This is an announcement for the paper "The Ascoli property for function spaces and the weak topology of Banach and Fr'echet spaces" by S. Gabriyelyan, J. Kakol, and G. Plebanek.
Abstract: Following [3] 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 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 prove also that a Fr'{e}chet lcs $F$ does not contain isomorphic copy of $\ell_1$ iff each closed and convex bounded subset of $F$ is Ascoli in the weak topology. However 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 Fr'{e}chet lcs $F$ which is a quojection is Ascoli in the weak topology iff either $F$ is finite dimensional or $F$ is isomorphic to the product $\mathbb{K}^{\mathbb{N}}$, where $\mathbb{K}\in{\mathbb{R},\mathbb{C}}$.
Archive classification: math.FA math.GN
Mathematics Subject Classification: 46A04, 46B03, 54C30
Submitted from: saak@bgu.ac.il
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1504.04202
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