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 K 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 ℓ 1 , we show that the following assertions are equivalent for a Banach space E: (i) E does not contain isomorphic copy of ℓ 1 , (ii) every real-valued sequentially continuous map on the unit ball B w with the weak topology is continuous,We prove also that a Fréchet lcs F does not contain isomorphic copy of ℓ 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 K N , where K ∈ {R, C}.

Topological properties of spaces of Baire functions

“…In Theorem 2.5, we prove that every κ-Fréchet-Urysohn space X is Ascoli. Applying Theorem 2.5 and some of the main results from [2,8,9,10,13] we characterize the κ-Fréchet-Urysohness in various important classes of locally convex spaces including strict (LF )-spaces and free locally convex spaces. In Section 3 we prove Theorem 1.3 using several more general results.…”

The Ascoli property for function spaces

“…On the other hand, it is proved in [10] that if X is a first-countable paracompact σ-space, then C k (X, I) is Ascoli if and only if C k (X) is Ascoli if and only if X is a locally compact metrizable space. However this result does not cover the case for X being a non-metrizable compact space X for which clearly the Banach space C k (X) is Ascoli.…”

“…Theorem 2.5 of [10] states in particular that, for a first-countable paracompact σ-space X, the space C k (X) is an Ascoli space if and only if C k (X) is a k R -space if and only if X is a locally compact metrizable space. In this section we prove an analogous result using the following proposition.…”

“…The question for which spaces X the space C p (X) is Ascoli or a k R -space is posed in [10]. It turned out that for spaces of the form C p (X), the Ascoli property is formally stronger than the κ-Fréchet-Urysohn one.…”

On the weak and pointwise topologies in function spaces II

“…In [GKP,Theorem 1.5] it was proved that for X as in the above corollary, the space (X, w) is not even an Ascoli space (which is weaker than being a k-space). Using Proposition 2.1 from this paper and above Propoition 6.4 one can easily prove a counterpart of this result for the weak * topology.…”