  The adiabatic theorem refers to a setup where an evolution equation contains
a time-dependent parameter whose change is very slow, measured by a vanishing
parameter $\epsilon$. Under suitable assumptions the solution of the
time-inhomogenous equation stays close to an instantaneous fixpoint. In the
present paper, we prove an adiabatic theorem with an error bound that is
independent of the number of degrees of freedom. Our setup is that of quantum
spin systems where the manifold of ground states is separated from the rest of
the spectrum by a spectral gap. One important application is the proof of the
validity of linear response theory for such extended, genuinely interacting
systems. In general, this is a long-standing mathematical problem, which can be
solved in the present particular case of a gapped system, relevant e.g.~for the
integer quantum Hall effect.
