On countable tightness and the Lindelöf property in non-Archimedean Banach spaces

0488913 - MÚ 2019 RIV DE eng J - Journal Article
Kąkol, Jerzy - Kubzdela, A. - Perez-Garcia, C.
On countable tightness and the Lindelöf property in non-Archimedean Banach spaces.
Journal of Convex Analysis. Roč. 25, č. 1 (2018), s. 181-199. ISSN 0944-6532. E-ISSN 0944-6532
R&D Projects: GA ČR GF16-34860L
Institutional support: RVO:67985840
Keywords : non-archimedean Banach spaces * weak topology * Lindelöf property
OECD category: Pure mathematics
Impact factor: 0.794, year: 2018
http://www.heldermann.de/JCA/JCA25/JCA251/jca25011.htm

Let K be a non-archimedean valued field and let E be a non-archimedean Banach space over K. By E-w we denote the space E equipped with its weak topology and by E-w*(*) the dual space E* equipped with its weak* topology. Several results about countable tightness and the Lindelof property for E-w and E-w*(*) are provided. A key point is to prove that for a large class of infinite-dimensional polar Banach spaces E, countable tightness of E-w or E-w*(*) implies separability of K. As a consequence we obtain the following two characterizations of K :
(a) A non-archimedean valued field K is locally compact if and only if for every Banach space E over K the space E-w has countable tightness if and only if for every Banach space E over K the space E-w*(*) has the Lindelof property.
(b) A non-archimedean valued separable field K is spherically complete if and only if every Banach space E over K for which E-w has the Lindelof property must be separable if and only if every Banach space E over K for which E-w*(*) has countable tightness must be separable. Both results show how essentially different are non-archimedean counterparts from the 'classical' corresponding theorems for Banach spaces over the real or complex field.
Permanent Link: http://hdl.handle.net/11104/0283429