Open AccessS-nuclearity and n-diameters of infinite Cartesian products of bounded subsets in Banach spaces
Author(s) -
Nashat Faried,
Mona Fathey
Publication year - 2003
Publication title -
acta et commentationes universitatis tartuensis de mathematica./acta et commentationes universitatis tartuensis de mathematica
Language(s) - English
Resource type - Journals
eISSN - 2228-4699
pISSN - 1406-2283
DOI - 10.12697/acutm.2003.07.05
Subject(s) - cartesian product , mathematics , sequence (biology) , ideal (ethics) , banach space , bounded function , product (mathematics) , compact space , combinatorics , zero (linguistics) , space (punctuation) , uniform boundedness , discrete mathematics , pure mathematics , mathematical analysis , geometry , computer science , philosophy , linguistics , genetics , epistemology , biology , operating system
In this paper we classify compact subsets of a normed space according to the rate of convergence to zero of its sequence {δn(B)} of Kolmogorov diameters. We introduce σ-compact sets to satisfy that {δn(B)}∈σ where σ is an ideal of convergent to zero sequences. Examples of sequence ideals are the ideals of rapidly decreasing sequences {λn} satisfying limn→∞λnnα=0 or radically decreasing sequences satisfying limn→∞(|λn|)1/n=0. In the case that σ is the ideal of rapidly decreasing sequences, this notion is identical to the S-nuclearity introduced by K. Astala and M. S. Ramanujan in 1987. We show that the infinite Cartesian product ∏i=1∞Bi of compact sets Bi is ℓp-compact in ℓp(Xi), for all p>0, if (δ0(Bi))∈S. In this case, we give upper estimates for the n-th diameters of ∏i=1∞Bi in ℓp(Xi) for any p>0.