 X is the space X^\star of all linear continuous functionals f:X\to\mathbb{C} endowed with the topology of uniform convergence on totally bounded sets in X.

References[edit]

  • Schaefer, Helmuth H. (1966). Topological vector spaces. New York: The MacMillan Company. ISBN 0-387-98726-6. 
  • Robertson, A.P.; Robertson, W.J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics 53. Cambridge University Press. 
  • Brauner, K. (1973). "Duals of Frechet spaces and a generalization of the Banach-Dieudonne theorem". Duke Math. Jour. 40 (4): 845–855. doi:10.1215/S0012-7094-73-04078-7.