A functional ([dot], [dot]) : X x X [right arrow] K is said to be a Hermitian form on X if
We use the following notations related to a given Hermitian form ([dot], [dot]) on X:
The following result concerning the functional properties of [sigma] as a function depending on the nonnegative hermitian form ([dot], [dot]) has been obtained in [?
2,1] is a nonnegative hermitian form and thus, by the above property one has,
If ([dot], [dot]) is a nonnegative Hermitian form on X, x, y [member of] X and ||y|| [not equal to] 0, then
Since, by Bessel's inequality the hermitian form ([dot], [dot])[.
then, using Hadamard's inequality, we conclude that q ([dot], [dot]) is also a nonnegative hermitian form.
Classical and recent inequalities for Hermitian forms on real or complex linear spaces are surveyed.
The following fundamental facts concerning Hermitian forms hold [?
His account relies on the basic results in the linear representations of finite groups in quadratic, symplectic, and Hermitian forms
and in involutions over simple algebras, which are outlined, mainly without proofs, in the first chapter.