A Canonical Decomposition for Quadratic Forms with Applications to Monotone Convergence Theorems

In this note, we wish to prove precise theorems for monotone convergence of quadratic forms on a complex Hilbert space, X. For convenience we only consider positive forms although semibounded forms can be treated. In order to describe the existing theorems and to establish some notation, we first review some of the main ideas in the theory [2, 41: a positive quadratic form is a sesquilinear form t(-, *) on a domain D(t) x D(t) with t(v, CJJ) > 0 for all y E D(t). Until Section 4, we follow the standard theory and consider only the case of densely defined forms where D(t) = X. t is called closed if and only if D(t) with the norm