Migdad Ismailov - Baku State University, Institute of Mathematics and Mechanics of the NAS of Azerbaijan.

Some generalizations of Besselian, Hilbertian systems and frames in nonseparable Banach spaces with respect to some nonseparable Banach space K of systems of scalars are considered in this work. The concepts of  uncountable K-Bessel, K-Hilbert systems, K-frames and  K^{*} -Riesz bases in nonseparable Banach spaces are introduced. Criteria of uncountable K-Besselianness, K-Hilbertianness for systems, K-frames and unconditional K^{*} -Riesz basicity are found, and the relationship between them is studied. Unlike before, these new facts about Besselian and Hilbertian systems in Hilbert and Banach spaces are proved without using a conjugate system and, in some cases, a completeness of a system. Examples of K-Besselian systems which are not minimal are given. It is proved that every K-Hilbertian systems is minimal. The case where K is an space of systems of coefficients of uncountable unconditional basis of some space is also considered.

Nonseparable Banach space, Uncountable unconditional basis, K-Bessel and K-Hilbert systems, K-frames, uncountable unconditional K-Riesz bases