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Separable determination in Banach spaces

Volume 243 / 2018

Marek Cúth Fundamenta Mathematicae 243 (2018), 9-27 MSC: Primary 46B26; Secondary 46B20, 03C30. DOI: 10.4064/fm480-11-2017 Published online: 18 June 2018

Abstract

We study a relation between three different formulations of theorems on separable determination: one using the concept of rich families, another via the concept of suitable models, and a third, new one, suggested in this paper, using the notion of $\omega $-monotone mappings. In particular, we show that in Banach spaces all those formulations are in a sense equivalent, and we give a positive answer to two questions of O. Kalenda and the author. Our results enable us to obtain new statements concerning separable determination of $\sigma $-porosity (and of similar notions) in the language of rich families; thus, without using any terminology from logic or set theory.

Moreover, we prove that in Asplund spaces, generalized lushness is separably determined.

Authors

  • Marek CúthCharles University
    Faculty of Mathematics and Physics
    Department of Mathematical Analysis
    Sokolovská 83
    186 75 Praha 8, Czech Republic
    e-mail

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