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## Studia Mathematica

2016 | 233 | 2 | 119-139
Tytuł artykułu

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

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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}.
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Numer
Strony
119-139
Opis fizyczny
Daty
wydano
2016
Twórcy
autor
• Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, P.O. 653, Israel
autor
• 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
• Instytut Matematyczny, Uniwersytet Wrocławski, 50-384 Wrocław, Poland
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