Topological Quasilinear Spaces

Yılmaz Yılmaz,Sümeyye Çakan,Şahika Aytekin

doi:10.1155/2012/951374

Abstract

We introduce, in this work, the notion of topological quasilinear spaces as a generalization of the notion of normed quasilinear spaces defined by Aseev (1986). He introduced a kind of the concept of a quasilinear spaces both including a classical linear spaces and also nonlinear spaces of subsets and multivalued mappings. Further, Aseev presented some basic quasilinear counterpart of linear functional analysis by introducing the notions of norm and bounded quasilinear operators and functionals. Our investigations show that translation may destroy the property of being a neighborhood of a set in topological quasilinear spaces in contrast to the situation in topological vector spaces. Thus, we prove that any topological quasilinear space may not satisfy the localization principle of topological vector spaces.

Highlights

In 1, Aseev introduced the concept of quasilinear spaces both including classical linear spaces and modelling nonlinear spaces of subsets and multivalued mappings
We show later that the minimality is a property of 0 and is shared by the other regular elements
In a real linear space, equality is the only way to define a partial ordering such that conditions 2.1 hold

Summary

Introduction

In 1 , Aseev introduced the concept of quasilinear spaces both including classical linear spaces and modelling nonlinear spaces of subsets and multivalued mappings.

Results

Conclusion