Functional Analysis and Applications Seminar 

13/07/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
João de Deus Marques, Universidade Nova de Lisboa

On vectorial inner product spaces

Let $E$ be a linear space and $p$ of $E$ into a Riesz space $Y$ a vectorial norm. The space $E$ with $p$ 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 $p$ is defined in terms of a vectorial inner product. Here we consider $Y$ to be a $B$-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 $B$-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 a $B$-regular and norm complete Yosida algebra.