K-orthonormal and K-Riesz Bases

Let 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 the  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.