  For the solution of operator equations, Stevenson introduced a definition of
frames, where a Hilbert space and its dual are {\em 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.

Functional Analysis