In this paper, we begin the study of Bousfield classes for cohomology theories
defined on spectra. Our main result is that a map $f: X \rightarrow Y$ 
induces an isomorphism on $E(n)$-cohomology if and only if it induces
an isomorphism on $E(n)$-homology. We also prove this for variants of
$E(n)$ such as elliptic cohomology and real K-theory. We also show
that there is a nontrivial map from a spectrum $Z$ to the $K(n)$-local
sphere if and only if $K(n)_{*}(Z) \neq 0$.