g-Orthonormal bases in Hilbert spaces

g-Orthonormal basis is a natural generalization of orthonormal basis in Hilbert spaces. In this paper,firstly, we give a general condition such that the g-orthonormal bases exist. Then we consider the properties of g-orthonormal basis. In particular, we establish the g-orthonormal basis versions of some classical properties of orthonormal basis, such as Bessel inequality and Bessel equality etc. We also give some new characterizations of g-Riesz bases. Finally, we discuss the redundancy problem of g-frames.