This paper introduces the concepts of M-basis and multi dimension in multi vector spaces, exploring their properties to advance understanding of their structure.
Contribution
It presents new definitions of M-basis and multi dimension, and investigates their properties within multi vector spaces.
Findings
01
Defined M-basis and multi dimension for multi vector spaces
02
Analyzed properties of these new concepts
03
Provided foundational results for future research
Abstract
In the present paper, a notion of M-basis and multi dimension of a multi vector space is introduced and some of its properties are studied.
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Taxonomy
TopicsFuzzy and Soft Set Theory · Fixed Point Theorems Analysis · Approximation Theory and Sequence Spaces
Full text
Some results on multi vector space
Moumita Chiney1, S. K. Samanta2
Abstract
In the present paper, a notion of M-basis and multi dimension of a
multi vector space is introduced and some of its properties are studied.
Department of Mathematics, Visva-Bharati, Santiniketan-731235.
Theory of Multisets is an important generalization of classical set
theory which has emerged by violating a basic property of classical
sets that an element can belong to a set only once. Synonymous terms
of multisets viz. list, heap, bunch, bag, sample, weighted set, occurrence
set and fireset (finitely repeated element set) are used in different
contexts but conveying the same idea. It is a set where an 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 structures arising in many areas of mathematics and computer
science such as database queries, multicriteria decision making, knowledge
representation in data based systems, 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 algebraic
structures in modern mathematics and this has been extended in different
setting 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 space
and 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 of multi vector space and to study
their properties.
2 Preliminaries
In this section definition of a multiset (mset in short) and some
of its properties are given. Unless otherwise stated, X will be
assumed to be an initial universal set and MS(X) denotes the set
of all mset over X.
Definition2.1 [8] An mset M drawn from
the set X is represented by a count function CM:X→N
where N represents the set of non negative integers.
Here CM(x) is the number of occurrence of the element x in
the mset M. The presentation of the mset M drawn from X={x1,x2,....,xn}
will be as M={x1/m1,x2/m2,....,xn/mn} where
mi is the number of occurrences of the element xi,i=1,2,...,n
in the mset M.
Also here for any positive integer ω,[X]ω is the
set of all msets whose elements are in X such that no element in
the mset occurs more than ω times and it will be referred
to as mset spaces.
The algebraic operations of msets are considered as in [8].
Definition2.2 [19] Let M be a mset over
a set X. Then a set Mn={x∈X:CM(x)≥n}, where n
is a natural number, is called n− level set of M.
Proposition2.3 [19] Let A,B be msets over
X and m,n∈N.
(1) If A⊆B, then An⊆Bn;
(2) If m≤n, then Am⊇An;
(3)(A∩B)n=An∩Bn;
(4)(A∪B)n=An∪Bn;
(5)A=B iff An=Bn,∀n∈N.
Definition2.4 [19] Let P⊆X. Then
for each n∈N, we define a mset nP over X, where
CnP(x)=n,∀x∈P.
Definition2.5 [19] Let X and Y be two
nonempty sets and f:X→Y be a mapping. Then
(1) the image of a mset M∈[X]ω under the mapping f
is denoted by f(M) or f[M], where
[TABLE]
(2) the inverse image of a mset N∈[Y]ω under the mapping
f is denoted by f−1(N) or f−1[N] where Cf−1(N)(x)=CN[f(x)].
The properties of functions, which are used in this paper, are as
in [19].
Definition2.6 [3] Let A1,A2,...,An∈[X]ω. Then we define A1+A2+...+An
as follows:
(b) For all scalars λ∈K and for all x∈X, we have
CλB(λx)≥CB(x).
Definition2.8 [3] A multiset V
in [X]ω is said to be a multi vector space or multi 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 MV(X).
Remark2.9 [3] For a multi vector space V
in [X]ω , V+V+.....ntimes=V, i. e., nV=V.
Remark2.10 [3] If V∈MV(X) with dimX=m,
then ∣CV(X)∣≤m+1, where ∣CV(X)∣ represents
the cardinality of CV(X).
Proposition2.11 [3] (Representation
theorem) Let V∈MV(X) with dimX=m and range of CV={n0,n1,....,nk}⊆{0,1,2,...,ω},k≤m,n0=CV(θ) and ω≥n0>n1>...>nk≥0.
Then there exists a nested collection of subspaces of X as
{θ}⊆Vn0⫋Vn1⫋Vn2⫋....⫋Vnk=X
such that V=n0Vn0∪n1Vn1∪.....∪nkVnk.
Also
(1) If n,m∈(ni+1,ni], then Vn=Vm=Vni.
(2) If n∈(ni+1,ni] and m∈(ni,ni−1], then
Vn⫌Vm.
Definition2.12 [3] Let X be a finite dimensional
vector space with dimX=m and V∈MV(X). Consider Proposition2.11.
Let Bni be a basis on Vni,i=0,1,...,k such that
Bn0⫋Bn1⫋Bn2⫋...⫋Bnk…..(iii)
Define a multi subset β of X by
Cβ(x)={∨{ni:x∈Bni}0,otherwise
Then β is called a multi basis of V corresponding to (iii).
We denote the set of all multi bases of V by BM(V).
Corollary2.13 [3] Let β be a multi
basis of V obtained by (iii). Then
(1) If n,m∈(ni+1,ni], then βn=βm=Bni.
(2) If n∈(ni+1,ni] and m∈(ni,ni−1], then
βn⫌βm.
(3)βn is a basis of Vn, for all n∈{1,2,....,ω}.
3 Some results on multi vector space
Lemma3.1 Let s,t∈R and A,A1 and
A2 be multisets on a vector space X. Then
(1)s.(t.A)=t.(s.A)=(st).A and
(2)A1≤A2⇒t.A1≤t.A2.
Proposition3.2 Let V∈MV(X). Then x∈X,a=0⇒CV(ax)=CV(x).
Proposition3.3 Let V∈MV(X) and u,v∈X such
that CV(u)>CV(v). Then CV(u+v)=CV(v).
Proposition3.4 Let V∈MV(X) and v,w∈X with
CV(v)=CV(w). Then CV(v+w)=CV(v)∧CV(w).
4 Multi linear independence
Definition4.1 Let V∈MV(X) and dimX=n. We
say that a finite set of vectors {xi}i=1n is multi linearly
independent in V if and only if {xi}i=1n is linearly
independent in X and for all {ai}i=1n⊂R
with ai=0 , CV(i[=1]n∑aixi)
=i[=1]n∧CV(aixi).
The following example shows that every linearly independent set is
not multi linearly independent.
Example4.2 Let X=R2 and ω=4.
We define a multi vector space CV:X→N by
If we take the vectors x=(1,0) and y=(−1,1), then they are linearly
independent but not multi linearly independent. As here CV(x)=CV(y)=1,
but CV(x+y)=2>(CV(x)∧CV(y))=1.
Proposition4.3 Let V∈MV(X) and dimX=m. Then
any set of vectors {xi}i=1N(N≤m), which have distinct
counts is linearly and multi linearly independent.
Proof. The proof follows by method of induction.
Note4.4 Converse of the above proposition is not true.
Let X=R2 and ω=6. We define a multi vector
space CV:X→N by
CV(x)={6,ifx=(0,0)1,otherwise.
Then we have {θ}=V6⊂V1=R2. Let
e1=(1,0),e2=(0,1). Then {e1,e2} are multi
linearly independent in V , although, CV(e1)=CV(e2).
5 M-basis
Definition5.1 A M-basis for a multi vector space V∈MV(X)
is a basis of X which is multi linearly independent in V.
We denote the set of all M-bases of V by \text{\mathscr{B}}(V).
Lemma5.2 If V∈MV(X) and Y is a proper subspace
of X, then for any t∈X∖Y with CV(t)=sup[CV(X∖Y)],
CV(t+y)=CV(t)∧CV(y), for all y∈Y.
Proof. Since ω is finite, such a t exists. Let y∈Y.
If CV(y)=CV(t) then by Proposition3.4,CV(t+y)=CV(t)∧CV(y).
If CV(y)=CV(t) then by Definition2.8,CV(t+y)≥CV(t)∧CV(y).
Since t+y∈X∖Y and CV(t)=sup[CV(X∖Y)],
we must have CV(t+y)≤CV(t)=CV(y) and thus CV(t+y)=CV(t)∧CV(y).
Lemma5.3 Let V∈MV(X), Y be a proper subspace
of X and CV∣Y=CV′. 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}\mid_{\prec B^{+}\text{\succ}}
and \text{\prec}B^{+}\text{\succ} is
the vector space spanned by B+.
Proof. Pick t∈X∖Y such that CV(t)=sup[CV(X∖Y)].
Then by Lemma5.2,B+=B∗∪{t} is a multi linearly
independent and hence a M-basis for W, where C_{W}=C_{V}\mid_{\prec B^{+}\text{\succ}}.
Proposition5.4 All multi vector spaces V∈MV(X)
with dimX=m have M-basis.
Proof. The proof follows by mathematical induction.
Proposition5.5 Let V∈MV(X) where dimX=m
and CV(X∖{θ})={n0,n1,n2,...,nk},k≤m.
Then a basis B={e1,e2,...,em} of X is a M-basis
for V if and only if B∩Vni is a basis of Vni
for any i=0,1,...,k.
Proof. Let ω≥n0>n1>....>nk≥0.
Then {θ}⫋Vn0⫋Vn1⫋Vn2⫋....⫋Vnk=X
. Let Bni=B∩Vni,i=0,1,...,k.
First suppose that B∩Vni=Bni is a basis of Vni
for any i=0,1,...,k. Then Bn0⫋Bn1⫋......⫋Bnk=B.
Let Bn0={en01,en02,...,en0j},j≤m.
Then CV(i[=1]j∑aien0i)≥i[=1]j∧CV(en0i)=n0.
Since n0 is the highest count,
CV(i[=1]j∑aien0i)=n0=i[=1]j∧CV(en0i).
Hence Bn0 is multi linearly independent.
Next let Bn1=Bn0∪{en11,en12,...,en1s},j+s≤m.
Consider the sum
i[=1]j∑bien0i+i[=1]s∑cien1i,
where some ci=0. Then CV(i[=1]j∑bien0i+i[=1]s∑cien1i)
If CV(i[=1]j∑bien0i+i[=1]s∑cien1i)>n1,
then
CV(i[=1]j∑bien0i+i[=1]s∑cien1i)=n0⇒i[=1]j∑bien0i+i[=1]s∑cien1i∈Vn0⇒ci=0,
for all i=1,2,..,s, a contradiction. Thus Bn1 is multi
linearly independent. Proceeding in the similar way it can be proved
that Bnk=B is multi linearly independent and hence a M-basis
for V.
Conversely, let B be a M-basis for V. Then either Bni=ϕ
or Bni=ϕ.
Let Bni=ϕ and x∈Vni. Then obviously Bnj=ϕ,j<i.
Since B is a basis of X, there exists some B′⊆B
such that x=ej∈B∑bjej,bj=0.
Then CV(x)=ej∈B∧CV(ej)≤ni+1,
a contradiction. So, Bni=ϕ.
Then Bn0⫋Bn1⫋......⫋Bnk=B.
Let x∈Vni and Bni is not a basis of Vni.
Choose x=ei∈Bni∑aiei+ei′∈/Bni∑biei′,
for all bi=0.
=(ei∈Bni′∧CV(ei))∧(ei′∈/Bni∧CV(ei′)),
[where Bni′={ei∈Bni:ai=0}]
=(ei′∈/Bni∧CV(ei′))<ni,
a contradiction to the fact that x∈Vni. Thus x=ei∈Bni∑aiei
and Bni is a basis of Vni.
Hence proved.
Proposition5.6 Let V be a multi vector space over
X where dimX=m. Then there is an one-to-one correspondence
between BM(V) and \text{\mathscr{B}}(V).
Proposition5.7 Let V∈MV(X) with dimX=m and
range of CV(X∖{θ})={n0,n1,....,nk}⊆{0,1,2,...,ω},k≤m. If a basis B={e1,e2,...,em} of X is a
M-basis, then CV(B)={n0,n1,....,nk}.
Remark5.8 Converse of the above Proposition is not
true. For example, suppose X=R4,ω=5. Define
multi vector space V with count functions CV as follows:
Then B={(0,0,0,1),(−1,1,1,1),(1,−1,1,1),(1,1,−1,1)} is a basis
of R4 and CV(B)={2,5}=CV(R4).
But B is not a M-basis as B is not multi linearly independent.
In fact, CV((−1,1,1,1))=CV(1,−1,1,1))=2. But CV((−1,1,1,1)+(1,−1,1,1))=CV((0,0,1,1))=5>[CV((−1,1,1,1))∧CV((−1,1,1,1))]=2.
Definition5.9 Let V∈MV(X) with dimX=m, range
of C_{V}(X\setminus\{\theta\})=\{n_{0},n_{1},$$...,n_{k}\}$$\subseteq\{0,1,2,...,\omega\},k≤m and B0 be any M-basis for V. Then CV(B0)={n0,n1,....,nk}.
We define multi index of a multi M-basis B0 with respect to
V by [B0]M={ri:ri is the number of element
of B0 taking the value n_{i}$$\}.
Proposition5.10 For a multi vector space V, multi
index of M-basis with respect to V is independent of M-basis.
Proof. Let V∈MV(X) with dimX=m, range of CV(X∖{θ})={n0,n1,....,nk}
⊆{0,1,2,...,ω},k≤m and ω≥n0>n1>...>nk≥0.
Then for any two M-bases B0,B0′ of V,CV(B0)=CV(B0′)={n0,n1,....,nk}.
Let [B0]M={ri} and [B0′]M={ri′}.
Now, ∣B0∩Vni∣=j[=0]i∑rj
and ∣B0′∩Vni∣=j[=0]i∑rj′,
for i=0,1,2,...,k. As B0∩Vni and B0′∩Vni
are both basis of Vni, ∣B0∩Vni∣
=∣B0′∩Vni∣, for all i=0,1,2,...,k.
Thus [B0]M=[B0′]M.
Note5.11 As multi index of M-basis with respect to a
multi vector space V is independent of M-basis, we can use only
the term multi index of a multi vector space V.
Definition5.12 Let V∈MV(X) with dimX=m,
CV(X)={n0,n1,....,nk}
⊆{0,1,2,...,ω},k≤m and B be any basis
for X. We define index of a basis B with respect to V by
[B]={ri:niri is the number of element of B
taking the value n_{i}$$\}.
Proposition5.13 Let V∈MV(X) with dimX=m,
CV(X∖{θ})={n0,n1,....
,n_{k}\}$$\subseteq\{0,1,2,...,\omega\},k≤m and B be
any basis of X with CV(B)={n0,n1,....
,nk}. 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≥n0>n1>...>nk≥0.
Then {θ}⫋Vn0⫋Vn1⫋Vn2⫋....⫋Vnk=X.
Suppose that [B]M={ri:i=0,1,2,...k}. Then dimVni=j[=0]i∑rj=∣B∩Vni∣,
for all i=0,1,2,...,k. Hence, B∩Vni becomes a basis
for Vni for each i=0,1,2,..,k. Thus by Proposition5.5,B is a M-basis for V.
6 Dimension of multi vector space
Definition6.1 We define the dimension of a multi vector
space V over X by dim(V)=BabaseforXsup(x∈B∑CV(x)).
Clearly dim is a function from the set of all multi vector spaces
to N.
Proposition6.2 Let V∈MV(X) where dimX=m<∞.
Then if B is a M-basis for V and B∗ is any basis for X
then
x∈B∗∑CV(x)≤x∈B∑CV(x).
Proposition6.3 If V is a multi vector space over
a finite dimensional vector space X, then dim(V)=x∈B∑CV(x),
where B is any M-basis for V.
Note6.4 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 Proposition5.5 and Proposition5.7.
Proposition6.5 Let X be any finite dimensional vector
space and V,W∈MV(X) such that CV(θ)≥sup[CW(X∖{θ})]
and CW(θ)≥sup[CV(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 A1={x∈X:CV(x)<CW(x)},A2=X∖A1, then for all v∈B∩A1,
(CV∩W)(v)=CV(v) and CV+W(v)=CW(v)
and for all v∈B∩A2,
(CV∩W)(v)=CW(v) and CV+W(v)=CV(v).
Proof. We prove this by induction on dimX. In case
dimX=1 the statement is clearly true.
Now suppose that the theorem is true for all the multi vector space
with dimension of the underlying vector space equal to n.
Let V and W be two multi vector spaces over X with dimX=n+1>1.
Let B1={vi}i=1n+1 be any M-basis for V. We may
assume that CV(v1)≤CV(vi) for all i={2,3,...,n+1}.
Let H=≺{vi}i=2n+1≻. Since n+1>1,H={θ}.
Clearly dimH=n. Define the following two multi vector spaces:
V1 with count function CV1=CV∣H and W1
with the count function CW1=CW∣H. By inductive hypothesis
the exists a basis B∗ for H which is also a M-basis for V1,W1,V1∩W1 and V1+W1. Also for all v∈B∗∩A1,
(CV1∩W1)(v)=CV1(v) and CV1+W1(v)=CW1(v)
and for all v∈B∗∩A2,
(CV1∩W1)(v)=CW1(v) and CV1+W1(v)=CV1(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∩A1,
Suppose that sup{CV(x1)∧CW(x−x1):x1∈H}<sup{CV(x2)∧CW(x−x2):x2∈X∖H}.………..(4)
This means that there exists x′∈X∖H such that
sup{CV(x1)∧CW(x−x1):x1∈H}<CV(x′)∧CW(x−x′).
In view of (2) we must have
CV(x)∨CW(x)<CV(x′)∧CW(x−x′).…….(5)
Since x′∈X∖H and CV(X∖H)=CV(v1)≤CV(vi)
for all i∈{2,3,...,n+1}, we must have CV(x)≥CV(x′).
Thus (4) becomes CV(x)∨CW(x)<CV(x)∧CW(x−x′).
It is not possible (Using the properties of ∨,∧ and <).
This means that our assumption (3) is false. Therefore we must have
From (1), we have for all x∈H,C(V+W)∣H(x)=sup{CV(x1)∧CW(x−x1):x1∈H}
=sup{CV∣H(x1)∧CW∣H(x−x1):x1∈H}
=CV1+W1(x).
This establishes (1).
Since B∗ is a M-basis of V_{1}+W_{1},$$(1) implies that
B∗ is multi linearly independent in V+W.
Step - 2: Let v∗∈X∖H such that CW(v∗)=sup[CW(X∖H)].
By Lemma5.2 and Lemma5.3, v∗ is an extension of M-basis
B∗ for W1 to B=B∗∪{v∗} a M-basis for W.
Step - 3: Since CV(X∖H)=CV(v1),CV(v1)=CV(v∗)
and then v∗is also an extension of M-basis B∗ for V1
to B a M-basis for V.
Step - 4: Now we shall show that v∗ is an extension
of M-basis B∗ for V1∩W1 to B a M-basis for V∩W.
If v∗∈A1, then (CV∧CW)(A1∩(X∖H))=CV(v∗)
and for all z∈A2∩(X∖H),(CV∧CW)(z)≤CV(v∗),
by definition of A2.
From this we may conclude that if v∗∈A1 then
(CV∧CW)(v∗)=sup[(CV∧CW)(X∖H)].
If v∗∈A2 then CW(v∗)≤CV(v∗). Since
CW(v∗)=sup[CW(X∖H)] and CV is constant
on X∖H we must have A1∩(X∖H)=ϕ. Therefore
we have that if v∗∈A2, then (CV∧CW)(v∗)=sup[(CV∧CW)(X∖H)].
By Lemma5.3, we may now conclude that v∗ extends M-basis
B∗ for V1∩W1 to B a M-basis for V∩W.
Step - 5: Now we shall show that v∗ is also an extension
of B∗ a M-basis for V1+W1 to B a M-basis for V+W.
Suppose that there exists z∈X∖H such that 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
If x1∈H then since v∈H we must have v−x1∈H.
Again v∗∈X∖H. So, by Lemma5.2,CW(v∗+v−x1)=CW(v∗)∧CW(v−x1)
and so (9) becomes CW(v∗)<CV(x1)∧CW(v∗)∧CW(v−x1),
which is impossible. Thus x1∈X∖H. Let x′=v∗
in (5). Since CW(θ)≥sup[CV(X∖{θ})]
we have
CV(v∗)<CV(x1)∧CW(v∗+v−x1).....(10)
Recall that CV(X∖H)=CV(v1) and thus CV(v1)=CV(v∗)=CV(x1),
as x1∈X∖H. This again means that the inequality
(10) is false. This means that for all z∈X∖H,C(V+W)(v∗)≥C(V+W)(z).
Therefore by Lemma5.3,v∗ is an extension of B∗ a
M-basis for V1+W1 to B a M-basis for V+W.
Step - 6: Now we shall show that if v∗∈A1 then
CV+W(v∗)=CW(v∗) and if v∗∈A2 then CV+W(v∗)=CV(v∗).
From the definition we have:
From equation (12) , we have v∗∈A1 then CV+W(v∗)=CW(v∗)
and if v∗∈A2 then CV+W(v∗)=CV(v∗).
This completes the proof.
Corollary6.6 If V and W are two multi vector spaces
over X such that the dimension of X is finite and CV(θ)≥sup[CW(X∖{θ})]
and CW(θ)≥sup[CV(X∖{θ})], then
dim(V+W)=dimV+dimW−dim(V∩W).
Example6.7 Suppose X=R2,ω=6. Define
two multi vector spaces V and W with count functions CV
and CW respectively as follows:
It is easily checked that V and W are multi vector spaces and
CV(θ)≥
sup[CW(X∖{θ})] and CW(θ)≥sup[CV(X∖{θ})].
It is also easy to check that
CV∩W((0,0))=5, CV∩W({(x,x):x∈R∖{0}})=1,
CV∩W(X∖{(x,x):x∈R})=1,CV+W((0,0))=5;
CV+W((0,R∖{0}))=3;CV+W(X∖(0,R))=2
and B={(0,1),(1,1)} 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,
dimV=3+1=4, dimW=2+1=3.
So, dimV+dimW−dim(V∩W)=4+3−2=5=dim(V+W).
Definition6.8 Let V be a multi vector space over
X and f:X→Y be a linear map. Then we define f(V)
as
Proposition6.9 If V be a multi vector space over
X where dimX is finite and f:X→Y is a linear
map, then
dim(kerf~)+dim(imf~)=dim(V).
Proof. Suppose that kerf={θ}. If kerf={θ}
then the proof is similar. Now let BKerf be a M-basis for kerf~
and BEx be an extension of BKer to a M-basis for V
(this is clearly possible by repeated application of Lemma5.3).
Then BKerf∪BEx=B is M-basis for V and BKerf∩BEx=ϕ.
We first show that f(BEx)=BIm is a M-basis for imf~.
Clearly BIm is a basis for imf. Let v1,v2,..,vk∈BEx
and a1,...,ak∈R not all zero. By definition we
have
If x∈kerf then x=θ or x=i[=1]p∑biui,ui∈Bkerf
where not all bi are zero; so if x∈kerf+i[=1]k∑aivi
then either CV(x)=CV(θ+i[=1]k∑aivi)
or CV(x)=CV(i[=1]p∑biui+i[=1]k∑aivi)
and thus
CV(x)=min(i[=1]p∧CV(biui),i[=1]k∧CV(aivi)),
[ As uiand vi are M-basis element of V]
which is clearly smaller than or equal to CV(i[=1]k∑aivi).
Thus
But by the above we have if z\in\text{\prec}B_{Ex}\text{\succ},
then Cf(V)(f(z))=CV(z), and thus
dim(V)=v∈BKer∑CV(v)+v∈BEx∑Cf(V)(f(v))
=v∈BKer∑CV(v)+v∈BIm∑Cf(V)(v)
=dim(kerf~)+dim(imf~).
7 Conclusion
There is a future scope of study of infinite dimensional multi vector
space and behavior of linear operators in multi vector space context.
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