  We present a form convergence theorem for sequences of sectorial forms and
their associated semigroups in a complex Hilbert space. Roughly speaking, the
approximating forms $a_n$ are all `bounded below' by the limiting form $a$, but
in contrast to the previous literature there is no monotonicity hypothesis on
the sequence. Moreover, the forms are not supposed to be closed or densely
defined.
  For a sectorial form one obtains an associated linear relation, whose
negative generates a degenerate strongly continuous semigroup of linear
operators. Our hypotheses on the sequence of forms imply strong resolvent
convergence of the associated linear relations, which in turn implies
convergence of the corresponding semigroups. The result is illustrated by two
examples, one of them closely related to the Galerkin method of numerical
analysis.
