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Brauner space
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
Let M be a \sigma-compact locally compact topological space, and \({\mathcal C}(M) t\)he 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
K.Brauner (1973).
S.S.Akbarov (2003).
S.S.Akbarov (2009).
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
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.
Akbarov, S.S. (2003). "Pontryagin duality in the theory of topological vector spaces and in topological algebra". Journal of Mathematical Sciences 113 (2): 179–349. doi:10.1023/A:1020929201133.
Akbarov, S.S. (2009). "Holomorphic functions of exponential type and duality for Stein groups with algebraic connected component of identity". Journal of Mathematical Sciences 162 (4): 459–586. doi:10.1007/s10958-009-9646-1. (subscription required (help)).
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