aa r X i v : . [ m a t h . G M ] J un Some results on multi vector space
Moumita Chiney , S. K. Samanta June 9, 2017
Department of Mathematics, Visva-Bharati, Santiniketan-731235.Email: [email protected], [email protected],[email protected] Abstract
In the present paper, a notion of M-basis and multi dimension of amulti vector space is introduced and some of its properties are studied.
Theory of Multisets is an important generalization of classical set theory whichhas emerged by violating a basic property of classical sets that an element canbelong to a set only once. Synonymous terms of multisets viz. list, heap, bunch,bag, sample, weighted set, occurrence set and fireset (finitely repeated elementset) are used in different contexts but conveying the same idea. It is a set wherean element can occur more than once. Many authors like Yager [23], Miyamoto[16, 17], Hickman [9], Blizard [1], Girish and John [7, 8], Chakraborty [2] etc.have studied the properties of multisets. Multisets are very useful structuresarising in many areas of mathematics and computer science such as databasequeries, multicriteria decision making, knowledge representation in data basedsystems, biological systems and membrane computing etc. [5, 6, 11, 12, 13, 17,18, 20]. Again the theory of vector space is one of the most important algebraicstructures in modern mathematics and this has been extended in different set-ting such as fuzzy vector space [10, 14, 22], intuitionistic fuzzy vector space ,soft vector space [4] etc. In [3], we introduced a notion of multi vector spaceand studied some of its basic properties. As a continuation of our earlier paper[3], here we have attempted to formulate the concept of basis and dimension ofmulti vector space and to study their properties.
In this section definition of a multiset (mset in short) and some of its propertiesare given. Unless otherwise stated, X will be assumed to be an initial universal1et and M S ( X ) denotes the set of all mset over X. Definition 2 . [8] An mset M drawn from the set X is represented by a countfunction C M : X → N where N represents the set of non negative integers.Here C M ( x ) is the number of occurrence of the element x in the mset M. The presentation of the mset M drawn from X = { x , x , ...., x n } will be as M = { x /m , x /m , ...., x n /m n } where m i is the number of occurrences of theelement x i , i = 1 , , ..., n in the mset M. Also here for any positive integer ω, [ X ] ω is the set of all msets whose ele-ments are in X such that no element in the mset occurs more than ω times andit will be referred to as mset spaces.The algebraic operations of msets are considered as in [8]. Definition 2 . [19] Let M be a mset over a set X . Then a set M n = { x ∈ X : C M ( x ) ≥ n } , where n is a natural number, is called n − level set of M. Proposition 2 . [19] Let A, B be msets over X and m, n ∈ N . (1) If A ⊆ B, then A n ⊆ B n ;(2) If m ≤ n, then A m ⊇ A n ;(3) ( A ∩ B ) n = A n ∩ B n ;(4) ( A ∪ B ) n = A n ∪ B n ;(5) A = B iff A n = B n , ∀ n ∈ N . Definition 2 . [19] Let P ⊆ X. Then for each n ∈ N , we define a mset nP over X, where C nP ( x ) = n, ∀ x ∈ P. Definition 2 . [19] Let X and Y be two nonempty sets and f : X → Y bea mapping. Then (1) the image of a mset M ∈ [ X ] ω under the mapping f is denoted by f ( M ) or f [ M ] , where C f ( M ) ( y ) = ∨ f ( x )= y C M ( x ) if f − ( y ) = φ otherwise (2) the inverse image of a mset N ∈ [ Y ] ω under the mapping f is denoted by f − ( N ) or f − [ N ] where C f − ( N ) ( x ) = C N [ f ( x )] .The properties of functions, which are used in this paper, are as in [19]. Definition 2 . [3] Let A , A , ..., A n ∈ [ X ] ω . Then we define A + A + ... + A n as follows: C A + A + ... + A n ( x ) = ∨{ C A ( x ) ∧ C A ( x ) ∧ ... ∧ C A n ( x n ) : x , x , ..., x n ∈ X and x + x + ... + x n = x } . Let λ ∈ K and B ∈ [ X ] ω . Then λB is defined as follows:2 λB ( y ) = ∨{ C B ( x ) : λx = y } . Lemma 2 . [3] Let λ ∈ K and B ∈ [ X ] ω . Then ( a ) For λ = 0 , C λB ( y ) = C B ( λ − y ) , ∀ y ∈ X. For λ = 0 ,C λB ( y ) = , y = 0 ,sup x ∈ X C B ( x ) , y = 0 . ( b ) For all scalars λ ∈ K and for all x ∈ X, we have C λB ( λx ) ≥ C B ( x ) . Definition 2 . [3] A multiset V in [ X ] ω is said to be a multi vector space ormulti linear space(in short mvector space) over the linear space X if ( i ) V + V ⊆ V ;( ii ) λV ⊆ V, for every scalar λ. We denote the set of all multi vector space over X by M V ( X ) . Remark 2 . [3] For a multi vector space V in [ X ] ω , V + V + .....n times = V, i. e., nV = V. Remark 2 . [3] If V ∈ M V ( X ) with dim X = m, then | C V ( X ) |≤ m + 1 , where | C V ( X ) | represents the cardinality of C V ( X ) . Proposition 2 . [3] ( Representation theorem ) Let V ∈ M V ( X ) with dim X = m and range of C V = { n , n , ...., n k } ⊆ { , , , ..., ω } , k ≤ m,n = C V ( θ ) and ω ≥ n > n > ... > n k ≥ . Then there exists a nestedcollection of subspaces of X as { θ } ⊆ V n $ V n $ V n $ .... $ V n k = X such that V = n V n ∪ n V n ∪ ..... ∪ n k V n k . Also (1) If n, m ∈ ( n i +1 , n i ] , then V n = V m = V n i . (2) If n ∈ ( n i +1 , n i ] and m ∈ ( n i , n i − ] , then V n % V m . Definition 2 . [3] Let X be a finite dimensional vector space with dim X = m and V ∈ M V ( X ) . Consider
P roposition . . Let B n i be a basis on V n i , i =0 , , ..., k such that B n $ B n $ B n $ ... $ B n k ..... ( iii ) Define a multi subset β of X by C β ( x ) = ( ∨{ n i : x ∈ B n i } , otherwise Then β is called a multi basis of V corresponding to ( iii ) . We denote the set ofall multi bases of V by B M ( V ) . Corollary 2 . [3] Let β be a multi basis of V obtained by ( iii ) . Then (1) If n, m ∈ ( n i +1 , n i ] , then β n = β m = B n i . (2) If n ∈ ( n i +1 , n i ] and m ∈ ( n i , n i − ] , then β n % β m . (3) β n is a basis of V n , for all n ∈ { , , ...., ω } .3 Some results on multi vector space
Lemma 3 . Let s, t ∈ R and A, A and A be multisets on a vector space X. Then (1) s. ( t.A ) = t. ( s.A ) = ( st ) .A and (2) A ≤ A ⇒ t.A ≤ t.A . Proposition 3 . Let V ∈ M V ( X ) . Then x ∈ X, a = 0 ⇒ C V ( ax ) = C V ( x ) . Proposition 3 . Let V ∈ M V ( X ) and u, v ∈ X such that C V ( u ) > C V ( v ) .Then C V ( u + v ) = C V ( v ) . Proposition 3 . Let V ∈ M V ( X ) and v, w ∈ X with C V ( v ) = C V ( w ) . Then C V ( v + w ) = C V ( v ) ∧ C V ( w ) . Definition 4 . Let V ∈ M V ( X ) and dim X = n. We say that a finite set ofvectors { x i } ni =1 is multi linearly independent in V if and only if { x i } ni =1 is lin-early independent in X and for all { a i } ni =1 ⊂ R with a i = 0 , C V ( n P i =1 a i x i )= n ∧ i =1 C V ( a i x i ) . The following example shows that every linearly independent set is not multilinearly independent.
Example 4 . Let X = R and ω = 4 . We define a multi vector space C V : X → N by C V ( x ) = , if x = (0 , , if x = (0 , a ) , a = 01 , otherwise. If we take the vectors x = (1 , and y = ( − , , then they are linearly inde-pendent but not multi linearly independent. As here C V ( x ) = C V ( y ) = 1 , but C V ( x + y ) = 2 > ( C V ( x ) ∧ C V ( y )) = 1 . Proposition 4 . Let V ∈ M V ( X ) and dim X = m. Then any set of vec-tors { x i } Ni =1 ( N ≤ m ) , which have distinct counts is linearly and multi linearlyindependent. P roof.
The proof follows by method of induction.4 ote 4 . Converse of the above proposition is not true. Let X = R and ω = 6 . We define a multi vector space C V : X → N by C V ( x ) = ( , if x = (0 , , otherwise. Then we have { θ } = V ⊂ V = R . Let e = (1 , , e = (0 , . Then { e , e } are multi linearly independent in V , although, C V ( e ) = C V ( e ) . Definition 5 . A M-basis for a multi vector space V ∈ M V ( X ) is a basis of X which is multi linearly independent in V .We denote the set of all M-bases of V by B ( V ) . Lemma 5 . If V ∈ M V ( X ) and Y is a proper subspace of X, then for any t ∈ X \ Y with C V ( t ) = sup [ C V ( X \ Y )] , C V ( t + y ) = C V ( t ) ∧ C V ( y ) , for all y ∈ Y.P roof.
Since ω is finite, such a t exists. Let y ∈ Y. If C V ( y ) = C V ( t ) then by P roposition . , C V ( t + y ) = C V ( t ) ∧ C V ( y ) . If C V ( y ) = C V ( t ) thenby Def inition . , C V ( t + y ) ≥ C V ( t ) ∧ C V ( y ) . Since t + y ∈ X \ Y and C V ( t ) = sup [ C V ( X \ Y )] , we must have C V ( t + y ) ≤ C V ( t ) = C V ( y ) and thus C V ( t + y ) = C V ( t ) ∧ C V ( y ) . Lemma 5 . Let V ∈ M V ( X ) , Y be a proper subspace of X and C V | Y = C V ′ . If B ∗ is a M-basis for V ′ , then there exists t ∈ X \ Y such that B + = B ∗ ∪ { t } is a M-basis for W, where C W = C V | ≺ B + ≻ and ≺ B + ≻ is the vector spacespanned by B + .P roof. Pick t ∈ X \ Y such that C V ( t ) = sup [ C V ( X \ Y )] . Then by
Lemma . ,B + = B ∗ ∪{ t } is a multi linearly independent and hence a M-basis for W, where C W = C V | ≺ B + ≻ . Proposition 5 . All multi vector spaces V ∈ M V ( X ) with dim X = m haveM-basis. P roof.
The proof follows by mathematical induction.
Proposition 5 . Let V ∈ M V ( X ) where dim X = m and C V ( X \ { θ } ) = { n , n , n , ..., n k } , k ≤ m . Then a basis B = { e , e , ..., e m } of X is a M-basisfor V if and only if B ∩ V n i is a basis of V n i for any i = 0 , , ..., k. Proof . Let ω ≥ n > n > .... > n k ≥ . Then { θ } $ V n $ V n $ V n $ .... $ V n k = X . Let B n i = B ∩ V n i , i = 0 , , ..., k. B ∩ V n i = B n i is a basis of V n i for any i = 0 , , ..., k. Then B n $ B n $ ...... $ B n k = B. Let B n = { e n , e n , ..., e jn } , j ≤ m. Then C V ( j P i =1 a i e in ) ≥ j ∧ i =1 C V ( e in ) = n . Since n is the highest count, C V ( j P i =1 a i e in ) = n = j ∧ i =1 C V ( e in ) . Hence B n is multi linearly independent.Next let B n = B n ∪ { e n , e n , ..., e sn } , j + s ≤ m. Consider the sum j P i =1 b i e in + s P i =1 c i e in , where some c i = 0 . Then C V ( j P i =1 b i e in + s P i =1 c i e in ) ≥ (cid:18) ∧ i ∈ J C V ( e in ) (cid:19) ∧ (cid:18) ∧ i ∈ J C V ( e in ) (cid:19) , [where J = { i : b i = 0 } , J = { i : c i = 0 } ] = n . If C V ( j P i =1 b i e in + s P i =1 c i e in ) > n , then C V ( j P i =1 b i e in + s P i =1 c i e in ) = n ⇒ j P i =1 b i e in + s P i =1 c i e in ∈ V n ⇒ c i = 0 , forall i = 1 , , .., s, a contradiction. Thus B n is multi linearly independent. Pro-ceeding in the similar way it can be proved that B n k = B is multi linearlyindependent and hence a M-basis for V. Conversely, let B be a M-basis for V. Then either B n i = φ or B n i = φ. Let B n i = φ and x ∈ V n i . Then obviously B n j = φ, j < i. Since B is a ba-sis of X, there exists some B ′ ⊆ B such that x = P e j ∈ B b j e j , b j = 0 . Then C V ( x ) = ∧ e j ∈ B C V ( e j ) ≤ n i +1 , a contradiction. So, B n i = φ. Then B n $ B n $ ...... $ B n k = B. Let x ∈ V n i and B n i is not a basis of V n i .Choose x = P e i ∈ B ni a i e i + P e ′ i / ∈ B ni b i e ′ i , for all b i = 0 .Now, C V ( x ) = C V ( P e i ∈ B ni a i e i + P e ′ i / ∈ B ni b i e ′ i )= ∧ e i ∈ B ′ ni C V ( e i ) ! ∧ (cid:18) ∧ e ′ i / ∈ B ni C V ( e ′ i ) (cid:19) , [where B ′ n i = { e i ∈ B n i : a i = 0 } ] = (cid:18) ∧ e ′ i / ∈ B ni C V ( e ′ i ) (cid:19) < n i , a contradiction to the fact that x ∈ V n i . Thus x = P e i ∈ B ni a i e i and B n i is a basis of V n i . Hence proved.
Proposition 5 . Let V be a multi vector space over X where dim X = m. Then there is an one-to-one correspondence between B M ( V ) and B ( V ) . Proposition 5 . Let V ∈ M V ( X ) with dimX = m and range of C V ( X \{ θ } ) = { n , n , ...., n k } ⊆ { , , , ..., ω } , k ≤ m. If a basis B = { e , e , ..., e m } of X is aM-basis, then C V ( B ) = { n , n , ...., n k } . Remark 5 . Converse of the above
P roposition is not true. For example,6uppose X = R , ω = 5 . Define multi vector space V with count functions C V as follows: C V ((0 , , , ; C V ((0 , , , R \ { } )) = 5; C V ((0 , , R \ { } , R )) = 5 , C V ((0 , R \ { } , R , R )) = 2 , C V ( R \ (0 , R , R , R )) = 2 . Then B = { (0 , , , , ( − , , , , (1 , − , , , (1 , , − , } is a basis of R and C V ( B ) = { , } = C V ( R ) . But B is not a M-basis as B is not multilinearly independent. In fact, C V (( − , , , C V (1 , − , , . But C V (( − , , ,
1) + (1 , − , , C V ((0 , , , > [ C V (( − , , , ∧ C V (( − , , , . Definition 5 . Let V ∈ M V ( X ) with dim X = m , range of C V ( X \ { θ } ) = { n , n ,..., n k }⊆ { , , , ..., ω } , k ≤ m and B be any M-basis for V. Then C V ( B ) = { n , n , ...., n k } . We define multi index of a multi M-basis B withrespect to V by [ B ] M = { r i : r i is the number of element of B taking the value n i } . Proposition 5 . For a multi vector space V , multi index of M-basis withrespect to V is independent of M-basis. Proof . Let V ∈ M V ( X ) with dimX = m , range of C V ( X \{ θ } ) = { n , n , ...., n k }⊆ { , , , ..., ω } , k ≤ m and ω ≥ n > n > ... > n k ≥ . Then for any twoM-bases B , B ′ of V, C V ( B ) = C V ( B ′ ) = { n , n , ...., n k } . Let [ B ] M = { r i } and [ B ′ ] M = { r ′ i } . Now, | B ∩ V n i | = i P j =0 r j and | B ′ ∩ V n i | = i P j =0 r ′ j , for i = 0 , , , ..., k. As B ∩ V n i and B ′ ∩ V n i are both basis of V n i , | B ∩ V n i | = | B ′ ∩ V n i | , for all i = 0 , , , ..., k. Thus [ B ] M = [ B ′ ] M . Note 5 . As multi index of M-basis with respect to a multi vector space V isindependent of M-basis, we can use only the term multi index of a multi vectorspace V. Definition 5 . Let V ∈ M V ( X ) with dim X = m , C V ( X ) = { n , n , ...., n k }⊆ { , , , ..., ω } , k ≤ m and B be any basis for X. We define index of a basis B with respect to V by [ B ] = { r i : n i r i is the number of element of B takingthe value n i } . Proposition 5 . Let V ∈ M V ( X ) with dimX = m , C V ( X \{ θ } ) = { n , n , ...., n k }⊆ { , , , ..., ω } , k ≤ m and B be any basis of X with C V ( B ) = { n , n , ...., n k } . If index [ B ] of B with respect to V is equal to the multi index of V , then B becomes a M-basis. Proof . Let us assume that ω ≥ n > n > ... > n k ≥ . Then { θ } $ V n $ V n $ V n $ .... $ V n k = X . Suppose that [ B ] M = { r i : i = 0 , , , ...k } . Then dim V n i = i P j =0 r j = | B ∩ V n i | , for all i = 0 , , , ..., k. Hence, B ∩ V n i becomes a7asis for V n i for each i = 0 , , , .., k. Thus by
P roposition . , B is a M-basisfor V. Definition 6 . We define the dimension of a multi vector space V over X by dim ( V ) = sup B a base for X (cid:18) P x ∈ B C V ( x ) (cid:19) .Clearly dim is a function from the set of all multi vector spaces to N . Proposition 6 . Let V ∈ M V ( X ) where dim X = m < ∞ . Then if B is aM-basis for V and B ∗ is any basis for X then P x ∈ B ∗ C V ( x ) ≤ P x ∈ B C V ( x ) . Proposition 6 . If V is a multi vector space over a finite dimensional vec-tor space X , then dim ( V ) = P x ∈ B C V ( x ) , where B is any M-basis for V. Note 6 . If V is a multi vector space over a finite dimensional vector space X ,then dim ( V ) is independent of M-basis for V. It follows from
P roposition . and P roposition . . Proposition 6 . Let X be any finite dimensional vector space and V, W ∈ M V ( X ) such that C V ( θ ) ≥ sup [ C W ( X \ { θ } )] and C W ( θ ) ≥ sup [ C V ( X \ { θ } )] .Then there exists a basis B for X which is also a M-basis for V, W , V ∩ W and V + W. In addition, if A = { x ∈ X : C V ( x ) < C W ( x ) } , A = X \ A , then forall v ∈ B ∩ A , ( C V ∩ W )( v ) = C V ( v ) and C V + W ( v ) = C W ( v ) and for all v ∈ B ∩ A , ( C V ∩ W )( v ) = C W ( v ) and C V + W ( v ) = C V ( v ) . Proof . We prove this by induction on dim X.
In case dim X = 1 the statementis clearly true.Now suppose that the theorem is true for all the multi vector space with dimen-sion of the underlying vector space equal to n.Let V and W be two multi vector spaces over X with dim X = n + 1 > . Let B = { v i } n +1 i =1 be any M-basis for V. We may assume that C V ( v ) ≤ C V ( v i ) for all i = { , , ..., n + 1 } . Let H = ≺ { v i } n +1 i =2 ≻ . Since n + 1 > , H = { θ } . Clearly dim H = n. Define the following two multi vector spaces: V with countfunction C V = C V | H and W with the count function C W = C W | H . Byinductive hypothesis the exists a basis B ∗ for H which is also a M-basis for V ,W , V ∩ W and V + W . Also for all v ∈ B ∗ ∩ A , ( C V ∩ W )( v ) = C V ( v ) and C V + W ( v ) = C W ( v ) and for all v ∈ B ∗ ∩ A , C V ∩ W )( v ) = C W ( v ) and C V + W ( v ) = C V ( v ) . We shall now show that B ∗ can be extended to B such that B is a M-basis for V, W , V ∩ W and V + W. Furthermore, for all v ∈ B ∩ A , ( C V ∩ W )( v ) = C V ( v ) and C V + W ( v ) = C W ( v ) and for all v ∈ B ∩ A , ( C V ∩ W )( v ) = C W ( v ) and C V + W ( v ) = C V ( v ) . Step - 1:
First we have to show that for all x ∈ H,C ( V + W ) | H ( x ) = C V + W ( x ) . ..........(1)Let x ∈ H. Then we have C ( V + W ) | H ( x ) = sup { C V ( x ) ∧ C W ( x − x ) : x ∈ X } = sup { C V ( x ) ∧ C W ( x − x ) : x ∈ H } ∨ sup { C V ( x ) ∧ C W ( x − x ) : x ∈ X \ H } . ........(2)If x ∈ H \ { θ } , we have C V ( x ) ∧ C W ( x − x ) = C V ( x ) ∧ C W ( θ ) ≤ sup { C V ( x ) ∧ C W ( x − x ) : x ∈ H } ,C V ( θ ) ∧ C W ( x − θ ) = C V ( θ ) ∧ C W ( x ) ≤ sup { C V ( x ) ∧ C W ( x − x ) : x ∈ H } . Since C V ( θ ) ≥ sup [ C W ( H \ { θ } )] and C W ( θ ) ≥ sup [ C V ( H \ { θ } )] ,C V ( x ) ∧ C W ( θ ) = C V ( x ) and C V ( θ ) ∧ C W ( x ) = C W ( x ) , and this leads to the following inequality: C V ( x ) ∨ C W ( x ) ≤ sup { C V ( x ) ∧ C W ( x − x ) : x ∈ H } . .........(3)Suppose that sup { C V ( x ) ∧ C W ( x − x ) : x ∈ H } < sup { C V ( x ) ∧ C W ( x − x ) : x ∈ X \ H } . ...........(4)This means that there exists x ′ ∈ X \ H such that sup { C V ( x ) ∧ C W ( x − x ) : x ∈ H } < C V ( x ′ ) ∧ C W ( x − x ′ ) . In view of (2) we must have C V ( x ) ∨ C W ( x ) < C V ( x ′ ) ∧ C W ( x − x ′ ) . .......(5)Since x ′ ∈ X \ H and C V ( X \ H ) = C V ( v ) ≤ C V ( v i ) for all i ∈ { , , ..., n + 1 } , we must have C V ( x ) ≥ C V ( x ′ ) . Thus (4) becomes C V ( x ) ∨ C W ( x ) < C V ( x ) ∧ C W ( x − x ′ ) . It is not possible (Using the properties of ∨ , ∧ and < ). This meansthat our assumption (3) is false. Therefore we must have sup { C V ( x ) ∧ C W ( x − x ) : x ∈ H } ≥ sup { C V ( x ) ∧ C W ( x − x ) : x ∈ X \ H } ...... (6) If x = θ, C ( V + W ) | H ( θ ) = C V ( θ ) ∧ C W ( θ ) = sup { C V ( x ) ∧ C W ( θ − x ) : x ∈ H } ≥ sup { C V ( x ) ∧ C W ( x − x ) : x ∈ X \ H } .... (7) .Using equations (6) and (7) ,we have for all x ∈ H,sup { C V ( x ) ∧ C W ( x − x ) : x ∈ H } ≥ sup { C V ( x ) ∧ C W ( x − x ) : x ∈ X \ H } , From (1) , we have for all x ∈ H, C ( V + W ) | H ( x ) = sup { C V ( x ) ∧ C W ( x − x ) : x ∈ H } = sup { C V | H ( x ) ∧ C W | H ( x − x ) : x ∈ H } = C V + W ( x ) . This establishes (1).Since B ∗ is a M-basis of V + W , (1) implies that B ∗ is multi linearly inde-pendent in V + W. Step - 2:
Let v ∗ ∈ X \ H such that C W ( v ∗ ) = sup [ C W ( X \ H )] . By Lemma . Lemma . , v ∗ is an extension of M-basis B ∗ for W to B = B ∗ ∪ { v ∗ } aM-basis for W. Step - 3:
Since C V ( X \ H ) = C V ( v ) , C V ( v ) = C V ( v ∗ ) and then v ∗ is also anextension of M-basis B ∗ for V to B a M-basis for V. Step - 4:
Now we shall show that v ∗ is an extension of M-basis B ∗ for V ∩ W to B a M-basis for V ∩ W. If v ∗ ∈ A , then ( C V ∧ C W )( A ∩ ( X \ H )) = C V ( v ∗ ) and for all z ∈ A ∩ ( X \ H ) , ( C V ∧ C W )( z ) ≤ C V ( v ∗ ) , by definition of A . From this we may conclude that if v ∗ ∈ A then ( C V ∧ C W )( v ∗ ) = sup [( C V ∧ C W )( X \ H )] . If v ∗ ∈ A then C W ( v ∗ ) ≤ C V ( v ∗ ) . Since C W ( v ∗ ) = sup [ C W ( X \ H )] and C V is constant on X \ H we must have A ∩ ( X \ H ) = φ. Therefore we have thatif v ∗ ∈ A , then ( C V ∧ C W )( v ∗ ) = sup [( C V ∧ C W )( X \ H )] . By Lemma . , wemay now conclude that v ∗ extends M-basis B ∗ for V ∩ W to B a M-basis for V ∩ W. Step - 5:
Now we shall show that v ∗ is also an extension of B ∗ a M-basisfor V + W to B a M-basis for V + W. Suppose that there exists z ∈ X \ H suchthat C ( V + W ) ( v ∗ ) < C ( V + W ) ( z ) . Clearly vector z can be written in the form z = a ( v ∗ + v ) where a = 0 and v ∈ H. Therefore we have C ( V + W ) ( v ∗ ) < C ( V + W ) ( z ) = C ( V + W ) ( a ( v ∗ + v )) = C ( V + W ) ( v ∗ + v ) . This means that there exists x ∈ X such that for all x ′ ∈ X,C V ( x ′ ) ∧ C W ( v ∗ − x ′ ) < C V ( x ) ∧ C W ( v ∗ + v − x ) . ............(8)In particular this is true for x ′ = θ, i.e. C V ( θ ) ∧ C W ( v ∗ ) < C V ( x ) ∧ C W ( v ∗ + v − x ) . But since C V ( θ ) ≥ sup [ C W ( X \ { θ } )] we have C W ( v ∗ ) < C V ( x ) ∧ C W ( v ∗ + v − x ) . ........(9)If x ∈ H then since v ∈ H we must have v − x ∈ H. Again v ∗ ∈ X \ H. So,by
Lemma . , C W ( v ∗ + v − x ) = C W ( v ∗ ) ∧ C W ( v − x ) and so (9) becomes C W ( v ∗ ) < C V ( x ) ∧ C W ( v ∗ ) ∧ C W ( v − x ) , which is impossible. Thus x ∈ X \ H. Let x ′ = v ∗ in (5). Since C W ( θ ) ≥ sup [ C V ( X \ { θ } )] we have C V ( v ∗ ) < C V ( x ) ∧ C W ( v ∗ + v − x ) ..... (10) Recall that C V ( X \ H ) = C V ( v ) and thus C V ( v ) = C V ( v ∗ ) = C V ( x ) , as x ∈ X \ H. This again means that the inequality (10) is false. This means thatfor all z ∈ X \ H, C ( V + W ) ( v ∗ ) ≥ C ( V + W ) ( z ) . Therefore by Lemma . , v ∗ isan extension of B ∗ a M-basis for V + W to B a M-basis for V + W. Step - 6:
Now we shall show that if v ∗ ∈ A then C V + W ( v ∗ ) = C W ( v ∗ ) and if v ∗ ∈ A then C V + W ( v ∗ ) = C V ( v ∗ ) . From the definition we have: C ( V + W ) ( v ∗ ) = sup { C V ( x ) ∧ C W ( v ∗ − x ) : x ∈ X } . Let x ′ be such that sup { C V ( x ) ∧ C W ( v ∗ − x ) : x ∈ X } = C V ( x ′ ) ∧ C W ( v ∗ − x ′ ) . By substituting x = θ and then x = v ∗ and recalling that C V ( θ ) ≥ sup [ C W ( X \{ θ } )] and C W ( θ ) ≥ sup [ C V ( X \ { θ } )] , we obtain C V ( v ∗ ) ∨ C W ( v ∗ ) ≤ C V ( x ′ ) ∧ C W ( v ∗ − x ′ ) . C V ( v ∗ ) ∨ C W ( v ∗ ) < C V ( x ′ ) ∧ C W ( v ∗ − x ′ ) . .........(11)If x ′ ∈ H then by Lemma . (as B = B ∗ ∪ { v ∗ } is a M-basis for W ), (11)becomes C V ( v ∗ ) ∨ C W ( v ∗ ) < C V ( x ′ ) ∧ C W ( v ∗ ) ∧ C W ( x ′ ) . This is never true, and thus x ′ ∈ X \ H. But now since C V ( v ∗ ) = C V ( x ′ ) theinequality (11) never holds, and so, C V ( v ∗ ) ∨ C W ( v ∗ ) = C V ( x ′ ) ∧ C W ( v ∗ − x ′ ) = C ( V + W ) ( v ∗ ) . ........(12)From equation (12) , we have v ∗ ∈ A then C V + W ( v ∗ ) = C W ( v ∗ ) and if v ∗ ∈ A then C V + W ( v ∗ ) = C V ( v ∗ ) . This completes the proof.
Corollary 6 . If V and W are two multi vector spaces over X such thatthe dimension of X is finite and C V ( θ ) ≥ sup [ C W ( X \ { θ } )] and C W ( θ ) ≥ sup [ C V ( X \ { θ } )] , then dim ( V + W ) = dim V + dim W − dim ( V ∩ W ) . Example 6 . Suppose X = R , ω = 6 . Define two multi vector spaces V and W with count functions C V and C W respectively as follows: C V ((0 , ; C V ((0 , R \ { } )) = 3; C V ( X \ R ) = 1 , C W ((0 , C W ( { ( x, x ) : x ∈ R \ { }} ) = 2; C W ( X \ { ( x, x ) : x ∈ R } ) = 1 . It is easily checked that V and W are multi vector spaces and C V ( θ ) ≥ sup [ C W ( X \ { θ } )] and C W ( θ ) ≥ sup [ C V ( X \ { θ } )] . It is also easy to check that C V ∩ W ((0 , , C V ∩ W ( { ( x, x ) : x ∈ R \ { }} ) = 1 , C V ∩ W ( X \ { ( x, x ) : x ∈ R } ) = 1 , C V + W ((0 , ; C V + W ((0 , R \ { } )) = 3; C V + W ( X \ (0 , R )) = 2 and B = { (0 , , (1 , } is a M-basis for V, W, V ∩ W and V + W. Thus dim ( V + W ) = 3 + 2 = 5 , dim ( V ∩ W ) = 1 + 1 = 2 ,dim V = 3 + 1 = 4 , dim W = 2 + 1 = 3 .So, dim V + dim W − dim ( V ∩ W ) = 4 + 3 − dim ( V + W ) . Definition 6 . Let V be a multi vector space over X and f : X → Y be alinear map. Then we define f ( V ) as C f ( V ) ( x ) = ( sup { C V ( z ) : z ∈ f − ( x ) } if f − ( x ) = φ otherwise and ˜ kerf = ( kerf, C V | kerf ) , ˜ imf = ( imf, C V | imf ) . Proposition 6 . If V be a multi vector space over X where dim X is finiteand f : X → Y is a linear map, then dim ( ˜ kerf ) + dim ( ˜ imf ) = dim ( V ) .P roof. Suppose that kerf = { θ } . If kerf = { θ } then the proof is similar.Now let B Kerf be a M-basis for ˜ kerf and B Ex be an extension of B Ker to aM-basis for V (this is clearly possible by repeated application of Lemma . ).Then B Kerf ∪ B Ex = B is M-basis for V and B Kerf ∩ B Ex = φ. We first show that f ( B Ex ) = B Im is a M-basis for ˜ imf . Clearly B Im is a basis11or imf. Let v , v , .., v k ∈ B Ex and a , ..., a k ∈ R not all zero. By definition wehave C f ( V ) ( k P i =1 a i f ( v i ))= sup { C V ( x ) : x ∈ f − ( k P i =1 a i f ( v i )) } if f − (cid:18) k P i =1 a i f ( v i ) (cid:19) = φ otherwise Since k P i =1 a i f ( v i ) ∈ imf we have C f ( V ) ( k P i =1 a i f ( v i )) = sup { C V ( x ) : x ∈ f − ( k P i =1 a i f ( v i )) } . By linearity of f and by the property of f − we get C f ( V ) ( k P i =1 a i f ( v i )) = sup { C V ( x ) : x ∈ kerf + k P i =1 a i v i } . If x ∈ kerf then x = θ or x = p P i =1 b i u i , u i ∈ B kerf where not all b i arezero; so if x ∈ kerf + k P i =1 a i v i then either C V ( x ) = C V ( θ + k P i =1 a i v i ) or C V ( x ) = C V ( p P i =1 b i u i + k P i =1 a i v i ) and thus C V ( x ) = min (cid:18) p ∧ i =1 C V ( b i u i ) , k ∧ i =1 C V ( a i v i ) (cid:19) , [ As u i and v i are M-basis elementof V ]which is clearly smaller than or equal to C V ( k P i =1 a i v i ) . Thus C f ( V ) ( k P i =1 a i f ( v i )) = sup { C V ( x ) : x ∈ kerf + k P i =1 a i v i } = C V ( k P i =1 a i v i ) = k ∧ i =1 C V ( a i v i ) . By the same argument we get that C f ( V ) ( f ( v i )) = C V ( v i ) . Thus C f ( V ) ( k P i =1 a i f ( v i )) = k ∧ i =1 C f ( V ) ( a i v i ) . and therefore B Im is a M-basis for ˜ imf . Now by definition of multi dimension we get dim ( V ) = P v ∈ B Ker ∪ B Ex C V ( v ) = P v ∈ B Ker C V ( v ) + P v ∈ B Ex C V ( v ) .But by the above we have if z ∈ ≺ B Ex ≻ , then C f ( V ) ( f ( z )) = C V ( z ) , and thus dim ( V ) = P v ∈ B Ker C V ( v ) + P v ∈ B Ex C f ( V ) ( f ( v ))= P v ∈ B Ker C V ( v ) + P v ∈ B Im C f ( V ) ( v )= dim ( ˜ kerf ) + dim ( ˜ imf ) . Conclusion
There is a future scope of study of infinite dimensional multi vector space andbehavior of linear operators in multi vector space context.