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 |
---|---|
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