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

S. Gabriyelyan ¹   ; J. Kąkol ²   ; G. Plebanek ³

¹ Department of Mathematics Ben-Gurion University of the Negev Beer-Sheva, P.O. 653, Israel
² Faculty of Mathematics and Informatics A. Mickiewicz University 61-614 Poznań, Poland and Institute of Mathematics Czech Academy of Sciences Žitna 25, Praha 1, Czech Republic
³ Instytut Matematyczny Uniwersytet Wrocławski 50-384 Wrocław, Poland

Studia Mathematica, Tome 233 (2016) no. 2, pp. 119-139

Résumé

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 $\mathbb {K}\in \{\mathbb {R},\mathbb {C}\}$.

Détail

DOI : 10.4064/sm8289-4-2016

Keywords: following banakh gabriyelyan say tychonoff space ascoli space every compact subset mathcal evenly continuous notion closely related classical ascoli theorem every mathbb space hence k space ascoli metrizable space prove space ascoli mathbb space locally compact moreover endowed weak topology ascoli countable discrete using basic concepts probability theory measure theoretic properties ell following assertions equivalent banach space nbsp nbsp does contain isomorphic copy ell every real valued sequentially continuous map unit ball weak topology continuous iii mathbb space nbsp nbsp ascoli space prove chet lcs does contain isomorphic copy ell each closed convex bounded subset ascoli weak topology moreover banach space weak topology ascoli finite dimensional supplement result showing chet lcs which quojection ascoli weak topology either finite dimensional isomorphic mathbb mathbb where mathbb mathbb mathbb