@article { author = {Ismailov, Migdad}, title = {On Uncountable Frames and Riesz Bases in Nonseparable Banach Spaces}, journal = {Sahand Communications in Mathematical Analysis}, volume = {19}, number = {2}, pages = {149-170}, year = {2022}, publisher = {University of Maragheh}, issn = {2322-5807}, eissn = {2423-3900}, doi = {10.22130/scma.2021.523500.905}, abstract = {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.}, keywords = {Nonseparable Banach space,Uncountable unconditional basis,K-Bessel and K-Hilbert systems,K-frames,uncountable unconditional K-Riesz bases}, url = {https://scma.maragheh.ac.ir/article_252487.html}, eprint = {https://scma.maragheh.ac.ir/article_252487_22bde7068edb87cf8c0a534e9a8ac4aa.pdf} }