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

Volume 233 / 2016

S. Gabriyelyan, J. Kąkol, G. Plebanek Studia Mathematica 233 (2016), 119-139 MSC: Primary 46A04, 46B25, 54C35; Secondary 28C15. DOI: 10.4064/sm8289-4-2016 Published online: 19 May 2016


Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space $X$ is an Ascoli space if every compact subset $\mathcal {K}$ of $C_k(X)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_{\mathbb {R}}$-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 $k_{\mathbb {R}}$-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 $\ell _1$, we show that the following assertions are equivalent for a Banach space $E$: (i) $E$ does not contain an isomorphic copy of $\ell _1$, (ii) every real-valued sequentially continuous map on the unit ball $B_{w}$ with the weak topology is continuous, (iii) $B_{w}$ is a $k_{\mathbb {R}}$-space, (iv) $B_{w}$ is an Ascoli space.

We also prove that a Fréchet lcs $F$ does not contain an isomorphic copy of $\ell _1$ 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 $\mathbb {K}^{\mathbb {N}}$, where $\mathbb {K}\in \{\mathbb {R},\mathbb {C}\}$.


  • S. GabriyelyanDepartment of Mathematics
    Ben-Gurion University of the Negev
    Beer-Sheva, P.O. 653, Israel
  • J. KąkolFaculty of Mathematics and Informatics
    A. Mickiewicz University
    61-614 Poznań, Poland
    Institute of Mathematics
    Czech Academy of Sciences
    Žitna 25, Praha 1, Czech Republic
  • G. PlebanekInstytut Matematyczny
    Uniwersytet Wrocławski
    50-384 Wrocław, Poland

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