International Journal of Mathematics Trends and Technology (IJMTT) – Volume 48 Number 2 August 2017 On Multi Normed Linear Spaces Ranajoy Roy 1, Sujoy Das 2, S. K. Samanta 3 1 Department of Mathematics, Visva Bharati, Santiniketan, West Bengal, India. 2 Department of Mathematics, Suri Vidyasagar College, Suri, Birbhum -731101, West Bengal, India 3 Department of Mathematics, Visva Bharati, Santiniketan, West Bengal, India. Abstract — In the present paper notions of multi linear the idea of norm in such spaces. Furthermore we are space and multi normed linear space are presented and going to investigate some properties of such Multi some basic properties of such spaces have been normed linear spaces. investigated. Keywords — Multi vector space, multi vector, multi II. PRILIMINARIES norm, multi open set, multi limit point, multi derived Definition 2.1. [11] A multi set 푀 drawn from the set set. 푋 is represented by a function 퐶표푢푛푡푀 or 퐶푀 defined I. INTRODUCTION as 퐶푀 : 푋 → 푁where 푁 represents the set of non Multiset (bag) is a well established notion negative integers. both in mathematics and in computer science ([9], Definition 2.2. [11] Let 푀 and 푁 be two msets drawn [10], [22]). In mathematics, a multiset is considered to from a set 푋. Then, the following are defined: be the generalization of a set. In classical set theory, a 푖 푀 = 푁 푖푓 퐶푀 푥 = 퐶푁 푥 푓표푟 푎푙푙 푥 ∈ 푋. set is a well defined collection of distinct objects. If 푖푖 푀 ⊂ 푁 푖푓 퐶푀 푥 ≤ 퐶푁 푥 푓표푟 푎푙푙 푥 ∈ 푋. repeated occurrences of any object is allowed in a set, 푖푖푖 푃 = 푀 ∪ 푁 푖푓 퐶푃 푥 = 푀푎푥 퐶푀 푥 , 퐶푁 푥 then a mathematical structure, that is known as multiset (mset, for short), is obtained ([21], [23], [24]). 푓표푟 푎푙푙 푥 ∈ 푋. 푖푣 푃 = 푀 ∩ In various counting arguments it is convenient to 푁 푖푓 퐶푃 푥 = 푀푖푛 퐶푀 푥 , 퐶푁 푥 distinguish between a set like {푎, 푏, 푐} and a collection like {푎, 푎, 푎, 푏, 푐, 푐}.The latter, if viewed as a set, will 푓표푟 푎푙푙 푥 ∈ 푋. be identical to the former. However, it has some of its 푣 푃 = 푀 ⊕ 푁 푖푓 퐶푃 푥 = 퐶푀 푥 + 퐶푁 푥 elements purposely listed several times. We formalize 푓표푟 푎푙푙 푥 it by defining a multiset as a collection of elements, ∈ 푋: each considered with certain multiplicity. For the sake 푣푖 푃 = 푀 ⊝ 푁 푖푓 퐶푃 푥 of convenience a multiset is written as {푘1/푥1, 푘2/푥2, = 푀푎푥 퐶푀 푥 − 퐶푁 푥 , 0 … . , 푘푛 /푥푛 } in which the element 푥푖 occurs 푘푖 times. 푓표푟 푎푙푙 푥 ∈ 푋, where ⊕ and ⊝represents mset We observe that each multiplicity 푘푖 is a positive addition and mset subtraction respectively. integer. Let 푀 be an mset drawn from a set 푋. The support set From 1989 to 1991, Wayne D. Blizard made of 푀, denoted by 푀∗ , is a subset of X and 푀∗ = a thorough study of multiset theory, real valued ∗ {푥 ∈ 푋 ∶ 퐶푀 (푥) > 0}, i.e., 푀 is an ordinary set. multisets and negative membership of the elements of 푀∗ is also called root set. multisets ([1], [2],[3],[4]). K. P. Girish and S. J. John An mset M is said to be an empty mset if for all introduced and studied the concepts of multiset x X,CM (x) 0 . The cardinality of an mset 푀 topologies, multiset relations, multiset functions, drawn from a set 푋 is denoted by Card (M) or chains and antichains of partially ordered multisets ([11], [12],[13],[14],[15]). D. Tokat studied the |푀| and is given by 퐶푎푟푑 푀 = 푥∈ 푋 퐶푀 . concept of soft multi continuous function Definition 2.3. [20] Multi point: Let M be a multi set [25].Concepts of multigroups and soft multigroups are over a universal set 푋, 푥 ∈ 푋 and 푘 ∈ 푵 such that found in the studies of Sk. Nazmul and S. K. Samanta 푘 ≤ 퐶푀 푥 . Then a multi point of M is defined by a 푘 푘 ([17], [18]). Many other authors like Chakrabarty et al. mapping 푃푥 ∶ 푋 → 푵 such that 푃푥 푦 = 푘 if 푦 = 푥 ([5], [6], [7], [8]), S. P. Jena et al. ([16]), J. L. Peterson ([19]) also studied various properties and applications 0 , otherwise x and k will be referred to as the base and the of multisets. multiplicity of the multi point 푃푘 , respectively. In our previous paper we have introduced the 푥 Collection of all multi points of an mset M is denoted notion of Multi metric Space. In this paper we are by 푀 . going to introduce a concept of Multi linear space and 푝푡 ISSN: 2231-5373 http://www.ijmttjournal.org Page 111 International Journal of Mathematics Trends and Technology (IJMTT) – Volume 48 Number 2 August 2017 Definition 2.4. [20] The mset generated by a Definition 2.12. [20] Let (푀, 푑) be an M-metric collection 퐵 of multi points is denoted by 푀푆(퐵) and space. Then a collection B of multi points of M is said 푥 is defined by 퐶푀푆 퐵 푥 = 푆푢푝 푘 ∶ 푃푘 ∈ 퐵 . to be open if every multi point of B is an interior point + 푘 Definition 2.5. [20] Let 푚푹 denote the multi set over of B i.e., for each 푃푎 ∈ 퐵, there esists an open ball + 푘 1 푘 푹 (set of non-negative real numbers) having 퐵(푃푎 , 푃푟 ) with centre at 푃푎 and r > 0 such that 푘 1 multiplicity of each element equal to 푤, 푤 ∈ 푁. The 퐵(푃푎 , 푃푟 ) ⊂ 퐵. + members of 푚푹 푝푡 will be called non-negative 휙 is separately considered as an open set. multi real points. Definition 2.13. [20] Let (푀, 푑) be an M-metric 푎 푏 space. Then 푁 ⊂ 푀 is said to be multi open in (푀, 푑) Definition 2.6. [20] Let 푃푖 and 푃푗 be two multi real + 푎 푏 푎 iff there esists a collection B of multi points of N such points of 푚푹 . We define 푃푖 > 푃푗 if 푎 > 푏 or 푃푖 > that B is open and MS(B) = N. 푃푏 if 푖 > 푗 when 푎 = 푏. 푗 Definition 2.14. [20] A multi set N in an M-metric Definition 2.7. [20] (Addition of multi real points) space (푀, 푑) is said to be multi closed if its 푎 푏 푘 We define 푃푖 + 푃푗 = 푃푖+푗 where 푘 = 푀푎푥 푖, 푗 , complement 푁푐 is multi open in (푀, 푑). 푎 푏 + 푃푖 ; 푃푗 ∈ 푚푹 푝푡 . Definition 2.15. [20] Let (푀, 푑) be an M-metric space Definition 2.8. [20] (Multiplication of multi real and B be a collection of multi points of M. Then a 푙 points) We define multiplication of two multi real multi point 푃푥 of M is said to be a limit point of B if + 푙 1 푙 points in 푚푹 as follows: every open ball 퐵(푃푥 , 푃푟 ) (푟 > 0) containing 푃푥 in 1 푎 푏 1 푙 푎 푏 푃0 , if either 푃푖 표푟 푃푗 = 푃0 (푀, 푑) contains at least one point of B other than 푃푥 . 푃 × 푃 = 푖 푗 푘 푃푎푏 , otherwise where 푘 = 푀푎푥 푖, 푗 The set of all limit points of B is said to be the derived . set of B and is denoted by 퐵푑 . Definition 2.9. [20] Multi Metric: Definition 2.16. [20] Let (푀, 푑) be an M-metric space + 푙 Let 푑 ∶ 푀푝푡 × 푀푝 푡 → 푚푹 푝푡 (푀 being a multi set and 푁 ⊂ 푀. Then 푃푥 ∈ 푀푝푡 is said to be a multi over a Universal set 푋 having multiplicity of any limit point of N if it is a limit point of 푀푝푡 ie. if every 푙 1 푙 element at most equal to 푤 ) be a mapping which open ball 퐵(푃푥 , 푃푟 ) (푟 > 0) containing 푃푥 in (푀, 푑) 푙 satisfy the following: contains at least one point of 푁푝푡 other than 푃푥 . 푙 푚 1 푙 푚 푀1 푑 푃푥 , 푃푦 ≥ 푃0 , ∀ 푃푥 , 푃푦 ∈ 푀푝푡 Definition 2.17. [20] Let (푀, 푑) be an M-metric space 푙 푚 1 푙 푚 푙 푚 푀2 푑 푃푥 , 푃푦 = 푃0 푖푓푓 푃푥 , = 푃푦 ∀ 푃푥 , 푃푦 and 퐵 ⊂ 푀푝푡 . Then the collection of all points of B ∈ 푀푝푡 . together with all limit points of B is said to be the 푙 푚 푚 푙 푙 푚 푀3 푑 푃푥 , 푃푦 = 푑 푃푦 , 푃푥 ∀ 푃푥 , 푃푦 ∈ 푀푝푡 . closure of B in (푀, 푑) and is denoted by 퐵. Thus 푙 푚 푚 푛 푙 푛 퐵 = 퐵 ∪ 퐵푑 . 푀4 푑 푃푥 , 푃푦 + 푑 푃푦 , 푃푧 ≥ 푑 푃푥 , 푃푧 , 푙 푚 푛 Definition 2.18. [20] Let (푀, 푑) be an M-metric space ∀ 푃푥 , 푃푦 , 푃푧 and 푁 ⊂ 푀. Then the multi set generated by all multi ∈ 푀 . 푝푡 points and all multi limit points of N is said to be the 푀5 퐹표푟 푙 ≠ 푚, 푑 푃푙 , 푃푚 = 푃푘 ⟺ 푥 = 푦 푥 푦 0 multi closure of N and is denoted by 푁 . 푎푛푑 푘 = 푀푎푥 푙, 푚 . 푙 Definition 2.19. [20] A sequence {푃 푛 } of multi points Then d is said to be a multi metric on M and (푀, 푑) is 푥푛 in 푚푹+ is said to converge to 푃1 if for any 휖 > 0, called a Multi metric (or an M - metric)space. 0 ∃ 푛 ∈ 푁 such that 푃푙푛 < 푃1 ∀ 푛 ≥ 0. Definition 2.10. [20] Let 푀, 푑 be an M-metric 0 푥푛 ∈ 푛 푘 Since 푙 ≥ 1 ∀ 푛 ∈ 푁, 푃푙푛 < 푃1 ⟺ 푥 < ∈ .