We study the notions of acs, luacs and uacs Banach spaces which were introduced by Kadets et al. and form common generalisations of the usual rotundity and smoothness properties of Banach spaces. In particular, we are interested in (mainly infinite) ABSOLUTE SUMS of such spaces. We also introduce some new classes of spaces that lie in between those of acs and uacs spaces and study their behaviour under the formation of ABSOLUTE SUMS as well.

Purpose: The purpose of this paper is to develop a set of identities for Euler type SUMS of products of harmonic numbers and reciprocal binomial coefficients.Method: We use analytical methods to obtain our results.Results: We obtain identities for variant Euler SUMS of the type ¥Sn=1H2n/n (n+k)k, and its finite counterpart, which generalize some results obtained by other authors.Conclusions: Identities are successfully achieved for the SUMS under investigation. Some published results have been successfully generalized.

Let (X; N) be a fuzzy normed space and A be a fuzzy bounded subset of X. We de ne fuzzy ℓ ¥-SUMS and fuzzy c0-SUMS of fuzzy normed spaces. Then we will show that in these spaces, all fuzzy uniquely remotal sets are singletons.