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

Taras Banakh¹

¹

Studia Mathematica, Tome 138 (2000) no. 3, pp. 201-208

Résumé

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 $Q×ℝ^∞$.

Détail

DOI : 10.4064/sm-138-3-201-208

Keywords: dual space, nuclear (LF)-space, Montel space, direct limit, Hilbert cube