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

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

References in corpus (2)

• On topological properties of the weak topology of a Banach space
• Topological properties of function spaces $C_k(X,2)$ over zero-dimensional metric spaces $X$