Nonlinear Analysis 71 (2009) 1654–1661 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Approximation theory in 2-Banach spaces Mehmet Gürdal a,∗ , Ahmet Şahiner a , Işıl Açık b a Süleyman Demirel University, Department of Mathematics, 32260, Isparta, Turkey b Mehmet Akif Ersoy University, Education Faculty, 15100, Burdur, Turkey article info Article history: Received 26 October 2007 Accepted 7 January 2009 MSC: primary 40A05 46A70 secondary 41A36 41A65 Keywords: Convergence Cauchy sequence 2-Banach space T -convergence Approximation abstract In order to study the approximation theory in 2-Banach spaces, we define the concept of T -convergence by means of a sequence of linear operators in 2-Banach spaces, and we get some results by imposing the stability and approximation conditions on linear operators. Further, we consider some applications related to the subject. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction This paper was inspired by [1], where the concept of approximation of a Banach space by a sequence of Banach spaces is introduced. We will often quote some results from [1], that can be easily transferred to the concept of T -convergence by means of a sequence of linear operators in 2-Banach spaces. The concept of 2-normed spaces was initially introduced by Gähler [2–4] in the 1960’s. This notion, which is nothing but a two dimensional analogue of a normed space, got the attention of a wider audience after the publication of a paper by White [5]. Siddiqi delivered a series of lectures on this theme in various conferences. His joint paper with Gähler and Gupta [6] also provided valuable results related to the theme of this paper. Results up to 1977 were summarized in the survey paper by Siddiqi [7]. The idea of developing approximation theory in 2-normed (2-Banach) spaces was discussed through letters and long meetings in Berlin by Gähler and Siddiqi as early as 1972 (see, for instance [8–10]). Since then, this concept has been studied by many authors, see for instance [11–16]. Approximation theory has many important applications in various areas of functional analysis, the theory of polynomial approximation and numerical solutions of differential and integral equations [17–19]. Most of the classical approximation operators tend to converge to the value of the function being approximated. The study of approximation of a Banach space by a sequence of Banach spaces is a well-established area of research which deals with the problem of approximating the relationship between the spaces X n and X by the sequence of linear operators T n . Recently, it has been studied in different areas of functional analysis [20–22]. It is a well-known fact that 2-Banach spaces are in fact a special case of Banach spaces. What we offer in this paper is to study approximation theory in 2-Banach spaces organizing the paper as follows: in Section 2, T -convergence in 2-Banach ∗ Corresponding author. E-mail addresses: gurdal@fef.sdu.edu.tr (M. Gürdal), sahiner@fef.sdu.edu.tr (A. Şahiner), isilacik@yahoo.com (I. Açık). 0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.030