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

### Authors:

- S. Gabriyelyan,
- Jerzy Kąkol,
- G. Plebanek

### Abstract

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 ℓ_{1}, we show that the following assertions are equivalent for a Banach space E: (i) E does not contain an isomorphic copy of ℓ_{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_{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 ℓ_{1} 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}.

- Record ID
- UAMd09652b0458e479fb5ff5c3cfd4df00e
- Author
- Journal series
- Studia Mathematica, ISSN 0039-3223
- Issue year
- 2016
- Vol
- 233
- Pages
- 119-139
- ASJC Classification
- DOI
- DOI:10.4064/sm8289-4-2016 opening in a new tab
- Language
- en English
- Score (nominal)
- 25
- Score source
- journalList
- Score
- Publication indicators
- = 9.000; : 2016 = 1.047; : 2016 = 0.535 (2) - 2016=0.775 (5)

- Uniform Resource Identifier
- https://researchportal.amu.edu.pl/info/article/UAMd09652b0458e479fb5ff5c3cfd4df00e/

* presented citation count is obtained through Internet information analysis and it is close to the number calculated by the Publish or Perishopening in a new tab system.

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