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

Abstract

In this paper we classify compact subsets of a normed space according to the rate of convergence to zero of its sequence {δ_n(B)} of Kolmogorov diameters. We introduce σ-compact sets to satisfy that {δ_n(B)}∈σ where σ is an ideal of convergent to zero sequences. Examples of sequence ideals are the ideals of rapidly decreasing sequences {λ_n} satisfying lim_n→∞λ_nn^α=0 or radically decreasing sequences satisfying lim_n→∞(|λ_n|)^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^∞B_i of compact sets B_i is ℓ^p-compact in ℓ^p(X_i), for all p>0, if (δ₀(B_i))∈S. In this case, we give upper estimates for the n-th diameters of ∏_i=1^∞B_i in ℓ^p(X_i) for any p>0.