  We show that the isomorphism induced by the inclusion of pairs
$(X,\emptyset)\subset (X,Y)$ between the relative bounded cohomology of $(X,Y)$
and the bounded cohomology of $X$ is isometric in degree at least 2 if the
fundamental group of each connected component of $Y$ is amenable. As an
application we provide a self-contained proof of Gromov Equivalence theorem and
a generalization of a result of Fujiwara and Manning on the simplicial volume
of generalized Dehn fillings.
