On vectorial inner product spaces

Abstract

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 $B$-regular Yosida space, that is a Dedekind complete Yosida space such that $\mathop \cap \limits_{J \in B} J = \left\{ 0 \right\}$, where $B$ is the set of all hypermaximal bands in Y. In Theorem 2.1.1 we assert that any $B$-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 $B$-regular and norm complete Yosida algebra $(B(A),{\mathop {\sup }\limits_{\alpha \in A}} |{\kern 1pt} x(\alpha ){\kern 1pt} |)$.