On Uncountable Frames and Riesz Bases in Nonseparable Banach Spaces

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.