  In this paper, we study slow manifolds for infinite-dimensional evolution
equations. We compare two approaches: an abstract evolution equation framework
and a finite-dimensional spectral Galerkin approximation. We prove that the
slow manifolds constructed within each approach are asymptotically close under
suitable conditions. The proof is based upon Lyapunov-Perron methods and a
comparison of the local graphs for the slow manifolds in scales of Banach
spaces. In summary, our main result allows us to change between different
characterizations of slow invariant manifolds, depending upon the technical
challenges posed by particular fast-slow systems.
