The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces
Tom 233 / 2016
Streszczenie
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 $\mathbb {K}\in \{\mathbb {R},\mathbb {C}\}$.
