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

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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 푃0 , if either 푃 표푟 푃 = 푃0 (푀, 푑) contains at least one point of B other than 푃 . 푃푎 × 푃푏 = 푖 푗 푥 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 < ∈ . space, 푟 > 0 and 푃푎 ∈ 푀푝푡 . Then the open ball 푛 푥푛 ∈ 푛 푘 1 푙 with centre 푃푎 and radius 푃푟 푟 > 0 is denoted by ∴ 푃 푛 → 푃1 ⟺ 푥 → 0 푎푠 푛 → ∞ in 푹+. 푥푛 0 푛 퐵 푃푘 , 푃1 and is defined by 퐵 푃푘 , 푃1 = 푎 푟 푎 푟 Definition 2.20. [20] Let {푃푙푛 } be a sequence of multi 푃푙 ∶ 푑 푃푙 , 푃푘 < 푃1 . 푥푛 푥 푥 푎 푟 points in an M-metric space (푀, 푑) . The sequence 푀푆[퐵 푃푘 ,푃1 ] will be called a multi open ball with 푎 푟 푙푛 푙 푘 1 1 {푃푥 } is said to converge in (푀, 푑) if ∃ 푃푥 ∈ 푀푝푡 centre 푃푎 and radius 푃푟 > 푃0 . 푛 푙 푘 1 푙 such that 푑 푃 푛 , 푃푙 → 푃1 푎푠 푛 → ∞. Definition 2.11. [20] 퐵 푃푎 , 푃푟 = 푃푥 ∶ 푥푛 푥 0 푙 푘 1 푑 푃푥 , 푃푎 ≤ 푃푟 is called the closed ball with III. MULTI VECTOR SPACE 푘 1 푘 1 centre 푃푎 and radius 푃푟 푟 > 0 . 푀푆 퐵 푃푎 ,푃푟 Definition 3.1. Multi vector space : Let V be vector 푘 space over a field K (to be denoted by 푉 ). A multiset will be called a multi closed ball with centre 푃푎 and 퐾 1 X over V is said to be a multi vector space or a multi radius 푃푟 푟 > 0 . linear space or Mvector space of V over K if every

ISSN: 2231-5373 http://www.ijmttjournal.org Page 112 International Journal of Mathematics Trends and Technology (IJMTT) – Volume 48 Number 2 August 2017 element of X has the same multiplicity and the support of F has the same multiplicity and for any 푎, 푏 ∈ 퐾, 푋∗ of X is a subspace of V . 푎퐹 + 푏퐹 ⊂ 퐹. The multiplicity of every element of X will be denoted by 푤푋 . Proof. Let 퐹 ⊂ 푋 is an Msubspace of X. Then by Example 3.2. Let 푹3 be the Euclidean 3-dimensional definition, every element of F has the same ∗ space over 푹 . Let 푋 = {5/(푎, 푏, 0) ∶ 푎, 푏 ∈ 푹} . multiplicity 푤퐹 and 퐹 is a subspace of 푉퐾 . Then X is a multi vector space of 푹3 over 푹. Consequently ∀ 푎, 푏 ∈ 퐾, (푎퐹 + 푏퐹 ) = 퐹∗. Also Definition 3.3. (i) Addition of two multisets over V : for 푥 ∈ 푎퐹 + 푏퐹 ∗ = 퐹∗,

Let P and Q be two multisets over V . Then P + Q is a 퐶푎퐹+푏퐹 푥 = 푆푢푝 {퐶푎퐹 푢 퐶푏퐹 (푣) ∶ 푢 ∈ ∗ ∗ multiset over V such that 퐶푃+푄 푥 = 푆푢푝 {퐶푃 푢 ∨ 푎퐹 , 푣 ∈ 푏퐹 푎푛푑 푥 = 푢 + 푣} . ∗ 퐶푄(푣) ∶ 푥 = 푢 + 푣, ∀ 푢 ∈ 푃 , 푣 ∈ = 푆푢푝 {퐶푎퐹 (푎푝) 퐶푏퐹 (푏푞) ∶ 푝, 푞 ∈ 퐹_ 푎푛푑 푥 = 푄∗}. Clearly 푃 + 푄 ∗ = 푃∗ + 푄∗ = {푢 + 푣: 푢 ∈ 푎푝 + 푏푞} ∗ ∗ 푃 , 푣 ∈ 푄 } = 푆푢푝 {퐶퐹 푝 퐶퐹 (푞) ∶ 푝, 푞 ∈ 퐹_ 푎푛푑 푥 = 푎푝 + (ii) Let P be a multiset over V and 푎 ∈ 퐾, then for 푏푞}

푎 ≠ 0, we define 푎푃 as 퐶푎푃 (푥) = 퐶푃 (푦) where = 푆푢푝 푤퐹 푤퐹 = 푤퐹 = 퐶퐹 푥 . ∗ −1 푦 ∈ 푃 and 푥 = 푎푦 ie 퐶푎푃 (푥) = 퐶푃 (푎 푥) and for 푎 = 0, we define 푎푃 as 퐶푎푃 (푥) = Conversely, let 퐹 ⊂ 푋 such that every 0, if 푥 ≠ 휃 element of F has the same multiplicity and for any ∗ 푦∈푃∗ 퐶푃 (푦) , 푖푓 푥 = 휃 푎, 푏 ∈ 퐾, 푎퐹 + 푏퐹 ⊂ 퐹. ⇒ 푎퐹 + 푏퐹 = Clearly for 푎 ≠ 0, 푎푃 ∗ = 푃∗ and for a = 0, 푎퐹∗ + 푏퐹∗ ⊂ 퐹∗ ∀ 푎, 푏 ∈ 퐾 ∗ ∗ 푎푃 = 휃 . ⇒ 푎푢 + 푏푣 ∈ 퐹 ∀ 푎, 푏 ∈ 퐾 푎푛푑 ∀ 푢 , 푣 ∈ 푉퐾 ∗ (iii) If F is a multiset over V and 푥 ∈ 푉 , we define ⇒ 퐹 is a subspace of 푉퐾. 푥 + 퐹 to be a multiset over V defined as 푥 + Hence F is an Msubspace of X. ∗ ∗ ∗ 퐹 = 푥 + 퐹 = {푥 + 푓 ∶ 푓 ∈ 퐹 } and 퐶푥+퐹 (푢) = ∗ 퐶퐹 (푣) where 푣 ∈ 퐹 and 푢 = 푥 + 푣. Proposition 3.9. (i) If F and G are two Msubspaces of (iv) If 푈 ⊂ 푉 and F is a multiset over V. Then 푈 + 퐹 an Mvector space X over 푉퐾 , then 퐹 + 퐺 and ∀ a ∈ is a multiset over 푉 defined as 푈 + 퐹 = {푚/푢 + 푓 ∶ K, 푎퐹 are also Msubspaces of X over 푉퐾: ∗ 푢 ∈ 푈, 푓 ∈ 퐹 and 푚 = 퐶퐹 푓 } . (ii) If {퐹푖 ∶ 푖 ∈ ∆} be a family of Msubspaces of an Clearly 푈 + 퐹 = 푢∈푈 (푢 + 퐹). Mvector space X over 푉퐾 , then 푖∈∆ 퐹푖 is also an Msubspace of X. Theorem 3.4. If F and G are two multisets over a Proof: The proof is straight forward. vector space 푉퐾, then for 푎 ∈ 퐾, 푎(퐹 + 퐺) = 푎퐹 + 푎퐺. IV. MULTIVECTORS IN MVECTOR SPACE

Theorem 3.5. If , 퐺푖 , 푖 = 1, 2, 3 … … … … . 푛, are Definition 4.1. Multivectors: Let X be an Mvector multisets over a vector space 푉, 퐹 = 퐹1 + 퐹2 + ⋯ + space over a vector space 푉퐾. Then every multi point 퐹푛, 퐺 = 퐺1 + 퐺2 + ⋯ + 퐺푛 and 퐻 = 퐹1 + 퐹2 + of X ie., every element of 푋푝푡 will be called a ⋯ + 퐹푛 + 퐺2 + ⋯ + 퐺푛 , then 퐻 = 퐹 + 퐺. multivector or Mvector of X. Proof: The proof is straight forward. Definition 4.2. Multi scalar field: Let K be a field. Definition 3.6. Let X be an Mvector space over a Then a multi set L over K is called a multi scalar field vector space 푉퐾. Then 퐹 ⊂ 푋 is said to be a multi or Mscalar field if every element of L has the same ∗ subspace or Msubspace of X if F is an Mvector space multiplicity (denoted by 푤퐿 ) and the support 퐿 of L ∗ over 푉퐾 ie 퐹 is a subspace of 푉퐾 and every element of is a subfield of K. F has the same multiplicity. Multi points of L will be referred to as multi scalars or Mscalars of L. Example 3.7. 푌 = {4/(푎, 0, 0) ∶ 푎 ∈ 푹} is an Multiplicity of each element of L will be denoted by

Msubspace of the Mvector space defined in example 푤퐿. 3.2. Example 4.3. In example 3.2, 1 2 4 Theorem 3.8. Let X be an Mvector space over 푉퐾. 푃 1,1,0 , 푃 1,1,0 , 푃 1,5,0 etc. are Mvectors of the given Then 퐹 ⊂ 푋 is an Msubspace of X iff every element Mvector space.

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푌∗ ⇒ 퐿∗ is a subspace 푋∗_ and since every element

Definition 4.4. Let X be an Mvector space over 푉퐾. of Y has the same multiplicity, Y is an Msubspace of 푘 Then an Mvector 푃푥 of X will be called a null X. Mvector if its base 푥 = 휃 (휃 being the null vector of ∗ 푘 푘 푘 푋 ie 푉퐾) and it will be denoted by Θ , ie. 푃푥 = Θ Definition 4.9. Multi linear combination: Let X be if 푥 = 휃 . an Mvector space over a vector space 푉퐾 and L be an 푘 An Mvector푃푥 will be called non null if 푥 ≠ 휃. Mscalar field over K such that 푤퐿 ≤ 푤푋 . Then an 푙 Mvector 푃푥 ∈ 푋푝푡 is said to be a multi linear Note 4.5. Null Mvector of an Mvecotr space is not combination or Mlinear combination of the vectors unique. 푃푙1 , 푃푙2 , … … … … … .. 푃푙푛 ∈ 푋 over L if 푃푙 can 푥1 푥2 푥푛 푝푡 푥 be expressed as Definition 4.6. Let X be an Mvector space over a 푃푙 = 푃푖1 . 푃푙1 + 푥 푎푖 푥1 vector space 푉퐾, L be an Mscalar field over K such 푖2 푙2 푖푛 푙푛 푙 푚 푖 푃 . 푃 , + ⋯ … … … … . + 푃 . 푃 for some Mscalars that 푤퐿 ≤ 푤푋 , 푃푥 , 푃푦 ∈ 푋푝푡 and 푃푎 ∈ 퐿푝푡 .Then we 푎2 푥2 푎푛 푥푛 푃푖1 , 푃푖2 … … … … … … … . . 푃푖푛 ∈ 퐿 . define 푎푖 푎2 푎푛 푝푡 푃1 푖푓푓 푥 = − 푦 푎푛푑 푙 = 푚 푙 푚 휃 푃푥 + 푃푦 = 푙 푚 푃푥+푦 표푡ℎ푒푟푤푖푠푒 Example 4.10. Let us consider the Mvector space given in Example 3.2. Let L be an Mscalar field over and 푹 given by 퐿 = {3/푟 ∶ 푟 ∈ 푹}. 퐿푒푡 푥1 = 1 푖 1 푙 1 (1, 2, 0), 푥2 = (−2, 5, 0), 푥3 = (0, 1, 0). Then 푖 푙 푃휃 푖푓푓푃푎 = 푃0 표푟 푃푥 = 푃휃 푃푎 . 푃푥 = , 1 푖∨푙 푃푥 + 푐 , 푃푎푥 표푡ℎ푒푟푤푖푠푒 1 1 2 3 3 1 2 2 3 2 2 3 where 0 is the null element of K. 푃푥1 + 푃푥2 + 푃푥3 , 푃2 . 푃푥1 + 푃5 푃푥2 ; 푃5 . 푃푥2 + 푃7/8. 푃푥3 6 are Mlinear combinations of the Mvectors Proposition 4.7. Let X be an Mvector space over a 1 2 3 푃푥 , 푃푥 , 푃푥 over L. vector space 푉 and L be an Mscalar field over K 1 2 3 퐾 such that푤 ≤ 푤 . Then for 푃푙 ∈ 푋 and 푃푖 ∈ 퐿 퐿 푋 푥 푝푡 푎 푝푡 Definition 4.11. Multi linearly dependent and multi with 푙, 푖 > 1, linearly independent: Let X be an Mvector space 푖 푙 푛 (i) 푃푎 . 푃푥 = 훩 where 푛 = 푖 ∨ 푙. over a vector space 푉퐾. Then a finite collection of 푖 푙 푛 (ii) 푃푎 . 훩 = 훩 where 푛 = 푖 ∨ 푙. 푙1 푙2 푙푛 Mvectors 푃푥 , 푃푥 , … … … … … .. 푃푥 of X is said to (iii) 푃푖 푃푙 = . 푃푛 where 푛 = 푖 ∨ 푙. 1 2 푛 −1 . 푥 −푥 be multi linearly dependent or Mlinearly dependent or (iv) ∀ 푃푙 ∈ 푋 and 푃푖 ∈ 퐿 ; 푃푖푎, 푃푖 . 푃푙 = 훩푛 푥 푝푡 푎 푝푡 푎 푥 ML.D if for some ⇒ 푒푖푡ℎ푒푟 푎 = 0 표푟 푥 = 휃. ∗ Mscalar field L over K with 퐿 ≠ {0} and 푤퐿 ≤ 푤푋 , Proof: Proofs are straight forward. there exist Mscalars 푃푖1 , 푃푖2 … … … … … … … . . 푃푖푛 ∈ 푎푖 푎2 푎푛 for some i = 1, 2, ...... , n such that Theorem 4.8. Let X be an Mvector space over a 퐿푝푡 푃푖1 . 푃푙1 + 푃푖2 . 푃푙2 + ⋯ … … … … … … . . 푃푖푛 . 푃푙푛 = vector space 푉퐾. Then 푌 ⊂ 푋 is an Msubspace of X 푎푖 푥1 푎2 푥2 푎푛 푥푛 푙 iff every element of Y has the same multiplicity 푤푌 Θ . and for any Mscalar field L over K with 푤퐿 ≤ 푤푌, , 푖 푙 푗 푚 푖 푗 The collection of Mvectors 푃푎 . 푃푥 + 푃푏 . 푃푦 ∈ 푌푝푡 ∀ 푃푎 ; 푃푏 ∈ 퐿푝푡 and 푙 푚 푃푙1 , 푃푙2 , … … … … … .. 푃푙푛 of X is said to be multi 푃푥 , 푃푦 ∈ 푌푝푡 . 푥1 푥2 푥푛 Proof. Let 푌 ⊂ 푋 is an Msubspace of X. Then every linearly independent or Mlinearly independent or element of Y has the same multiplicity 푤푌 and the ML.Id it is not ML.D ie. if for any Mscalar field L ∗ ∗ ∗ support 푌 is a subspace of 푋 . Let L be an Mscalar over K with 퐿 ≠ 0 and 푤퐿 ≤ 푤푋 , a relation field over K such that 푃푖1 . 푃푙1 + 푃푖2 . 푃푙2 + ⋯ … … … … … … . . 푃푖푛 . 푃푙푛 = 푎푖 푥1 푎2 푥2 푎푛 푥푛 ∗ ∗ 푤퐿 ≤ 푤푌 . Then ∀ 푥, 푦 ∈ 푌 and 푎, 푏 ∈ 퐿 ⊂ 퐾, 푙 푖푘 Θ where 푃푎 ∈ 퐿푝푡 , 푘 = 1, 2, … … … . 푛, holds only ∗ 푖 푙 푗 푚 푘 푎푥 + 푏푦 ∈ 푌 ⇒ 푃푎 . 푃푥 + 푃푏 . 푃푦 ∈ when 푎푘 = 0 푖 푗 푙 푚 푌푝푡 ∀ 푃푎 ; 푃푏 ∈ 퐿푝푡 and 푃푥 , 푃푦 ∈ 푌푝푡 . ∀ 푘 = 1, 2, ...... 푛.

Conversely let the given conditions hold. Note 4.12. (i) Every superset of a finite collection of Then ∀ 푥, 푦 ∈ 푌∗ and 푎, 푏 ∈ 퐿∗ ⊂ 퐾, 푎푥 + 푏푦 ∈ ML.D Mvectors in an Mvector space is ML.D.

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(ii) Every nonempty subset of a finite collection of Theorem 4.18. Let X be an Mvector space over a

ML.Id Mvectors in an Mvector space is ML.Id. vector space 푉퐾 . Then a finite collection of Mvectors 푃푙1 , 푃푙2 , … … … … … .. 푃푙푛 of X is ML.D iff 푥1 푥2 푥푛 Definition 4.13. (i) An infinite collection of Mvectors 푥1, 푥2, … … … … … . . 푥푛 is L.D. in an Mvector space is ML.D if it has a finite ML.D Proof. The proof follows similarly as the previous subset. one. (ii) An infinite collection of Mvectors in an Mvector space is ML.Id if every nonempty finite subset of it is Definition 4.19. Linear span : Let 푋 be an Mvector ML.Id. space over a vector space 푉퐾 , L be an Mscalar field

over K such that 푤퐿 ≤ 푤푋 and Theorem 4.14. Let X be an Mvector space over a 푆 = 푃푙1 , 푃푙2 , … … … … … .. 푃푙푛 be a collection of 푥1 푥2 푥푛 vector space 푉퐾. If a collection of Mvectors Mvectors of 푋. Then the linear span of S over L 푃푙1 , 푃푙2 ,… … … … … .. 푃푙푛 of X is ML.Id (or 푥1 푥2 푥푛 denoted by 퐿푆(푆, 퐿) is defined as 퐿푆 푆, 퐿 = 푘1 푘2 푘푛 ML.D), then 푃 , 푃 , … … … … … .. 푃 is ML.Id 푖 푙 푖 푙 푖 푙 푥1 푥2 푥푛 1 1 2 2 푛 푛 푃푎 . 푃푥 + 푃푎 . 푃푥 + ⋯ … … … … … … . . 푃푎 . 푃푥 ∶ (or ML.D) 푖 1 2 2 푛 푛 푖1 푖2 푖푛 푃푎 , 푃푎 , … … … … … … … . . 푃푎 ∈ 퐿푝푡 . ∀ 푘1, 푘2, … … … … . , 푘푛 ∈ 1, 2, … … … . . , 푤푋 . 푖 2 푛 Proof. The proof is straight forward. 푀푆 퐿푆 푆, 퐿 will be referred to as the multi linear span or Mlinear span of S over L. Definition 4.15. An arbitrary multiset 퐺 ⊂ 푋 is said to be ML.D or ML.Id according as the collection Theorem 4.20. Let X be an Mvector space over a 푙 ∗ vector space 푉퐾, L be an Mscalar field over K such {푃푥 ∶ 푥 ∈ 퐺 } that 푤 ≤ 푤 and 푆 = 푃푙1 , 푃푙2 , … … … … … .. 푃푙푛 is ML.D or ML.Id. 퐿 푋 푥1 푥2 푥푛 be a collection of Mvectors of X. If 푀푆[퐿푆(푆, 퐿)] = ∗ Note 4.16. Any collection of Mvectors having at least 푋 then 푋 = 퐿푆 푥1, 푥2, … … … … … . . 푥푛 and either two elements with same base is ML.D. 푤퐿 = 푤푋 표푟 푙푘 = 푤푋 for some 푘 = 1, 2, … … … . 푛 and conversely. Theorem 4.17. Let X be an Mvector space over a Proof: Let 푀푆[퐿푆(푆, 퐿)] = 푋. Then for any ∗ vector space 푉퐾. Then a finite collection of Mvectors 푥 ∈ 푋 , 퐶푋 푥 = 푤푋 ⇒ 퐶푀푆 퐿푆 푆,퐿 푥 = 푤푋 ⇒ 푃푙1 , 푃푙2 ,… … … … … .. 푃푙푛 of X is ML.Id iff 푙 푤푋 푥1 푥2 푥푛 푆푢푝 {푙 ∶ 푃푥 ∈ 퐿푆 푆, 퐿 } = 푤푋 ⇒ 푃푥 ∈ 푖1 푖2 푖푛 푥1, 푥2, … … … … … . . 푥푛 is L.Id. 퐿푆(푆, 퐿) ⇒ ∃ 푃 , 푃 , … … … … … … … . . 푃 ∈ 푎푖 푎2 푎푛 푙1 푙2 푙푛 Proof: Let 푃 , 푃 ,… … … … … .. 푃 is ML.Id , 푤푋 푖1 푙1 푖2 푙2 푥1 푥2 푥푛 퐿 푠푢푐ℎ 푡ℎ푎푡 푃 = 푃 . 푃 + 푃 . 푃 + 푝푡 푥 푎푖 푥1 푎2 푥2 ∗ L be an Mscalar field over 푉 with 퐿 = 퐾, 푤 ≤ 푤 푖 푙 퐾 퐿 푋 ⋯ … … … … … … . . 푃 푛 . 푃 푛 and 푎 , 푎 , … … … … … … 푎 ∈ 퐾 = 퐿∗ such that 푎푛 푥푛 1 2 푛 ⇒ 푥 = 푎 . 푥 + 푎 . 푥 + ⋯ … … . + 푎 푥 = 휃 ⇒ 1 1 2 2 푛 푛 푎 . 푥 + 푎 . 푥 + ⋯ … … . + 푎 푥 푎푛푑 푤 = 푖1 푙1 푖2 푙2 푖푛 푙푛 1 1 2 2 푛 푛 푋 푃푎 . 푃푥 + 푃푎 . 푃푥 + ⋯ … … … … … … . . 푃푎 . 푃푥 = 푖 1 2 2 푛 푛 푆푢푝 {푖푘 , 푙푘 ∶ 푘 = 1, 2, … … … … . , 푛} ⇒ 푥 ∈ Θ푙 where 푃푖푘 ∈ 퐿 , 푘 = 1, 2, … … … . 푛 ⇒ 푎 = 푎푘 푝푡 1 퐿푆 푥1, 푥2,… … … … … . . 푥푛 and either 푖푘 = 푤푋 or 푎2 = ⋯ … … … … . = 푎푛 = 0 ⇒ 푙푘 = 푤푋 for some 푘 = 1, 2, … … … . 푛. Since 푥1,푥2, … … … … … . . 푥푛 is ML.Id. 푖푘 ≤ 푤퐿 ≤ 푤푋 ∀ 푘 = 1, 2, … … … . 푛, so for some 푘 = 1, 2, … … … . 푛, 푖푘 = 푤푋 ⇒ 푤퐿 = 푤푋 . ∗ Conversely let 푥1, 푥2, … … … … … . . 푥푛 is Also ∀ 푥 ∈ 푋 , 푥 ∈ 퐿푆 푥1, 푥2,… … … … … . . 푥푛 ⇒ ∗ ∗ ∗ L.Id and L be an Mscalar field over 푉퐾 with with 푋 ⊂ 퐿푆 푥1, 푥2, … … … … … . . 푥푛 ⊂ 푋 ⇒ 푋 = ∗ 퐿 = 퐾, 푤퐿 ≤ 푤푋 . Now for Mscalars 퐿푆 푥1, 푥2, … … … … … . . 푥푛 . 푃푖1 , 푃푖2 … … … … … … … . . 푃푖푛 ∈ 퐿 with 푃푖1 . 푃푙1 + 푎 푎2 푎푛 푝푡 푎 푥1 푖 푖 Conversely let 푃푖2 . 푃푙2 + ⋯ … … … … … … . . 푃푖푛 . 푃푙푛 = Θ푙 ⇒ 푎2 푥2 푎푛 푥푛 ∗ 푋 = 퐿푆 푥1, 푥2, … … … … … . . 푥푛 푎푛푑 푒푖푡ℎ푒푟 푙푘 = 푎1. 푥1 + 푎2. 푥2 + ⋯ … … . + 푎푛 푥푛 = 휃 ⇒ 푎1 = 푤푋 for some 푘 = 1, 2, … … … . 푛 or 푤퐿 = 푤푋 . Let 푎 = ⋯ … … … … . = 푎 = 0 2 푛 푥 ∈푤푋 푋 ⇒ 푥 ∈ 푋∗ = ⇒ 푃푙1 , 푃푙2 , … … … … … .. 푃푙푛 is ML.Id. 푥1 푥2 푥푛 퐿푆 푥1, 푥2,… … … … … . . 푥푛

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∗ + ⇒ ∃ 푎1, 푎2, … … , 푎푛 ∈ 퐿 such that 푥 = 푎1. 푥1 + 푚푹 ) and L as an Mscalar field over K with support ∗ 푎2. 푥2 + ⋯ … … . + 푎푛 푥푛 . If 푙푘 = 푤푋 for some 푘 = 퐿 = 퐾 and 푤퐿 ≤ 푤푋 . 1, 2, . … . , 푛, then 푙 푙 푙 + 푃1 . 푃 1 + 푃1 . 푃 2 + ⋯ … … … … … . . + 푃1 . 푃 푛 Definition 5.1. A mapping ∶ 푋푝푡 → 푚푹 푝푡 푎푖 푥1 푎2 푥2 푎푛 푥푛 푛 푙 will be called a multi norm or Mnorm on 푋 if it = 푃 푘=1 푘 = 푃푤푋 ⇒ 푃푤푋 푎1.푥1 + 푎2.푥2 +⋯…….+ 푎푛 푥푛 푥 푥 satisfies the following: ∈ 퐿푆 푆, 퐿 푙 1 푙 푁1 푃푥 ≥ 푃0 ∀ 푃푥 ∈ 푋푝푡 . ⇒ 퐶푀푆 퐿푆 푆,퐿 푥 = 푤푋 [since 푀푆[퐿푆(푆, 퐿)] ⊂ 푋] 푙 푘 푁2 푃푥 = 푃0 iff 푥 = 휃 and 푙 = 푘. ⇒ 푋 = 푀푆[퐿푆(푆, 퐿)]. 푖 푙 푖 푙 푖 푙 푁3 푃푎 . 푃푥 = 푃|푎| 푃푥 ∀ 푃푎 ∈ 퐿푝푡 , 푃푥 ∈ 푋푝푡. 푙 푚 푙 푚 푙 푚 푁4 푃푥 + 푃푦 ≤ 푃푥 + 푃푦 ∀ 푃푥 , 푃푦 ∈ 푋푝푡 . Definition 4.21. An Mvector space X over 푉퐾 is said to be finite dimensional if there is a finite set of ML.Id Mvectors in X that also generates X ie. there An Mvector space 푋 with an Mnorm on 푋 is exists a finite set called a multi normed linear space or Mnormed linear space and is denoted by 푆 = 푃푙1 , 푃푙2 , … … … … … .. 푃푙푛 of Mvectors of X 푥1 푥2 푥푛 (푋, ). (푁1), (푁2), (푁3) and (푁4) are called which is ML.Id and 푀푆[퐿푆(푆, 퐿)] = 푋 for some norm axioms. Mscalar field L over K with 푤퐿 ≤ 푤푋 . The number of elements of such a set S is Example 5.2. Let us consider the Mvector space called the dimension of X and is denoted by Dim(X). 푚푹 = {푤/푟 ∶ 푟 ∈ 푅} over R and L be an Mscalar + field over 푹. Also let ∶ 푚푹 푝푡 → 푚푹 푝푡 Theorem 4.22. Let X be an Mvector space over 푉 . 퐾 defined by 푃푖 = 푃푖 ∀ 푃푖 ∈ 푚푹 where | | Then 푑푖푚(푋∗) = 푛 iff there exists a collection of n 푎 |푎| 푎 푝푡 ML.Id Mvectors of X generating X. denotes the modulus of real numbers. Then : 푖 푁1 Clearly ∀ 푃푎 ∈ 푚푹 푝푡 , 푎 ∈ 푹 and 1 ≤ 푖 ≤ 푖 푖 1 Proof. Let there is a finite collection 푆 = 푤 ⇒ 푃푎 = 푃 푎 ≥ 푃0 . 푃푙1 , 푃푙2 ,… … … … … .. 푃푙푛 of n ML.Id Mvectors 푥1 푥2 푥푛 푖 푖 푘 푖 of (푁2) ∀ 푃푎 ∈ 푚푹 푝푡 , 푃푎 = 푃0 ⟺ 푃|푎| = 푘 X such that 푀푆[퐿푆(푆, 퐿)] = 푋 for some Mscalar 푃0 ⟺ |푎| = 0 and 푖 = 푘. field L over K with 푤퐿 ≤ 푤푋 . 푖 푚 푚 푖 Now as S is ML.Id, 푥1, 푥2, … … … … … . . 푥푛 is L.Id in (푁3) For 푃푎 ∈ 푚푹 푝푡 , 푃훼 ∈ 퐿푝푡 , 푃훼 푃푎 = 푋∗. Also 푀푆[퐿푆(푆, 퐿)] = 푋 ⇒ 푋∗ = 푚 푖 푚 푖 푚 푖 푚 푖 푚 푖 푃훼푎 = 푃 훼푎 = 푃 훼 푎 = 푃 훼 푃 푎 = 푃 훼 푃푎 . 퐿푆 푥1, 푥2, … … … … … . . 푥푛 ⇒ ∗ 푥1,푥2, … … … … … . . 푥푛 is a basis of 푋 ⇒ 푖 푗 푖 푗 푖 푗 (푁4) For 푃푎 , 푃 ∈ 푚푹 푝푡 , 푃푎 + 푃 = 푃 = 퐷푖푚 푋∗ = 푛. 푏 푏 푎+푏 푃푖 푗 ≤ 푃푖 푗 = 푃푖 + 푃푗 = 푃푖 + 푃푗 . Conversely let 퐷푖푚 푋∗ = 푛 and 푎+푏 푎 + 푏 푎 푏 푎 푏 ∗ 푥1, 푥2, … … … … … . . 푥푛 is a basis of 푋 . Then clearly 푤푋 Thus (푚푹, ) is an Mnormed linear space. 푆 = 푃푥 ∶ 푘 = 1,2, … … . 푛 ⊂ 푋푝푡 and

푀푆[퐿푆(푆)] = 푋 where LS(S) can be considered over Example 5.3. Let 푉, be a normed linear space any Mscalar field L over K such that 퐿∗ = 퐾 and over 퐾 = 푹/푪 and X be an Mvector space over V 푤 ≤ 푤 . 퐿 푋 with 푤 = 푤. Let ∶ 푋 → 푚푹+ such 푋 푚 푝푡 푝푡 푙 푙 푙 Note 4.23. Since 푑푖푚 푋∗ unique, it follows that that 푃푥 푚 = 푃 푥 ∀ 푃푥 ∈ 푋푝푡 . Then 푚 is an ∗ 퐷푖푚 푋 is also unique and 푑푖푚 푋 = 퐷푖푚 푋 . Mnorm over X and (푋, 푚 ) is an Mnormed linear space. V. MULTI NORMED LINEAR SPACE Note 5.4. Corresponding to every normed linear space, Notation: Throughout this section we shall consider there exists an Mnormed linear space. V as a vector space over 퐾 = 푹/푪, 푋 as an Mvector space over 푉퐾 with 푤푋 ≤ 푤 (w being the multiplicity Theorem 5.5. Let (푋, ) be an Mnormed linear of every element of space over a vector space 푉_퐾. Then 푑 ∶ 푋푝푡 ×

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+ 푙 푚 푙 1 푚 푙 푚 1 푋푝푡 → 푚푹 푝푡 defined by 푑(푃푥 , 푃푦 ) = (i) 퐵 푃푥 , 푃푟 = 푃푦 ∈ 푋푝푡 ∶ 푃푥 − 푃푦 < 푃푟 is 푙 푚 푙 푚 푙 1 푃푥 − 푃푦 ∀ 푃푥 , 푃푦 ∈ 푋푝푡 is a multimetric on 푋. called an open ball with center 푃푥 and radius 푃푟 . 푙 1 푚 푙 푚 1 (ii) 퐵 푃푥 , 푃푟 = 푃푦 ∈ 푋푝푡 ∶ 푃푥 − 푃푦 ≤ 푃푟 is 푙 푚 1 푙 1 Proof. 푀1 Clearly from 푁1 , 푑 푃푥 , 푃푦 ≥ 푃0 called a closed ball with center 푃푥 and radius 푃푟 . 푙 푚 푙 1 푚 푙 푚 1 ∀ 푃푥 , 푃푦 ∈ 푋푝푡 . (iii) 푆 푃푥 , 푃푟 = 푃푦 ∈ 푋푝푡 ∶ 푃푥 − 푃푦 = 푃푟 is 푙 1 called a sphere with center 푃푥 and radius 푃푟 . 푙 푚 푙 푚 1 (푀2) Let 푃푥 , 푃푦 ∈ 푋푝푡 , then 푑 푃푥 , 푃푦 = 푃0 ⟺ 푙 푚 1 푙 푚 푀푆 퐵 푃푙 , 푃1 , 푀푆[퐵 푃푙 , 푃1 ] and 푀푆[푆 푃푙 , 푃1 ] 푃푥 − 푃푦 = 푃0 ⟺ 푃푥 − 푃푦 = 푥 푟 푥 푟 푥 푟 1 푙 푚 are respectively called an Mopen ball, an Mclosed 푃휃 퐹푟표푚 푁2 ⟺ 푃푥 = 푃푦 . 푙 1 ball and an Msphere with center 푃푥 and radius 푃푟 .

(푀3) Let 푃푙 , 푃푚 ∈ 푋 . If 푃푙 = 푃푚 , then there is 푥 푦 푝푡 푥 푦 Definition 6.2. Convergence of a sequence: A nothing to prove. Let 푃푙 ≠ 푃푚 .Then 푑 푃푙 , 푃푚 = 푥 푦 푥 푦 sequence 푃푙푛 of Mvectors in an Mnormed linear 푃푙 − 푃푚 = 푃푙 푚 = 푃푙 푚 = 푥푛 푥 푦 푥−푦 −(푦−푥) space 푋, over 푉 is said to be convergent and 1 푙 푚 1 푙 푚 1 푙 푚 퐾 푃−1. 푃푦−푥 = 푃|−1| 푃푦−푥 = 푃|1| 푃푦−푥 = 푙 푙푛 푙 converges to an Mvector 푃푥 if 푃푥 − 푃푥 → 1 푙 푚 푙 푚 푚 푙 푚 푙 푛 푃1 . 푃푦−푥 = 푃푦−푥 = 푃푦 − 푃푥 = 푑 푃푦 ,푃푥 . 1 푃0 as 푛 → ∞ which means, for any

휖 > 0, ∃ 푛 ∈ 푵 such thatk 푃푙푛 − 푃푙 < 푃1 푙 푚 푧 푙 푧 0 푥푛 푥 휖 푀4 Let 푃푥 , 푃푦 , 푃푛 ∈ 푋푝푡 . Then 푃푥 − 푃푛 = ∀ 푛 ≥ 푛 ie. 푛 ≥ 푛 ⇒ 푃푙푛 ∈ 퐵 푃푙 , 푃1 . We 푙 푚 푚 푛 푙 푚 0 0 푥푛 푥 휖 푃푥 − 푃푦 + 푃푦 − 푃푧 ≤ 푃푥 − 푃푦 + 푙푛 푙 푚 푛 denote this by 푃푥 → 푃푥 as 푛 → ∞ or by 푃푦 − 푃푧 푛 푙 푙 푛 푙 푚 lim 푃 푛 = 푃푙 . 푃푙 is said to be the limit of [퐹푟표푚 (푁4)] ) ⇒ 푑 푃푥 , 푃푧 ≤ 푑 푃푥 , 푃푦 + 푛→∞ 푥푛 푥 푥 푚 푛 푃푙푛 as 푛 → ∞ . 푑( 푃푦 , 푃푧 ). 푥푛

푙 푚 푀5 Let 푃푥 , 푃푦 ∈ 푋푝푡 with 푙 ≠ 푚. Now Example 6.3. In Example 5.2, let us consider a 푙 푚 푘 푙 푚 푘 sequence 푃푙푛 of Mvectors in 푚푹, where 푑 푃푥 , 푃푦 = 푝0 ⟺ 푃푥 − 푃푦 = 푝0 ⟺ 푥푛 푙 푚 푘 푙 푚 푙 푃푥−푦 = 푝0 [ since 푙 ≠ 푚 ⇒ 푃푥 ≠ 푃푦 ⇒ 푃푥 − 푥푛 = 1/푛 and 푙푛 = 푤 ∀ 푛 ∈ 푁. Then for any 푚 푙 푚 푙 푚 ∈ > 0 ∃ 푛 ∈ 푵 such that 푃푙푛 − 푃푤 < 푃1 푃푦 = 푃푥 + 푃−푦 = 푃푥−푦 ] 0 푥푛 0 휖 ⟺ 푥 − 푦 = 휃 and 푙 푚 = 푘 퐵푦 푁2 ∀ 푛 ≥ 푛 ⇒ 푃푙푛 → 푃푤 푎푠 푛 → ∞ . 0 푥푛 0 ⟺ 푥 = 푦 and 푙 푚 = 푘. Thus Note 6.4. All convergent sequences of Mvectors 푙 푚 푙 푚 if 푃푥 , 푃푦 ∈ 푋푝푡 with 푙 ≠ 푚, then 푑 푃푥 , 푃푦 = having the same base will converge to Mvectors 푘 푙푛 푝0 ⟺ 푥 = 푦 and 푙 푚 = 푘. having the same base ie. if for a sequence 푃 of 푥푛 ∴ d is a multi metric on 푋. 푙 Mvectors, 푃 푛 → 푃푙 , then 푥푛 푥 푘푛 푘 푃푥 → 푃푥 for any 1 ≤ 푘 ≤ 퐶푋 (푥) and for any Definition 5.6. Mnorm subspace: Let (푋, 푋 ) be 푛 sequence {푘 } of natural numbers with 푘 ≤ 퐶 (푥). an Mnormed linear space over 푉퐾 and 푌 ⊂ 푋 is an 푛 푛 푋 + Msubspace of X. Then 푌 ∶ 푌푝푡 → 푚푹 푝푡 To prove this, let 휖 > 0 be taken arbitrarily. Then as 푙 푙 푙 푙 푙 푃 푛 → 푃푙 , ∃ 푛 ∈ 푵 such that 푃 푛 − 푃푙 < 푃1 defined by 푃푥 푌 = 푃푥 푋 ∀ 푃푥 ∈ 푌푝푡 is an 푥푛 푥 0 푥푛 푥 휖/3 Mnorm on Y. This Mnorm is known as the relative ∀ 푛 ≥ 푛0 ⇒ for any sequence {푘푛 } of natural numbers

Mnorm on Y induced by 푋. The Mnormed linear and 푘 ∈ 푵 with 푘푛 ≤ 퐶푋 (푥) and 1 ≤ 푘 ≤ 퐶푋 푥 , space ( , 푌 ) is called a an Mnorm subspace or 푃푘푛 − 푃푘 = 푃푘푛 − 푃푙푛 + 푃푙푛 − 푃푙 + 푃푙 − 푌 푥푛 푥 푥푛 푥푛 푥푛 푥 푥 simply an Msubspace of the Mnormed linear space 푘 푘푛 푙푛 푙푛 푙 푙 푘 푃푥 ≤ 푃푥 − 푃푥 + 푃푥 − 푃푥 + 푃푥 − 푃푥 < (푋, ) . 푛 푛 푛 푋 1 1 1 1 푘푛 푘 푃휖 + 푃휖 + 푃휖 = 푃 ∀ 푛 ≥ 푛 ⇒ 푃 → 푃 휖 0 푥푛 푥 3 3 3 VI. SEQUENCE AND THEIR CONVERGENCE IN AN 푛 → ∞. MNORMED LINEAR SPACE Definition 6.6. Boundedness : (i) In an Mnormed

Definition 6.1. Let (푋, ) be an Mnormed linear linear space 푋, , a multi subset 푌 ⊂ 푋 is said 푙 to be bounded if ∃ 푟 > 0 such that 푃푥 < space over a vector space 푉퐾 and 푟 > 0. We define 1 푙 the following: 푃푟 ∀ 푃푥 ∈ 푌푝푡 .

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(ii) A sequence 푃푙푛 of Mvectors in an Mnormed Proof. Let 푃푙 , 푃푚 ∈ 푀 = 푀 . Then 푃푙 , 푃푚 ∈ 푥푛 푥 푦 푝푡 푝푡 푥 푦 linear space 푋, is bounded if ∃ 푟 > 0 such 푀 푝푡 that ∀ 푙, 푚 ∈ 1, 2, … … … … . , 푤푋 ie. every element of M 푃푙푛 − 푃푙푚 < 푃1 ∀ 푚, 푛 ∈ 푵. has the same multiplicity equal to 푤 . Since 푥푛 푥푚 푟 푋 푃푙 , 푃푚 ∈ 푀 = 푀 , for any 휖 > 0, ∃ 푃푙1 , 푥 푦 푝푡 푝푡 푥1 Definition 6.7. Cauchy sequence : A sequence 푃푙푛 푃푚1 ∈ 푀 such that 푥푛 푦1 푝푡 of Mvectors in an Mnormed linear space 푋, is 푃푙 − 푃푙1 < 푃1 , 푃푚 − 푃푚1 < 푃1. 푥 푥1 휖 푦 푦1 휖 푝 푞 said to be Cauchy if for any ∈ > 0, ∃ 푛0 ∈ 푵 such Let 푃푎 , 푃푏 ∈ 퐿푝푡 [ 퐿 being an Mscalar field over 퐾 that with 퐿∗ = 퐾 ]. Since M is an Msubspace of 푃푙푛 − 푃푙푚 < 푃1 ∀ 푚, 푛 ≥ 푛 푖푒. 푃푙푛 − 푃푙푚 → 푥푛 푥푚 ∈ 0 푥푛 푥푚 푋, , 1 푝 푙 푞 푚 푝 푙 푃0 푃 . 푃 1 + 푃 . 푃 1 ∈ 푀 . Now 푃 . 푃 1 + 푎 푥1 푏 푦1 푝푡 푎 푥1 푎푠 푚, 푛 → ∞. 푃푞 . 푃푚1 − (푃푝 . 푃푙 + 푃푞 . 푃푚 ) ≤ 푃푝 푃푙1 − 푃푙 + 푏 푦1 푎 푥 푏 푦 푎 푥1 푥 푃푞 푃푚1 − 푃푚 ≤ 푃푝 . 푃1 + 푃푞 . 푃1 = Theorem 6.8. Every convergent sequence in an 푏 푦1 푦 푎 휖 푏 휖 푝 푞 1 Mnormed linear space is Cauchy and every Cauchy 푃 푎 + 푏 휖 < 푃휂 where sequence is bounded. 0 < 푎 + 푏 휖 < 휂 . Since 휖 > 0 is arbitrary so Proof. Since Mnorm induces multi metric, the result 푝 푙1 푞 푚1 푝 푙 is 휂 and hence 푃 푃 + 푃 . 푃 ∈ 퐵(푃 . 푃푥 + follows obviously. 푎 푥1 푏 푦1 푎 푃푞 . 푃푚 , 푃1 ) for any arbitrary 휂 > 0 ⇒ 푏 푦 휂 푝 푙 푞 푚 1 Definition 6.9. Completeness: An Mnormed linear 퐵 푃푎 . 푃푥 + 푃푏 . 푃푦 , 푃휂 ∩ 푀푝푡 ≠ 휙for any 휂 > 0 ⇒ 푝 푙 푞 푚 space 푋, is said to be complete if every 푃푎 . 푃푥 + 푃푏 . 푃푦 ∈ 푀푝푡 = 푀 푝푡 . Cauchy sequence of Mvectors in 푋, converges to an Mvector of X. VII. CONCLUSIONS

Example 6.10. (푚푹; ) is complete where 푚푹 is Functional analysis is an important branch of the multiset over 푹 having multiplicity of every Mathematics and it has many applications in Mathematics and Sciences. Metric space is the element equal to 푤 and 푃푙 = 푃푙 ∀ 푃푙 ∈ 푚푹 . 푥 |푥| 푥 푝푡 beginning of functional analysis and it has several applications in many branch of functional Theorem 6.11. In an Mnormed linear space 푋, , analysis. In this paper an extension of the concept if of metric is made by using multi set and multi 푃푙푛 → 푃푙 and 푃푘푛 → 푃푘 , then 푃푙푛 + 푃푘푛 → 푃푙 + number instead of crisp set and crisp real number. 푥푛 푥 푦푛 푦 푥푛 푦푛 푥 There is an ample scope for further research on 푘 푃푦 . . multi metric space. Research on Multi norm and multi inner product can be of special interest. Theorem 6.12. In an Mnormed linear space 푙 Acknowledgements. The present work is 푋, over a vector space 푉 , if {푃 푛 } be a 퐾 푥푛 supported by the Minor Research Project (MRP) sequence of Mvectors such that 푃푙푛 → 푃푙 and {푃푘푛 } 푥푛 푥 푎푛 of UGC, New Delhi, India [Grant No. be a sequence of Mscalars such PSW{203/13{14(ERO)] and Department of Mathematics, Visva-Bharati under the Special that 푃푘푛 → 푃푘 , then 푃푘푛 .푃푙푛 → 푃푘 . 푃푙 . 푎푛 푎 푎푛 푥푛 푎 푥 Assistance Programme (SAP) of UGC, New Delhi, India [Grant No. F 510/3/DRS-III/2015 Theorem 6.13. In an Mnormed linear space 푋, (SAP-I)]. over a vector space 푉 , if 푃푙푛 , {푃푚푛 } are Cauchy 퐾 푥푛 푦푛 VIII. REFERENCES sequences of Mvectors and {푃푘푛 } is a Cauchy 푎푛 푙 푚 푘 푙 sequence of Mscalars, then 푃 푛 + 푃 푛 , {푃 푛 .푃 푛 } [1] Wayne D. Blizard, Multiset theory, Notre Dame Journal of 푥푛 푦푛 푎푛 푥푛 are Cauchy sequences of Mvectors. Formal Logic 30(1), 36 - 66 (1989).

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