%0 Journal Article
%T Some Properties of the Sequence Space
%A Kuldip Raj
%A Sunil K. Sharma
%J Journal of Operators
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/904838
%X We introduce the sequence space defined by a Musielak-Orlicz function . We also study some topological properties and prove some inclusion relations involving this space. 1. Introduction and Preliminaries The concept of 2-normed spaces was initially developed by G£¿hler [1] in the mid-1960s, while one can see that of -normed spaces in Misiak [2]. Since then, many others have studied this concept and obtained various results; see Gunawan [3, 4] and Gunawan and Mashadi [5]. Let and be a linear space over the field , where is the field of real or complex numbers of dimension , where . A real valued function on satisfying the following four conditions: (1) if and only if are linearly dependent in ; (2) is invariant under permutation; (3) for any ; and (4) is called an -norm on , and the pair is called an -normed space over the field . For example, we may take being equipped with the -norm = the volume of the -dimensional parallelopiped spanned by the vectors which may be given explicitly by the formula where for each . Let be an -normed space of dimension and be linearly independent set in . Then the following function on defined by defines an -norm on with respect to . A sequence in an -normed space is said to converge to some if A sequence in an -normed space is said to be Cauchy if If every Cauchy sequence in converges to some , then is said to be complete with respect to the -norm. Any complete -normed space is said to be -Banach space. An Orlicz function is a function which is continuous, nondecreasing, and convex with , for and as . Lindenstrauss and Tzafriri [6] used the idea of Orlicz function to define the following sequence space. Let be the space of all real or complex sequences ; then which is called an Orlicz sequence space. The space is a Banach space with the norm It is shown in [6] that every Orlicz sequence space contains a subspace isomorphic to . The -condition is equivalent to for all values of and for . A sequence of Orlicz function is called a Musielak-Orlicz function; see [7, 8]. A sequence defined by is called the complementary function of a Musielak-Orlicz function . For a given Musielak-Orlicz function , the Musielak-Orlicz sequence space and its subspace are defined as follows: where is a convex modular defined by We consider equipped with the Luxemburg norm or equipped with the Orlicz norm Let be a linear metric space. A function : is called paranorm if (1) for all , (2) for all , (3) for all , (4)if is a sequence of scalars with as , and is a sequence of vectors with as ; then as . A paranorm for which implies is called total
%U http://www.hindawi.com/journals/joper/2013/904838/