p-frames in separable Banach spaces

Let X be a separable Banach space with dual X*. A countable family of elements {g(i)} subset of X* is a p-frame (1 < p < infinity) if the norm parallel to . parallel to (X) is equivalent to the l(p)-norm of the sequence {g(i)(.)}. Without further assumptions, we prove that a p-frame allows every g is an element of X* to be represented as an unconditionally convergent series g = Sigma d(i) g(i) for coefficients {d(i)} is an element of l(q), where 1/p + 1/q = 1. A p-frame {g(i)} is not necessarily linear independent, so {g(i)} is some kind of overcomplete basis for X*. We prove that a q-Riesz basis for X* is a p-frame for X and that the associated coefficient functionals {f(i)} constitutes a p-Riesz basis allowing us to expand every f is an element of X (respectively g is an element of X*) as f = Sigma g(i)(f)f(i) (respectively g = Sigma g(f(i))g(i)). In the general case of a p-frame such expansions are only possible under extra assumptions.