  We present three versions of the Lax-Milgram theorem in the framework of
Hilbert C*-modules, two for those over W*-algebras and one for those over
C*-algebras of compact operators. It is remarkable that while the Riesz theorem
is not valid for certain Hilbert C*-modules over C*-algebras of compact
operators, our Lax-Milgram theorem turns out to be valid for all of them. We
also give several examples to illustrate our results, in particular, we show
that the main theorem is not true for Hilbert modules over arbitrary
C*-algebras.
