IJPAM: Volume 36, No. 4 (2007)

ON A GENERALIZATION OF
THE HAHN-BANACH THEOREM

João de Deus Marques
Department of Mathematics
Faculty of Sciences and Technology
New University of Lisbon
Quinta da Torre, Caparica, 2829-516, PORTUGAL
e-mail: jdm@fct.unl.pt

Abstract.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 ( $\cal B$ -regular Yosida space). Let be a linear space and $\cal B$ -regular Yosida spaces. In Theorem 2.2.1 we define a vectorial norm on the linear space ${\cal L}(E,Y)$ of all bounded linear operators from into and with range in the partially ordered linear space ${\cal L}(X,Y)$ of all continuous linear operators from into . Next, in Theorem 3.1 we establish the following result: If is a bounded linear operator on a linear subspace of into , then there exists a bounded linear operator defined on that is an extension of and with the same vectorial norm, i.e. . We finish with some consequences of this result.