Brauner space
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.
- for any 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).
- 1 2 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
- 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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