In this paper, we study conditional shadowing for a nonautonomous system in a Banach space assuming that the linear part admits a family of invariant subspaces (scale) with different behavior of trajectories. Conditions of shadowing are formulated in terms of smallness of the projections of one-step errors to the scale and of smallness of Lipschitz constants of the projections of nonlinear terms. We also give conditions under which a system has the conditional property of inverse shadowing (dual to the shadowing property).