Frames for the solution of operator equations in Hilbert spaces with fixed dual pairing

For the solution of operator equations, Stevenson introduced a definition of frames, where a Hilbert space and its dual are not  identified. This means that the Riesz isomorphism is not used as an identification, which, for example, does not make sense for the Sobolev spaces $H_0^1(Omega)$ and $H^{-1}(Omega)$ . In this article, we are going to revisit the concept of Stevenson frames and introduce it for Banach spaces. This is equivalent to $ell^2$ -Banach frames. It is known that, if such a system exists, by defining a new inner product and using the Riesz isomorphism, the Banach space is isomorphic to a Hilbert space. In this article, we deal with the contrasting setting, where $mathcal{H}$ and $mathcal{H}'$ are not identified, and equivalent norms are distinguished, and show that in this setting the investigation of $ell^2$ -Banach frames make sense.