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Abstract and Applied Analysis
Volume 2012, Article ID 951374, 10 pages
http://dx.doi.org/10.1155/2012/951374
Research Article

Topological Quasilinear Spaces

Department of Mathematics, Inonu University, 44280 Malatya, Turkey

Received 14 March 2012; Accepted 22 July 2012

Academic Editor: Jing Ping Wang

Copyright © 2012 Yılmaz Yılmaz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.