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

Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset K of C_k(X) is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every k_{double-struck 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_{double-struck 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_{double-struck 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 double-struck K^{double-struck N}, where double-struck K ∈ {double-struck R, double-struck C}.