Before revision:  The forward operator, mapping from a sequence space to an arbitrary Banach space, typically an L^2-space, is assumed to satisfies a two-sided Lipschitz condition with respect to a weighted l ^2-norm and the norm of the image space.
After revision:  The forward operator, mapping from a sequence space to an arbitrary Banach space, typically an L^2-space, is assumed to satisfies a two-sided Lipschitz condition with respect to a weighted \ell ^2-norm and the norm of the image space.