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

• Autorzy: S. Gabriyelyan , J. Kąkol , G. Plebanek
• Języki publikacji: EN

Abstrakty

EN

Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset 𝒦 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 $𝕂^{ℕ}$, where 𝕂 ∈ {ℝ,ℂ}.

Kategorie tematyczne

• 54C35: Function spaces
• 46A04: Locally convex Fr\'echet spaces and (DF)-spaces
• 28C15: Set functions and measures on topological spaces (regularity of measures, etc.)
• 46B25: Classical Banach spaces in the general theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2016
• Tom: 233
• Numer: 2
• Strony: 119-139

Daty

wydano

2016

Twórcy

autor

S. Gabriyelyan

• Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, P.O. 653, Israel

autor

J. Kąkol

• Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Poznań, Poland
• Institute of Mathematics, Czech Academy of Sciences, Žitna 25, Praha 1, Czech Republic

autor

G. Plebanek

• Instytut Matematyczny, Uniwersytet Wrocławski, 50-384 Wrocław, Poland

Identyfikatory

DOI

10.4064/sm8289-4-2016