S-nuclearity and n-diameters of infinite Cartesian products of bounded subsets in Banach spaces

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.