## 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 (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

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

Original language | English |
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Pages (from-to) | 427-442 |

Number of pages | 16 |

Journal | International Journal Of Pure And Applied Mathematics |

Volume | 36 |

Issue number | 4 |

Publication status | Published - 1 Jan 2007 |

## Keywords

- Yosida space
- vectorial norm
- family of seminorms
- extension of bounded linear operators