The Fundamental Theorems in the framework of Bicomplex Topological Modules
aa r X i v : . [ m a t h . F A ] S e p The Fundamental Theorems in the framework ofBicomplex Topological Modules
Rajeev Kumar, Romesh Kumar and Dominic Rochon
Abstract
In this paper, we generalize the fundamental theorems of functional analysis tothe framework of bicomplex topological modules.
This section summarizes a number of known results on the set of bi-complex numbers.The set T of bicomplex numbers is defined as T := { w = z + ι z : z , z ∈ C ( ι ) } , where ι and ι are independent imaginary units such that ι = ι = − . The hyper-bolic number is denoted as j = ι ι such that j = 1 . Under the usual addition andmultiplication of bicomplex numbers, T is a commutative ring. Note that the spaces C ( ι k ) := { a + bι k ; a, b ∈ R } , k = 1 , D := { a + dj ; a, d ∈ R } . are the subrings of the space T . The norm on T is defined as | w | := p | z | + | z | = √ a + b + c + d , where w = a + bι + cι + dι ι , for a, b, c, d ∈ R . The norm | . | is such that | s · t |≤ √ | s | · | t | Mathematics Subject Classification . Primary 16D10, 30G35; Secondary 46C05, 46C50.
Key words and phrases . Bi-complex numbers, bounded linear operator, norm, topological modules,F- modules. dempotent basis. We introduce two bicomplex numbers e and e defined as e = 1 + j e = 1 − j e k = e k , e + k = e k , e + e = 1 , e .e = 0 , for k = 1 , . (1.1)Any bicomplex number w can be written as w = z + z ι = z ˆ1 · e + z ˆ2 · e , where z ˆ1 = z − z ι and z ˆ2 = z + z ι both are the elements of C ( ι ) . We then have another representation of the modulus(norm) on T as under: | w | = p | z ˆ1 | + | z ˆ2 |√ . Note that | e | = 1 / √ | e | . Definition 1.
A number w = z + z ι ∈ T is said to have a multiplicative inverse in T , if there exists a number w = ν + ν i ∈ T such that w w = 1 . Such elements of T are called nonsingular elements, otherwise they are called singular.By equation (1.1), we see that T is not even an integral domain so that it is not afield.We have two interesting principal ideals I and I defined as I = { w · e ; w ∈ T } and I = { w · e ; w ∈ T } . Note that I ∩ I = { } and the set ˆ O of all singular elements in T is nothing butˆ O = I ∪ I . The set set ˆ O is also caled Null cone.Recently, Lavoie, Marchildon and Rochon, [4] [5] have introduced bicomplex Hibertspaces and studied some of their basic properties. In this paper we introduce the conceptof bicomplex topologial modules and extend the Principle of uniform boundedness, openmapping theorem, closed graph theorem and Hahn Banach Theorem to this framework.For basic properties of bicomplex analysis one can refer to [4] , [5], [6], [9] and referencestherein. 2 Principle of Uniform Boundedness
In this section we study the Uniform Boundedness Principle on bicomplex topologicalmodules.
Definition 2.
A module M = ( M + , · ) over a ring R is called a topological R -module ifthere exists a topology τ M on M such that the corresponding operations + : M × M → M and · : M → M are continuous. Remark.
In particularly, if we take R = T , the ring of Bicomplex numbers, we call M =( M, + , · ) as a topological T -module or a topological Bicomplex module. Throughoutthis paper we assume that M is a topological T module. With this assumption, we arein the framework of free modules. Example.
For each n ∈ N , the set T n := T × T × . . . × T , is a topological T -module. Definition 3.
Let M be a topological T -module. A map T : M → M is said to be a T -linear map or operator if it satisfies the following properties:( i ) T ( x + y ) = T ( x ) + T ( y ) , for each x, y ∈ M. ( ii ) T ( αx ) = αT ( x ) , for each α ∈ T . We say that such a T -linear map is bounded if it takes bounded sets into boundedones.The proofs of the following lemmas are straightforward. Lemma 2.1
For any a ∈ M, the map T a : M → M defined by T a ( x ) = a + x, for each x ∈ M is a homeomorphism. (cid:3) Lemma 2.2
For each λ ∈ T , the map M λ : M → M defined by M λ ( x ) = λ · x iscontinuous. In case λ ˆ O then the map M λ is a homeomorphism. (cid:3) Lemma 2.3
The closure of a submodule of a topological T -module M is a topological T -module. (cid:3) Remark. If M = ( M, + , · ) is topological T -module. Then it can be seen that V = { e w ; w ∈ M } and V = { e w ; w ∈ M } F = C ( ι ) . We see that M ′ = V ⊕ V isalso a topological vector space over the field F = C ( ι ) . In this case note that M = M ′ as a set.From Lemma 2.3, we conclude that for any set B ⊆ M, the set sp ( B ) is a topological T -submodule of M. Moreover if sp ( B ) = M, we can say that B is a fundamental set. Definition 4. If V is spanned by a set B == { x i : i ∈ ∧} in M and V is spanned bya set B = { y i : i ∈ ∧} in M, where ∧ is an index set , then the set B = B ⊕ B := { x i + y i : x i ∈ B , y i ∈ B , i ∈ ∧} is said to be a spanning set denoted by sp ( B ) . By the Lemma 2.2, we see that for any set B ⊆ M, the set sp ( B ) is a topologicalsubmodule of M over T . If sp ( B ) = M, we say that B is a fundamental set.Note that here we have used the fact that dim ( V ) and dim ( V ) is same so that Card ( B ) = Card ( B ) = Card ( B ) . See [5] for details in the finite dimensional case.
Lemma 2.4
The closed topological submodule of M determined by a denumerable set B in a topological T -module is separable. Proof.
The set A = { e b : b ∈ B } generates a separable topological vector space in V and so does A = { e b : b ∈ B } in V . Clearly B = A ⊕ A does so in M. (cid:3) Definition 5.
A set B in a topological T -module M is said to be bounded if given anyneighbourhood V of 0 in M, there exists a number ǫ > αB ⊆ M, whenever | α |≤ ǫ, for α ∈ T . Lemma 2.5
A compact subset of a topological T -module M is bounded. Proof.
Let B be a compact set in topological T -module M, and let V be a neighbourhoodof 0 in M. Let B = e B and B = e B. Then using [3, Lemma 8, p-51], we see thatthe sets B and B are bounded in the topological vector spaces V and V , respectively.Thus the set B = B ⊕ B is bounded in M. (cid:3) Corollary 2.6
A convergent sequence in a topological T -module is bounded. (cid:3) Definition 6. An F -module space or a module space of type F, is a topological space M which is also a metric space under some metric ρ such that( i ) ρ is translation invariant, that is, ρ ( x, y ) = ρ ( x − y, . ( ii ) ( X, ρ ) is a complete metric space.In this case, we define an F -norm on X as: | x | = ρ ( x, . Note that it is not apparent whether M is a topological T -module until Theorem 2.8.The next theorem demonstrates the principle of uniform boundedness in the settingof topological T -modules. 4 heorem 2.7 For each a ∈ ∧ , where ∧ is an index set, let T a : M → M be a continuous T -linear map. If for each x ∈ M, the set B x = { T a x : a ∈ ∧} is bounded. Then lim x → T a x = 0 uniformly for a ∈ ∧ . Proof.
Proof is along the similar lines as in [3, Theorem 11, p- 52]. (cid:3)
Theorem 2.8 An F -module space is a topological T -module. Proof.
Proof is along the similar lines as in [3, Theorem 12, p- 52]. (cid:3)
Theorem 2.9 A T -linear map of one F -module space to another is continuous if andonly if it maps bounded sets into bounded ones. Proof.
Proof is along the similar lines as in [3, Theorem 14, p- 52]. (cid:3)
Corollary 2.10
Any continuous T -linear map from one topological T -module to anothersends bounded sets into bounded ones. (cid:3) Corollary 2.11
Any continuous T -linear map from one F -module space to anotherwhich sends sequences converging to into bounded sets is continuous. (cid:3) Theorem 2.12
Let ( T n ) ∞ n =1 be a sequence of continuous T -linear maps of one F -modulespace X into another F -module space Y, such that the limit T ( x ) = lim n →∞ T n x existsfor each x ∈ X, then lim x → T n x = 0 uniformly for n ∈ N and that T is a continuous T -linear map of X into Y. Proof.
Proof is along the similar lines as in [3, Theorem 17, p- 52]. (cid:3)
Remark.
The above result also holds in case we replace the sequence ( T n ) ∞ n =1 by a net( T a ) a ∈∧ . We state the result for more clarity.
Theorem 2.13
Let ( T a ) a ∈∧ be a generalized sequence of continuous T -linear maps of one F -module space X into another F -module space Y, such that the limit T ( x ) = lim n →∞ T n x exists for each x in a fundamental set B in X, and if for each x ∈ X the set { T a x } a ∈∧ isbounded, then the limit T ( x ) = lim n →∞ T n x exists for each x ∈ X, and is a continuous T -linear map of X into Y. (cid:3) The Interior Mapping Theorem
The interior mapping principle is stated in the following thoeorem.
Theorem 3.1
A continuous T -linear map of one F -module space X onto another F -module space Y is an open map. Proof.
Proof is along similar lines as in [3, Theorem 1, p-55]. (cid:3)
Theorem 3.2
A continuous T -linear bijective map of one F -module space X onto an-other F -module space Y has a continuous inverse. Proof.
Proof is obvious by the above arguments or results. (cid:3)
Note that the T -linear maps defined by T a x = ax, for a ∈ ˆ O is clearly not onto. Definition 7.
Let M and N be two topological T -modules. Let T be a T -linear mapwhose domain D ( T ) defined as D ( T ) = { x ∈ M : T x ∈ N } is a topological T -submodule in M and whose range lies in N. Then the graph of T isthe set of all points in M × N of the form [ x, T x ] with x ∈ D ( T ) . The T -linear operator T is said to be closed if its graph is closed in the product space M × N. An equivalentstatement is as follows:The T -linear operator T is closed if whenever x n ∈ D ( T ) , x n −→ x, T x n −→ y ⇒ x ∈ D ( T ) and T x = y. Note that the product M × N of two F -module spaces M and N over the same ring T is also an F -module space over T under the metric d ( · , · ) defined on M × N as follows: d ([ x, y ] , [ x ′ , y ′ ]) = | x − x ′ | + | y − y ′ | . The next result is closed graph theorem in the setting of F -module spaces. Theorem 3.3
A closed linear map defined on all of an F -module space M into an F -module space N is continuous. Proof.
Clearly the graph G of T is a closed T -submodule in the product F -modulespace M × N, hence G is a complete metric space. Thus G is an F -module space. Themap p M : [ x, T x ] x of G onto M is one-to-one, linear, and continuous. Hence, byTheorem 3.2, its inverse p − M is continuous. Thus T = p N p − M is continuous by [3, p-32]. (cid:3) Theorem 3.4
If a module space M is an F -module space under each of the two metrics ρ and ρ , and if one of the corresponding topologies contains the other, then the twotopologies coincide. roof. Let τ and τ be two metric topologies on the module space M such that M = ( M, τ ) and M = ( M, τ ) are F -module spaces over T . If τ ⊆ τ , then the T -linear map x x of M onto M is continuous. By Theorem 3.2, it is a homeomorphismso that τ = τ . Hence the theorem. (cid:3)
Definition 8.
A family F of functions which map one module space M into anothermodule space N over the same ring T is called total if f ( x ) = 0 , ∀ f ∈ F ⇒ x = 0 is theonly possibility. Theorem 3.5
Let
X, Y and Z be F -module spaces over the same ring T and let F bea total family of continuous T -linear maps of X into Y. Let T be a linear T -map from Z to X such that f ◦ T is continuous ∀ f ∈ F , then T is continuous. Proof.
Let w n −→ w and T w n −→ x. Thenlim n →∞ f ( T w n ) = f ( x ) ∀ f ∈ F , since each f ◦ T is continuous so that f ◦ T ( w ) = f ( x ) , ∀ f ∈ F ⇒ f ( T W − x ) = 0 , ∀ f ∈ F ⇒ T w = x. Therefore F is total. (cid:3) First we define T -normed modules as follows: Definition 9.
A topological T -module is said to be T -normed module space if thereexists a map k · k : M R + = [0 , ∞ ) called a T -norm on M if( i ) k · k : M → R + is a norm over the field C ( ι ) or the field C ( ι ) . ( ii ) k wx k ≤ √ | w | k x k , for each w ∈ T and x ∈ M. Note that M is a topological vector space over the field C ( ι ) or the field C ( ι ) . A complete T -normed module space is called a Bicomplex Banach module or a T -Banach module. See [4] and [5] for more light on this aspect.A T -Banach module is an F -module space over T with the properties as follows: k αx k = | α | k x k , ∀ α ∈ C ( ι ) , x ∈ M and k αx k ≤ √ | α | k x k , ∀ α ∈ T , x ∈ M. Lemma 4.1
A set B in a T -normed module space is bounded if and only if sup x ∈ B k x k < ∞ . roof. A neighbourhood of origin in M contains an η -neighbourhood S η = { x ∈ M : k x k < η } of 0 . If a = sup x ∈ B k x k < ∞ , and ǫ = η a , then αB ⊆ V whenever | α |≤ ǫ, so that B isbounded.For x ∈ B, we have k x k ≤ a, and so k αx k ≤ √ | α | k x k ≤ √ | α | a ≤ √ η a a = η √ < η. Conversely, if B is bounded, then there exists an ǫ > αB ⊆ δ = { x ∈ M : k x k < } ∀ | α |≤ ǫ. For x ∈ B, we have ǫ k x k = k ǫx k < ⇒ k x k < ǫ . This proves the result. (cid:3)
Lemma 4.2
For a T -linear map T between T -normed module spaces X and Y, the fol-lowing properties are equivalent. ( i ) T is a continuous. ( ii ) T is a continuous at a point in M. ( iii ) sup k x k≤ k T x k < ∞ . ( iv ) There exists some
M > , such that k T x k ≤ √ M k x k , for each x ∈ M. Proof. (i) ⇒ (ii) is obvious.(ii) ⇒ (i) follows from [3, Lemma 6, p-51]. Thus (i) ⇔ (ii).(i) ⇒ (iv). If T is continuous at 0 ∈ M, there exists ǫ > k T x k < k x k < ǫ. For any x = 0 , let y = ǫx k x k ∈ M, then k y k = ( ǫ k x k ) k x k = ǫ < ǫ, so that ǫ k x k k T x k = k T ( ǫx k x k ) k = k T y k < k T x k < √ √ ǫ ) k x k . This inequality also holds in case x = 0 . This proves the required implication.Clearly (iv) ⇒ continuity at 0 ∈ M. Thus we have (iv) ⇒ (ii).Thus we have (i) ⇔ (ii) ⇔ (iv). 8ow we show that (iii) ⇒ (iv).If M = sup k x k≤ k T x k < ∞ , then for any x = 0 , we have k T x k = k x kk T ( x k x k ) k ≤ M k x k ≤ √ M k x k . This inequality also holds in case x = 0 . This proves the implication. (cid:3)
Definition 10. If X and Y are topological T -modules, we define B ( X, Y ) = { T : X → X : T is a continuous T − linear map } . In case Y = T , we write X ∗ = B ( X, T ) . Note here we have used the fact that T is atopological T -module.Now we define a T -norm on T ∈ B ( X, Y ) as follows: k T k = k T k B ( X, Y ) = 1 √ k x k≤ k T x k . (4.1)We have another representation of norm of T when we express T as T = T ˆ1 e + T ˆ2 e k T k = r k T ˆ1 k + k T ˆ2 k , where each T ˆ k maps X into C ( ι ) . If k T k < ∞ , we say that T is a bounded T -linear operator.By lemma 4.2, we have the next result as an easy implication. Theorem 4.3 A T -linear operator T between two T -normed module spaces X and Y iscontinuous if and only if it is bounded in sense of Definition . (cid:3) If X, Y and Z are topological T -modules such that B : X → Y and A : Y → Z, thatis, A contains the range of B, then we have k AB k ≤ √ k A kk B k . Theorem 4.4
Let X and Y be two T -Banach modules and ( T n ) n ≥ be a sequence ofbounded operators from X into Y. Then the limit
T x = lim n →∞ T n x exists for each x ∈ X if and only if we have ( i ) lim T n x exists for each x in a fundamental set, and ( ii ) For each x ∈ X, we have sup n k T n x k < ∞ . roof. Outline: k T X k = lim n →∞ k T n x k ≤ √ n →∞ k T n kk x k which further implies that k T k ≤ √ n →∞ k T n k . (cid:3) Lemma 4.5
Let X be a T -normed module and Y be a T -Banach module, then B ( X, Y ) is a Banach bicomplex module under the T -norm as defined in the Definition 6. Proof.
Clearly k T k = 0 if and only if T = 0 . Further, k αT k = | α | k T k , when α ∈ C ( ι ) , and k αT k ≤ √ | α | k T k , for α ∈ T . Since k ( T + U )( x ) k ≤ k T x k + k U x k ≤ √ k T k + k U k ) k x k , we have k T + U k = 1 √ k x k≤ k ( T + U )( x ) k ≤ k T k + k U k . Choose a Cauchy sequence ( T n ) ∞ n =1 in B ( X, Y ) with k T n − T m k < ǫ for all n, m ≥ n ( ǫ ) . Then
T x = lim n →∞ T n ( x ) exists for each x, that is, we have k T x − T m x k < ǫ k T x − T n x k ≤ k T x − T m x k + √ k T m − T n kk x k < √ ǫ √ ǫ k x k ) , since the left side of this inequality is independent of m, it is seen by letting m → ∞ that k T − T n k ≤ √ ǫ for n ≥ n ( ǫ ) so that k T k < ∞ , and k T − T n k −→ . (cid:3) Corollary 4.6
The conjugate T -module space X ∗ of a T -normed module space is a T -Banach module space. (cid:3) The next result is clear and and we recall here for the completeness of our presenta-tion. Note that a module over the field R is obviously a vector space. Theorem 4.7
Let the real function p on module space X over the ring (field) R satisfy p ( x + y ) ≤ p ( x ) + p ( y ) , p ( αx ) = αp ( x ) , α ≥ , x, y ∈ X. Let f : Y → R be R -linear map on some submodule Y of X, with f ( x ) ≤ p ( x ) , ∀ x ∈ Y. Then there exists an R -linear map F : X → R such that F ( x ) = f ( x ) , x ∈ Y and F ( x ) ≤ p ( x ) , x ∈ X. The next result is the Hahn-Banach Theorem for T -normed modules. Theorem 4.8
Let Y be a submodule space of T -normed module space X. Then for each y ∈ Y ∗ , there exists x ∗ ∈ X ∗ with x ∗ | Y = y ∗ and k x ∗ k = k y ∗ k . Proof. If X is a real linear space, then proof is clear by Theorem 4.7 with p ( x ) = k y ∗ kk x k , for x ∈ X, and f = y ∗ . Assume that X is T -normed module.For each y ∈ Y, there exists real linear functions f , f , f and f such that y ∗ ( y ) = ( f ( y ) + ι f ( y )) + ι ( f ( y ) + ι f ( y )) , y ∈ Y. Then for α, β ∈ T , and x, y ∈ Y, we have f ( αx + βy ) = αf ( x ) + βf ( y ) , and | f ( y ) |≤| y ∗ ( y ) | . Regarding X as a real module space and so real vector space, we can apply Theorem4.7, to get a real linear extension F on X by F | Y = f and k F k ≤ k y ∗ k . The function x ∗ on T -module space X is defined by x ∗ ( x ) = F ( x ) − ι F ( ι x ) − ι F ( ι x ) + ι ι F ( ι ι x ) . Then clearly x ∗ is additive, and x ∗ ( ι x ) = F ( ι x ) − ι F ( − x ) − ι F ( ι ι x ) + ι ι F ( − ι x )= F ( ι x ) + ι F ( x ) − ι F ( ι ι x ) − ι ι F ( ι x )= − ι F ( ι x ) + ι F ( x ) + ι ι F ( ι ι x ) − ι ι F ( ι x )= ι [ F ( x ) − F ( ι x ) − ι F ( ι x ) + ι ι F ( ι ι x )]= ι x ∗ ( x ) . Similarly we can show that x ∗ ( ι x ) = ι x ∗ ( x )and that also we have x ∗ ( ι ι x ) = F ( ι ι x ) − ι F ( − ι x ) − ι F ( − ι x ) + ι ι F ( x )= ι ι F ( ι ι x ) + ι F ( ι x ) + ι F ( ι x ) + ι ι F ( x )= ι ι ι ι F ( ι ι x ) − ι ι F ( ι x ) − ι ι F ( ι x ) + ι ι F ( x )= ι ι [ F ( x ) − ι F ( ι x ) − ι F ( ι x ) + ι ι F ( ι ι x )]= ι ι x ∗ ( x ) . x ∗ ( wx ) = wx ∗ ( x ) , for each w ∈ T . This shows that x ∗ is a T -linear function and that x ∗ : X → T . Clearly x ∗ is an extension of y ∗ . For y ∈ Y, we have f ( ι y ) + ι yf ( ι y ) + ι f ( ι y ) + ι ι f ( ι y ) = y ∗ ( ι y ) = ι y ∗ ( y )= ι f ( y ) − f ( y ) + ι ι f ( ι y ) − ι f ( y )which shows that f ( y ) = − f ( ι y ) , and similarly we have f ( y ) = − f ( ι y ) , and f ( y ) = f ( ι ι y ) , and hence that y ∗ ( y ) = f ( y ) − ι f ( ι y ) − ι f ( ι y ) + ι ι f ( ι ι y ) . Thus x ∗ is an extension of y ∗ . Note that we can express x ∗ as x ∗ = ( x ∗ ) ˆ1 e + ( x ∗ ) ˆ2 e , where each ( x ∗ ) ˆ k : X → C ( ι ) for k = 1 , , is an extension of the corresponding maps( y ∗ ) ˆ k : Y → C ( ι ) , where y ∗ is given by y ∗ = ( y ∗ ) ˆ1 e + ( y ∗ ) ˆ2 e , and k ( x ∗ ) ˆ k k = k ( y ∗ ) ˆ k k (4.2)by the Hahn-Banach theorem for complex vector spaces or the C ( ι )-modules.Now, using equation 4.2, we have k x ∗ k = r k ( x ∗ ) ˆ1 k + k ( x ∗ ) ˆ2 k r k ( y ∗ ) ˆ1 k + k ( y ∗ ) ˆ2 k k y ∗ k This proves the desired equality and hence the theorem. (cid:3)
The following results are easy implications of the above interesting result.12 heorem 4.9
Let Y be a T -submodule of the T -normed module X. Let x ∈ X be suchthat inf y ∈ Y k y − x k = d > . Then there exists a continuous T -linear map x ∗ : X → T with x ∗ ( x ) = 1 , | x ∗ | = 1 d , x ∗ ( y ) = 0 , ∀ y ∈ Y. (cid:3) Corollary 4.10
Let x be a vector not in the closed T -submodule Y of the T -normedmodule X. Then there is a continuous T -linear map x ∗ : X → T with x ∗ ( x ) = 1 , x ∗ ( y ) = 0 , ∀ y ∈ Y. (cid:3) Corollary 4.11
For each x = 0 in a T -normed module X, there is a continuous T -linearmap x ∗ : X → T with k x ∗ k = 1 and x ∗ ( x ) = k x k . (cid:3) Remark.
Note that in this case also, X ∗ is non-trivial for a non-trivial T -normedmodule space X, but it is not so in case of F -module spaces over the ring T . Corollary 4.12
For each x in a T -normed module X, k x k = sup x ∗ ∈ S ∗ | x ∗ ( x ) | , where S ∗ is the closed unit module sphere in the space X ∗ conjugate to X. (cid:3) References [1] Lars V. Ahlfors,
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Marcel Dekker,3rd edition, New York 1987.Rajeev Kumar,Department of Mathematics,University of Jammu, Jammu, INDIA.
E-mail: [email protected]
Romesh Kumar,Department of Mathematics,University of Jammu, Jammu - 180 006, INDIA.
E-mails: romesh jammu @yahoo.com , romeshmath @gmail.com Dominic Rochon,D´epartement de math´ematiques et d’informatique, Universit´e du Qu´ebec,Trois-Rivi`eres, C.P. 500, Qu´ebec, Canada, G9A 5H7.