The Ascoli property for function spaces and the weak topology of Banach and Fr\'echet spaces
The Ascoli property for function spaces and the weak topology of Banach and Fr\'echet spaces
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函数空间的 Ascoli 性质以及 Banach 和 Frechet 空间的弱拓扑
DOI：
10.4064/sm8289-4-2016
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
G. Plebanek
中科院分区：
文献类型：
--
作者：
S. Gabriyelyan;J. Ka̧kol;G. Plebanek
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 $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$.
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 $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$.