  J. Mather characterized uniform hyperbolicity of a discrete dynamical system
as equivalent to invertibility of an operator on the set of all sequences
bounded in norm in the tangent bundle of an orbit. We develop a similar
characterization of nonuniform hyperbolicity and show that it is equivalent to
invertibility of the same operator on a larger, Fr\'echet space. We apply it to
obtain a condition for a diffeomorphism on the boundary of the set of Anosov
diffeomorphisms to be nonuniformly hyperbolic. Finally we generalise the
Shadowing lemma in the same context.
