  We extend the concept of locality to enclose a situation where a
tensor-product structure for the Hilbert space is not \textit {a priori}
assumed; rather, this locality is related to a given matrix representation of
the Hamiltonian associated to the system. As a result, we formulate a
Lieb-Robinson-like bound for Hamiltonians local in a given basis. In
particular, we employ this bound to obtain alternatively the adiabatic
condition, where adiabaticity is naturally ensued from a locality in energy
basis and a relatively small Lieb-Robinson bound.
