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), p. 119-139 / Harvested from The Polish Digital Mathematics Library

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.

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Publié le : 2016-01-01

Zbl 06586871

EUDML-ID : urn:eudml:doc:286369