Topological Quasilinear Spaces (original) (raw)

Normed proper quasilinear spaces

Journal of Nonlinear Sciences and Applications, 2015

The fundamental deficiency in the theory of quasilinear spaces, introduced by Aseev [S. M. Aseev, Trudy Mat. Inst. Steklov., 167 (1985), 25-52], is the lack of a satisfactory definition of linear dependence-independence and basis notions. Perhaps, this is the most important obstacle in the progress of normed quasilinear spaces. In this work, after giving the notions of quasilinear dependence-independence and basis presented by Banazılı[H. K. Banazılı, M.Sc. Thesis, Malatya, Turkey (2014)] and Çakan [S. Çakan, Ph.D. Seminar, Malatya, Turkey (2012)], we introduce the concepts of regular and singular dimension of a quasilinear space. Also, we present a new notion namely "proper quasilinear spaces" and show that these two kind dimensions are equivalent in proper quasilinear spaces. Moreover, we try to explore some properties of finite regular and singular dimensional normed quasilinear spaces. We also obtain some results about the advantages of features of proper quasilinear spaces.

A Note on the Quasi-normed Linear Space

2018

In this paper, an alternative way of proving the quasi-normed linear space is provided through binomial inequalities. The new quasi-boundedness constant K = (α + β) 1 n ≥ 1, provides various ways for selecting values of quasi-boundedness constant for a quasi-normed linear space. Notwithstanding, the establishment of a subspace, quasi-product normed linear space, of the quasi-normed linear space was observed. Also, we have showed that the quasi-normed linear space admits quasi-Banach dilation mapping which is a Lipschitzian mapping. AMS Subject Classification: 44B54, 44B55.

Riesz Lemma in Normed Quasilinear Spaces and Its an Application

Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 2017

Aseev (Proc Steklov Inst Math 2:23-52, 1986) started a new field in functional analysis by introducing the concept of normed quasilinear spaces which is a generalization of classical normed linear spaces. Then, we introduced the normed proper quasilinear spaces in addition to the notions of regular and singular dimension of a quasilinear space, Ç akan and Yılmaz (J Nonlinear Sci Appl 8:816-836, 2015). In this study, we classify the normed proper quasilinear spaces as ''solid-floored'' and ''non solid-floored''. Thus, some properties of normed proper quasilinear spaces become more comprehensible. Also we present the counterpart of classical Riesz lemma in normed quasilinear spaces. Keywords Quasilinear spaces Á Floor of an element Á Normed proper quasilinear spaces Á Solid-floored quasilinear spaces Á Riesz lemma for normed quasilinear spaces Mathematics Subject Classification 06B99 Á 32A70 Á 46A99 Á 46B40 Á 54F05 2 Preliminaries and Some Results About Quasilinear Spaces and Normed Proper Quasilinear Spaces Definition 2.1 [1] A set X is called a quasilinear space (qls, for short), if a partial order relation ''"'' , an algebraic sum operation and an operation of multiplication by real numbers are defined in it in such a way that the following conditions hold for any elements x; y; z; v 2 X and any a; b 2 R:

On the Subspace Problem for Quasi-normed Spaces

Asian Research Journal of Mathematics, 2016

A definitive positive answer to the so-called proper subspace problem (a.k.a. the atomic space problem) for quasi-normed spaces is given: every infinite dimensional quasi-normed space has a proper closed infinite-dimensional subspace.

A Generalization of the Hahn-Banach Theorem in Seminormed Quasilinear Spaces

Journal of Mathematics and Applications, 2019

The concept of normed quasilinear spaces which is a generalization of normed linear spaces gives us a new opportunity to study with a similar approach to classical functional analysis. In this study, we introduce the notion of seminormed quasilinear space as a generalization of normed quasilinear spaces and give various auxiliary results and examples. We present an analog of Hahn-Banach theorem, in seminormed quasilinear spaces.

Some new results on inner product quasilinear spaces

Cogent Mathematics, 2016

In this article, we research on the properties of the floor of an element taken from an inner product quasilinear space. We prove some theorems related to this new concept. Further, we try to explore some new results in quasilinear functional analysis. Also, some examples have been given which provide an important information about the properties of floor of an inner product quasilinear space.

Quasilinear Inner Product Spaces and Hilbert Quasilinear Spaces

International Journal of Analysis, 2014

Aseev launched a new branch of functional analysis by introducing the theory of quasilinear spaces in the framework of the topics of norm, bounded quasilinear operators and functionals (Aseev (1986)). Furthermore, some quasilinear counterparts of classical nonlinear analysis that lead to such result as Frechet derivative and its applications were examined deal with. This pioneering work causes a lot of results in such applications such as (Rojas-Medar et al. (2005), Talo and Başar (2010), and Nikol'skiĭ (1993)). His work has motivated us to introduce the concept of quasilinear inner product spaces. Thanks to this new notion, we obtain some new theorems and definitions which are quasilinear counterparts of fundamental definitions and theorems in linear functional analysis. We claim that some new results related to this concept provide an important contribution to the improvement of quasilinear functional analysis.

A Remark about Quasi 2-Normed Space

C. Park [6] did the generalization of the term of quasi-normed space, i.e. he introduced the term of a quasi 2-normed space. Moreover, C. Park proved few properties of quasi 2-norm, and M. Kir and M. Acikgoz [2] elaborated the procedure for completing the quasi 2-normed space. In this paper will be proven that each quasi 2-norm generates a family of quasi-norms and will be considered some properties of such derived quasi-normed spaces. Mathematics Subject Classification 46B20 2718 Aleksa Malčeski et al.

A Pseudo Quasi in a Linear -Banach Spaces

International Journal of Innovation in Science and Mathematics, 2015

In this paper, a pseudo-normed spaces and quasi-normed spaces in a linear-Banach spaces is obtained.