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

S. Gabriyelyan, J. Kąkol, and G. Plebanek. "The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces." Studia Mathematica 233.2 (2016): 119-139. <http://eudml.org/doc/286369>.

@article{S2016,
abstract = {Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of $C_\{k\}(X)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_\{ℝ\}$-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 $k_\{ℝ\}$-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 ℓ₁, we show that the following assertions are equivalent for a Banach space E: (i) E does not contain an isomorphic copy of ℓ₁, (ii) every real-valued sequentially continuous map on the unit ball $B_\{w\}$ with the weak topology is continuous, (iii) $B_\{w\}$ is a $k_\{ℝ\}$-space, (iv) $B_\{w\}$ is an Ascoli space. We also prove that a Fréchet lcs F does not contain an isomorphic copy of ℓ₁ 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 $^\{ℕ\}$, where ∈ ℝ,ℂ.},
author = {S. Gabriyelyan, J. Kąkol, G. Plebanek},
journal = {Studia Mathematica},
keywords = {Banach space; Fréchet space; weak topology; ascoli property},
language = {eng},
number = {2},
pages = {119-139},
title = {The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces},
url = {http://eudml.org/doc/286369},
volume = {233},
year = {2016},
}

TY - JOUR
AU - S. Gabriyelyan
AU - J. Kąkol
AU - G. Plebanek
TI - The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces
JO - Studia Mathematica
PY - 2016
VL - 233
IS - 2
SP - 119
EP - 139
AB - Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of $C_{k}(X)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_{ℝ}$-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 $k_{ℝ}$-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 ℓ₁, we show that the following assertions are equivalent for a Banach space E: (i) E does not contain an isomorphic copy of ℓ₁, (ii) every real-valued sequentially continuous map on the unit ball $B_{w}$ with the weak topology is continuous, (iii) $B_{w}$ is a $k_{ℝ}$-space, (iv) $B_{w}$ is an Ascoli space. We also prove that a Fréchet lcs F does not contain an isomorphic copy of ℓ₁ 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 $^{ℕ}$, where ∈ ℝ,ℂ.
LA - eng
KW - Banach space; Fréchet space; weak topology; ascoli property
UR - http://eudml.org/doc/286369
ER -