Paley type inequalities for orthogonal series with vector-valued coefficients

We investigate the extension to Banach-space-valued functions of the classical inequalities due to Paley for the Fourier coefficients with respect to a general orthonormal system Phi. This leads us to introduce the notions of Paley Phi -type and Phi -cotype for a Banach space and some related concepts. We study the relations between these notions of type and cotype and those previously defined. We also analyze how the interpolation spaces inherit these characteristics from the original spaces, and use them to obtain sharp coefficient estimates for functions taking values in Lorentz spaces.