The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

S. Gabriyelyan ; J. Kąkol ; G. Plebanek

Studia Mathematica, Tome 233 (2016) no. 2, p. 119.

Résumé

Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of C k ( X ) is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every k ℝ -space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space C k ( X ) is Ascoli iff C k ( X ) is a k ℝ -space iff X is locally compact. Moreover, C k ( X ) endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability theory and measure-theoretic properties of ℓ₁, we show that the following assertions are equivalent for a Banach space E: (i) E does not contain an isomorphic copy of ℓ₁, (ii) every real-valued sequentially continuous map on the unit ball B w with the weak topology is continuous, (iii) B w is a k ℝ -space, (iv) B w is an Ascoli space. We also prove that a Fréchet lcs F does not contain an isomorphic copy of ℓ₁ iff each closed and convex bounded subset of F is Ascoli in the weak topology. Moreover we show that a Banach space E in the weak topology is Ascoli iff E is finite-dimensional. We supplement the last result by showing that a Fréchet lcs F which is a quojection is Ascoli in the weak topology iff F is either finite-dimensional or isomorphic to ℕ , where ∈ ℝ,ℂ.

Détail

Classification : 46B25, 54C35, 46A04, 28C15
Mots-clés : Banach space, Fréchet space, weak topology, ascoli property