Topological classification of strong duals to nuclear (LF)-spaces

Taras Banakh

Studia Mathematica (2000)

  • Volume: 138, Issue: 3, page 201-208
  • ISSN: 0039-3223

Abstract

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We show that the strong dual X’ to an infinite-dimensional nuclear (LF)-space is homeomorphic to one of the spaces: ℝ ω , ℝ ∞ , Q × ℝ ∞ , ℝ ω × ℝ ∞ , or ( ℝ ∞ ) ω , where ℝ ∞ = l i m ℝ n and Q = [ - 1 , 1 ] ω . In particular, the Schwartz space D’ of distributions is homeomorphic to ( ℝ ∞ ) ω . As a by-product of the proof we deduce that each infinite-dimensional locally convex space which is a direct limit of metrizable compacta is homeomorphic either to ℝ ∞ or to Q × ℝ ∞ . In particular, the strong dual to any metrizable infinite-dimensional Montel space is homeomorphic either to ℝ ∞ or to Q × ℝ ∞ .

How to cite

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Banakh, Taras. "Topological classification of strong duals to nuclear (LF)-spaces." Studia Mathematica 138.3 (2000): 201-208. <http://eudml.org/doc/216699>.

@article{Banakh2000,
abstract = {We show that the strong dual X’ to an infinite-dimensional nuclear (LF)-space is homeomorphic to one of the spaces: $ℝ^ω$, $ℝ^∞$, $Q×ℝ^∞$, $ℝ^ω×ℝ^∞$, or $(ℝ^∞)^ω$, where $ℝ^∞ = lim ℝ^n$ and $Q=[-1,1]^ω$. In particular, the Schwartz space D’ of distributions is homeomorphic to $(ℝ^∞)^ω$. As a by-product of the proof we deduce that each infinite-dimensional locally convex space which is a direct limit of metrizable compacta is homeomorphic either to $ℝ^∞$ or to $Q×ℝ^∞$. In particular, the strong dual to any metrizable infinite-dimensional Montel space is homeomorphic either to $ℝ^∞$ or to $Q×ℝ^∞$.},
author = {Banakh, Taras},
journal = {Studia Mathematica},
keywords = {dual space; nuclear (LF)-space; Montel space; direct limit; Hilbert cube; nuclear LF space; strong dual space},
language = {eng},
number = {3},
pages = {201-208},
title = {Topological classification of strong duals to nuclear (LF)-spaces},
url = {http://eudml.org/doc/216699},
volume = {138},
year = {2000},
}

TY - JOUR
AU - Banakh, Taras
TI - Topological classification of strong duals to nuclear (LF)-spaces
JO - Studia Mathematica
PY - 2000
VL - 138
IS - 3
SP - 201
EP - 208
AB - We show that the strong dual X’ to an infinite-dimensional nuclear (LF)-space is homeomorphic to one of the spaces: $ℝ^ω$, $ℝ^∞$, $Q×ℝ^∞$, $ℝ^ω×ℝ^∞$, or $(ℝ^∞)^ω$, where $ℝ^∞ = lim ℝ^n$ and $Q=[-1,1]^ω$. In particular, the Schwartz space D’ of distributions is homeomorphic to $(ℝ^∞)^ω$. As a by-product of the proof we deduce that each infinite-dimensional locally convex space which is a direct limit of metrizable compacta is homeomorphic either to $ℝ^∞$ or to $Q×ℝ^∞$. In particular, the strong dual to any metrizable infinite-dimensional Montel space is homeomorphic either to $ℝ^∞$ or to $Q×ℝ^∞$.
LA - eng
KW - dual space; nuclear (LF)-space; Montel space; direct limit; Hilbert cube; nuclear LF space; strong dual space
UR - http://eudml.org/doc/216699
ER -

References

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  1. [An] R. D. Anderson, On topological infinite deficiency, Michigan Math. J. 14 (1967), 365-389. 
  2. [Ba] T. Banakh, On linear topological spaces (linearly) homeomorphic to ℝ ∞ , Mat. Stud. 9 (1998), 99-101. Zbl0932.46002
  3. [BP] C. Bessaga and A. Pełczyński, Selected Topics in Infinite-Dimensional Topology, PWN, Warszawa, 1975. Zbl0304.57001
  4. [Ch] T. A. Chapman, Lectures on Hilbert Cube Manifolds, CBMS Regional Conf. Ser. in Math. 28, Amer. Math. Soc., 1976. 
  5. [Di] J. Dieudonné, Sur les espaces de Montel métrisables, C. R. Acad. Sci. Paris 238 (1954), 194-195. Zbl0055.09801
  6. [En] R. Engelking, General Topology, PWN, Warszawa, 1977. 
  7. [Ke] O. M. Keller, Die homoimorphie der kompakten konvexen Mengen in Hilbertschen Raum, Math. Ann. 105 (1931), 748-758. Zbl57.1523.01
  8. [Ma] P. Mankiewicz, On topological, Lipschitz, and uniform classification of LF-spaces, Studia Math. 52 (1974), 109-142. Zbl0328.46005
  9. [Sa] K. Sakai, On ℝ ∞ -manifolds and Q ∞ -manifolds, Topology Appl. 18 (1984), 69-79. 
  10. [Sch] H. H. Schaefer, Topological Vector Spaces, Macmillan, New York, 1966. 

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