REPRESENTATIONS OF IDEALS IN POLISH GROUPS AND IN BANACH SPACES
Copyright policy )REPRESENTATIONS OF IDEALS IN POLISH GROUPS AND IN BANACH SPACES
AbstractWe investigate ideals of the form {A⊆ω: Σn∈Axnis unconditionally convergent} where (xn)n∈ωis a sequence in a Polish group or in a Banach space. If an ideal onωcan be seen in this form for some sequence inX, then we say that it is representable inX.After numerous examples we show the following theorems: (1) An ideal is representable in a Polish Abelian group iff it is an analytic P-ideal. (2) An ideal is representable in a Banach space iff it is a nonpathological analytic P-ideal.We focus on the family of ideals representable inc0. We characterize this property via the defining sequence of measures. We prove that the trace of the null ideal, Farah’s ideal, and Tsirelson ideals are not representable inc0, and that a tallFσP-ideal is representable inc0iff it is a summable ideal. Also, we provide an example of a peculiar ideal which is representable inℓ1but not in ℝ.Finally, we summarize some open problems of this topic.
- University of Vienna Austria
classical sequence space, 101013 Mathematische Logik, summable-like ideal, Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces, submeasure, ideal, Analytic P-ideal, Classical sequence space, nonpathological submeasure, CARDINAL INVARIANTS, density-like ideal, FOS: Mathematics, summable ideal, Null set, Polish group, Banach space, trace ideal, Nonpathological submeasure, Density-like ideal, Mathematics - Logic, null set, Ideal, Other combinatorial set theory, Unconditional convergence, Trace ideal, 101013 Mathematical logic, Summable-like ideal, unconditional convergence, analytic P-ideal, Submeasure, Density ideal, Summable ideal, density ideal, Logic (math.LO), Descriptive set theory
classical sequence space, 101013 Mathematische Logik, summable-like ideal, Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces, submeasure, ideal, Analytic P-ideal, Classical sequence space, nonpathological submeasure, CARDINAL INVARIANTS, density-like ideal, FOS: Mathematics, summable ideal, Null set, Polish group, Banach space, trace ideal, Nonpathological submeasure, Density-like ideal, Mathematics - Logic, null set, Ideal, Other combinatorial set theory, Unconditional convergence, Trace ideal, 101013 Mathematical logic, Summable-like ideal, unconditional convergence, analytic P-ideal, Submeasure, Density ideal, Summable ideal, density ideal, Logic (math.LO), Descriptive set theory
selected citations These citations are derived from selected sources.This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).20 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Top 10% influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Top 10% impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Top 10%
Citations provided by BIP!Popularity provided by BIP!