University of MaraghehSahand Communications in Mathematical Analysis2322-580718120210201$K$-orthonormal and $K$-Riesz Bases59724711410.22130/scma.2020.130958.827ENAhmadAhmdiDepartment of Mathematics, Faculty of Science, University of Hormozgan, P.O.Box 7916193145, Bandar Abbas, Iran.0000-0001-9870-2374AsgharRahimiDepartment of Mathematics, Faculty of Science, University of Maragheh, P.O.Box 55136-553, Maragheh, Iran.0000-0003-2095-6811Journal Article20200714Let $K$ be a bounded operator. $K$-frames are ordinary frames for the range $K$. These frames are a generalization of ordinary frames and are certainly different from these frames. This research introduces a new concept of bases for the range $K$. Here we define theĀ $K$-orthonormal basis and the $K$-Riesz basis, and then we describe their properties. As might be expected, the $K$-bases differ from the ordinary ones mentioned in this article.https://scma.maragheh.ac.ir/article_47114_8902cf1717995f11537b68de849abfb0.pdf