Abstract: Duality principles in Gabor theory such as the Ron-Shen duality principle and the Wexler-Raz biorthogonality relations play a fundamental role for analyzing Gabor systems. For each sequence in a Banach space X, we define a corresponding sequence dependent only on two $p$-Riesz bases in the Banach space X. Then we characterize exactly properties of the first sequence in terms of the associated one. We generate some results that were obtained by P. G. Casazza, G. Kutyniok and M. C. Lammers about duality principles of frames in a separable Hilbert space H.

Key words: Duality of frame, p-Riesz basis, q-frame.