  We consider divergence-form scalar elliptic equations and vectorial equations
for elasticity with rough ($L^\infty(\Omega)$, $\Omega \subset \R^d$)
coefficients $a(x)$ that, in particular, model media with non-separated scales
and high contrast in material properties. We define the flux norm as the $L^2$
norm of the potential part of the fluxes of solutions, which is equivalent to
the usual $H^1$-norm. We show that in the flux norm, the error associated with
approximating, in a properly defined finite-dimensional space, the set of
solutions of the aforementioned PDEs with rough coefficients is equal to the
error associated with approximating the set of solutions of the same type of
PDEs with smooth coefficients in a standard space (e.g., piecewise polynomial).
We refer to this property as the {\it transfer property}.
  A simple application of this property is the construction of finite
dimensional approximation spaces with errors independent of the regularity and
contrast of the coefficients and with optimal and explicit convergence rates.
This transfer property also provides an alternative to the global harmonic
change of coordinates for the homogenization of elliptic operators that can be
extended to elasticity equations. The proofs of these homogenization results
are based on a new class of elliptic inequalities which play the same role in
our approach as the div-curl lemma in classical homogenization.
