Mediterr. J. Math.
Springer International Publishing AG,
part of Springer Nature 2018
On The Spaces of Linear Operators Acting
Between Asymmetric Cone Normed Spaces
Abstract. An asymmetric norm is a positive sublinear functional p on
a real vector space X satisfying x = θ
Since the space of all lower semi-continuous linear functionals of an
asymmetric normed space is not a linear space, the theory is diﬀerent
in the asymmetric case. The main purpose of this study is to deﬁne
bounded and continuous linear operators acting between asymmetric
cone normed spaces. After examining the diﬀerences with symmetric
case, we give some results related to Baire’s characterization of com-
pleteness in asymmetric cone normed spaces.
Mathematics Subject Classiﬁcation. 46B40, 54E50, 46A32.
Keywords. Asymmetric norm, Cone norm, Bounded linear operators,
In , by deﬁning the notion of cone metric as a mapping valued in an
ordered Banach space, the authors described convergence and completeness
in cone metric spaces. Also, by giving an example of a mapping which is a
contraction in a cone metric space but not a contraction in a metric space,
they show that metric spaces do not ensure enough theory for ﬁxed point
theorems having a signiﬁcant place in the study of metric spaces. After the
work of Long-Guang and Xian , many papers appeared especially dealing
with ﬁxed point theorems as well as topological concepts on those spaces.
For some of them, we refer to [1,21–24] and references therein. A real vector
space with a cone norm which is a mapping deﬁned by replacing the set of
real numbers with an ordered real Banach space is called cone normed space
[2,12]. The main properties of cone normed spaces and some theorems of
weighted means in cone normed spaces are studied in .
After the paper of Wilson , the study of asymmetric metrics became
a subject of extensive research in topology and theoretical computer science.
An asymmetric metric (also known as a quasi metric) is a mapping satisfying