On vectorial inner product spaces

Let E be a real linear space. A vectorial inner product is a mapping from E×E into a real ordered vector space Y with the properties of a usual inner product. Here we consider Y to be a -regular Yosida space, that is a Dedekind complete Yosida space such that , where is the set of all hypermaximal bands in Y. In Theorem 2.1.1 we assert that any -regular Yosida space is Riesz isomorphic to the space B(A) of all bounded real-valued mappings on a certain set A. Next we prove Bessel Inequality and Parseval Identity for a vectorial inner product with range in the -regular and norm complete Yosida algebra .