On uncountable $K$-Bessel and $K$-Hilbert systems in nonseparable Banach spaces

Abstract

We consider the uncountable K-Besselness and K-Hilbertness of systems in nonseparable Banach spaces with respect to nonseparable Banach space of systems of scalars K. The criteria for K-Besselness and K-Hilbertness of systems in case where the space K is a space of systems consisting of coefficients of some uncountable unconditional basis are found. The relationship between the K-Besselness and K-Hilbertness of the system and the existence of the uncountable unconditional basis with the space of coefficients K is established.