# IJPAM: Volume 36, No. 4 (2007)

**ON A GENERALIZATION OF**

THE HAHN-BANACH THEOREM

THE HAHN-BANACH THEOREM

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 (-regular Yosida space). Let be a linear
space and -regular Yosida spaces. In Theorem 2.2.1
we define a vectorial norm on the linear space
of all bounded linear operators from into and with range
in the partially ordered linear space 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.

**Received: **February 8, 2007

**AMS Subject Classification: **46A22, 46A40

**Key Words and Phrases: **Yosida space, vectorial norm, family of seminorms, extension of bounded linear operators

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2007

**Volume:** 36

**Issue:** 4