On a Generalization of the Hanh-Banach Theorem

A vectorial norm is a mapping from a linear space into a real
ordered vector space with the properties of a usual norm. Here we consider the
ordered vector space to be a unitary Archimedean-Riesz space (Yosida space),
Dedekind complete and such that the intersection of all its hypermaximal bands
is the zeroelement of the space (B-regular Yosida space). Let E be a linear space
and X, Y B-regular Yosida spaces. In Theorem 2.2.1 we define a vectorial norm
G on the linear space L(E, Y ) of all bounded linear operators from E into Y
and with range in the partially ordered linear space L(X, Y ) of all continuous
linear operators from X into Y . Next, in Theorem 3.1 we establish the following
result: If t is a bounded linear operator on a linear subspace F of E into Y ,
then there exists a bounded linear operator T defined on E that is an extension
of t and with the same vectorial norm, i.e. G(T) = G(t). We finish with some
consequences of this result