On vectorial inner product spaces
Let be a linear space and of into a Riesz space a vectorial norm. The space with is named a vectorially normed space. Of special interest among vectorially normed spaces are the vectorial inner product spaces, the theory of which is richer and retains many features of Euclidean spaces. In these spaces the vectorial norm is defined in terms of a vectorial inner product. Here we consider to be a -regular Yosida space, that is a unitary Archimedean and Dedekind complete Riesz space such that the intersection of all its hypermaximal bands is the zeroelement. In Yosida Theorem we assert that any -regular Yosida space is Riesz isomorphic to the space of all bounded real-valued mappings on a certain set . Next we prove Bessel Inequality and Parseval Identity for a vectorial inner product with range in a -regular and norm complete Yosida algebra.
