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Uğur Kadak, "Determination of the Köthe-Toeplitz Duals over the Non-Newtonian Complex Field", The Scientific World Journal, vol. 2014, Article ID 438924, 10 pages, 2014. https://doi.org/10.1155/2014/438924
Determination of the Köthe-Toeplitz Duals over the Non-Newtonian Complex Field
The important point to note is that the non-Newtonian calculus is a self-contained system independent of any other system of calculus. Therefore the reader may be surprised to learn that there is a uniform relationship between the corresponding operators of this calculus and the classical calculus. Several basic concepts based on non-Newtonian calculus are presented by Grossman (1983), Grossman and Katz (1978), and Grossman (1979). Following Grossman and Katz, in the present paper, we introduce the sets of bounded, convergent, null series and p-bounded variation of sequences over the complex field and prove that these are complete. We propose a quite concrete approach based on the notion of Köthe-Toeplitz duals with respect to the non-Newtonian calculus. Finally, we derive some inclusion relationships between Köthe space and solidness.
It is certainly not unusual to measure deviations by ratios rather than differences. For instance, during the Renaissance, many scholars, including Galileo, discussed the following problem. Two estimates, 10 and 1000, are proposed as the value of a horse, which estimates, if any, deviates more from the true value of 100? The scholars who maintained that deviations should be measured by differences concluded that the estimate of 10 was closer to the true value. However, Galileo eventually maintained that the deviations should be measured by ratios, and he concluded that two estimates deviated equally from the true value. From the story, the question comes out this way, what if we measure by ratios? The answer is the main idea of non-Newtonian calculus which consists of many calculuses such as the classical, geometric, anageometric, and bigeometric calculus.
Bashirov et al. [1, 2] have recently concentrated on the non-Newtonian calculus and gave the results with applications corresponding to the well-known properties of derivatives and integrals in the classical calculus. Quite recently, Uzer  has extended the non-Newtonian calculus to the complex-valued functions and was interested in the statements of some fundamental theorems and concepts of multiplicative complex calculus and demonstrated some analogies between the multiplicative complex calculus and classical calculus by theoretical and numerical examples. In particular, Bashirov et al.  have studied the multiplicative differentiation for complex-valued functions and established the multiplicative Cauchy-Riemann conditions. Further, Tekin and Başar have introduced some certain sequence spaces over the non-Newtonian complex field by using -calculus in  and many authors have introduced multiplicative calculus in biomedical image analysis and have derived non-Newtonian calculus as an alternative to the quantum calculus in [5, 6].
Following Tekin and Başar  we can construct the sets , , and consisting of the sets of all bounded, convergent, null series based on the non-Newtonian calculus, as follows: where . One can conclude that the sets , , and are complete non-Newtonian metric spaces with the metric defined by
Secondly, we introduce several sets , , and of bounded variation sequences in the sense of non-Newtonian calculus, as follows: One can easily see that the sets , , and are complete with corresponding metrics on the right-hand side with , and for all .
2. β-Arithmetics and Some Related Applications
A generator is a one-to-one function whose domain is and whose range is a subset of , the set of real numbers. Each generator generates exactly one arithmetic, and conversely each arithmetic is generated by exactly one generator. As a generator, we choose the function such that its basic algebraic operations are defined as follows: for all , where the non-Newtonian real field as in .
The -positive real numbers, denoted by , are the numbers in such that ; the -negative real numbers, denoted by , are those for which . The -zero, , and the -one, , turn out to be and . Further, . Thus the set of all -integers turns out to be the following:
Definition 1. Let be a nonempty set and let be a function such that, for all , the following axioms hold: (NM1) if and only if , (NM2) , (NM3) . Then, the pair and are called a non-Newtonian metric space and a non-Newtonian metric on , respectively.
Definition 2 (see ). Let be a non-Newtonian metric space. Then the basic notions can be defined as follows. (a)A sequence is a function from the set into the set . The -real number denotes the value of the function at and is called the term of the sequence.(b)A sequence in a metric space is said to be -convergent if for every given there exist an and such that for all and is denoted by or , as .(c)A sequence in is said to be non-Newtonian Cauchy (-Cauchy) if for every there is an such that for all .
Remark 3. Let . Then the number is called the -square and is denoted by . Let be a nonnegative number in . Then is called the -square root of and is denoted by . Further for each we can write .
The -absolute value of a number in is defined as and is denoted by . For each number in , . Then we say The non-Newtonian distance between two real numbers and is defined by . Similarly by taking into account the definition for -generator in (6) one can conclude that the equality holds for all .
Now, we give a new type calculus for non-Newtonian complex terms, denoted by -calculus, which is a branch of non-Newtonian calculus. From now on we will use -calculus type with respect to two arbitrarily selected generator functions.
2.1. -Arithmetics with respect to the Complex Field
Suppose that and are two arbitrarily selected generators and (“star-”) also are the ordered pair of arithmetics (-arithmetic and -arithmetic). The sets and are complete ordered fields and -generator generates -arithmetics, respectively. Definitions given for -arithmetic are also valid for -arithmetic.
The important point to note here is that -arithmetic is used for arguments and -arithmetic is used for values; in particular, changes in arguments and values are measured by -differences and -differences, respectively. The operators of this calculus type are applied only to functions with arguments in and values in . The -limit of a function with two generators and is defined by A function is -continuous at a point in if and only if is an argument of and . When and are the identity function , the concepts of -limit and -continuity are identical with those of classical limit and classical continuity.
The isomorphism from -arithmetic to -arithmetic is the unique function (iota) that possesses the following three properties:(i) is one to one.(ii) is from onto .(iii)For any numbers, and in , It turns out that for every in and that for every integer . Since, for example, , it should be clear that any statement in -arithmetic can readily be transformed into a statement in -arithmetic.
Let and be arbitrarily chosen elements from corresponding arithmetics. Then the ordered pair is called a -point. The set of all -points is called the set of -complex numbers and is denoted by ; that is, Define the binary operations addition and multiplication of -complex numbers and : where and .
Theorem 4 (see ). is a field.
Following Grossman and Katz  we can give the definition of -distance and some applications with respect to the -calculus which is a kind of calculi of non-Newtonian calculus.
The -distance between two arbitrary elements and of the set is defined by Up to now, we know that is a field and the distance between two points in is computed by the function , defined by (11).
Definition 5. Given a sequence of -complex numbers, the formal notation, is called an infinite series with -complex terms or simply complex -series. Also, for integers, , the finite -sums are called the partial sums of complex -series. If the sequence -converges to a complex number , then we say that the series -converges and write . The number is then called the -sum of this series. If -diverges, we say that the series -diverges or it is -divergent.
Proposition 6 (see ). For any , the following statements hold: (i) (-triangle inequality).(ii).(iii)Let and for . Then,
Following Tekin and Başar , we can give the -norm and next derive some required inequalities in the sense of non-Newtonian complex calculus.
Let be an arbitrary element. The distance function is called -norm of and is denoted by . In other words, where and . Moreover, since for all we have , is the induced metric from norm.
Definition 7 (complex conjugate). Let . We define the -complex conjugate of by . Conjugation changes the sign of the imaginary part of but leaves the real part the same. Thus
Remark 8. (i) Let , . We can give the -division as
(ii) Let and be the same generator functions and . Then the following condition holds
Corollary 10 (see ). is a Banach space with the -norm defined by ; .
3. Completeness of the Sets of Bounded, Convergent, and Null Series over the Geometric Complex Field
Quite recently Tekin and Başar  have introduced the sets , , , and of all bounded, convergent, null, and absolutely -summable sequences over the complex field which correspond to the sets , , , and over the complex field , respectively. That is to say, It is not hard to show that the sets , , , and are the subspaces of the space . This means that , , , and are classical sequence spaces over the field and complete metric spaces with corresponding metrics.
Theorem 11 (see ). The following statements hold.(a)The sets , , , and , , are sequence spaces.(b)Let denote any of the spaces , , and and , . Define on the space by . Then, is a complete metric space.(c)The spaces , , and are Banach spaces with the norm defined by (d)The space is Banach spaces with the norm defined by
In the present section, we introduce the sets , , and , , consisting of all bounded, convergent, null series and the sets of bounded variation sequences in the sense of non-Newtonian calculus which correspond to the sets , , and , , over the complex field , respectively.
Theorem 12. Let denote any of the spaces , , and and , . Define on the space by for arbitrarily chosen operators and corresponding function . Then, is a complete metric space.
Proof. Since the proof is similar to the spaces and , we prove the theorem only for the space . Let the -sums , , where , . Then the following metric axioms in Definition 1 are valid. (NM1) From (11) it can be easily obtained that
(NM2) It is trivial that the condition holds. (NM3) We show that triangle inequality in Definition 1 holds for , , . In fact by taking into account Proposition 6 (i)
Since the axioms (NM1)–(NM3) are satisfied, is a non-Newtonian metric space. It remains to prove the completeness of the space .
Let be a -Cauchy sequence in , where . Then, for every , there is an element such that, for all , A fortiori, for every fixed and for all Hence, for every fixed , the sequence is a -Cauchy sequence. Before that, by using the completeness of in Theorem 9, it -converges; that is, as . Using these infinitely many limits , we define and show that . From (25) letting and we have Since , there exists such that for all . Thus, (26) gives together with the -triangle inequality for It is clear that (26) holds for every whose right-hand side does not involve . Hence is a bounded sequence of geometric complex numbers; that is, . Also from (26) we obtain for Hence, as we have seen above that , as for an arbitrary -Cauchy sequence . Hence is complete.
Thus it is known by Theorem 12 that the spaces , , and are complete metric spaces with the metric induced by the norm or . Now, as a consequence of Theorem 12, the following corollary presents for these spaces to be Banach space.
Corollary 13. The spaces , , and are Banach spaces with the norm defined by
To avoid undue repetition in the statements we give the next theorem which is on the complete metric space without proof since the proof can be obtained similarly as Theorem 12.
Theorem 14. Let be defined on the space by where , , and . Then, is a complete metric space. Similarly, one can conclude that the other bounded variation sets and are also complete.
Corollary 15. The space is a Banach space with the norm defined by
4. The Duals of the Sets of Sequences over the Geometric Complex Field
The idea of dual sequence space which plays an important role in the representation of linear functionals and the characterization of matrix transformations between sequence spaces was introduced by Köthe and Toeplitz , whose main results concerned alpha-duals. An account of the duals of sequence spaces can be found in Köthe . One can also know about different types of duals of sequence spaces in Maddox .
In this section, we focus on the alpha-, beta-, and gamma-duals of the classical sequence spaces over non-Newtonian complex field. For , , the set , defined by is called the multiplier space of and for all . One can easily observe for a sequence space of -complex numbers that the inclusions if and if hold.
Firstly, we define the alpha-, beta-, and gamma-duals of a set which are, respectively, denoted by , , and , as follows: where is the coordinatewise product of -complex numbers and for all . Then is called beta-dual of or the set of all convergence factor sequences of in . Firstly, we give a remark concerning the -convergence factor sequences.
Throughout the text, we also use the notation “” for a non-Newtonian linear subspace which was created in .
Remark 16. Let . Then the following statements are valid. (a) is a sequence space if , where .(b)If , then .(c).(d) and .
Proof. Since the proof is trivial for the conditions (b) and (c), we prove only (a) and (d). Let , , and . (a)It is trivial that holds from the hypothesis. We show that for . Suppose that . Then and for all . We can deduce that Hence, . Now, we show that for any and , since for all . Combining this with for all we get . Therefore, we have proved that is a subspace of the space .(d)Using (a) we need only to show . Suppose that and are given with -division by if and otherwise. By taking into account the set from inclusion (a), then there exists an integer for all such that . Thus, we have Further, implies that . The rest is an immediate consequence of this part and we omitted the details.
Theorem 17. The following statements hold: (a).(b).
Proof. (a) Obviously by Remark 16 (b). Then we must show that and . Now, consider that and are given. By taking into account the cases (c)-(d) of Theorem 11, we have
which implies that . So the condition holds.
Conversely, for a given , we prove the existence of an with . According to we may confirm an index sequence which is strictly increasing with and . By taking into account Remark 8 (i), if we define by , the non-Newtonian complex signum function is defined by where is given complex conjugate in Definition 7 for all . Finally, by using Remark 8 (ii) taking the generators , we get for all . Therefore and thus . Hence .
(b) From the condition (c) of Remark 16 we have since . Now we assume the existence of a . Since is unbounded there exists a subsequence of and we can find a real number such that for all . The sequence is defined by if and otherwise. Then . However, Hence, , which contradicts our assumption and . This step completes the proof.
In addition to the statements in Remark 16 we make the following remarks which are immediate consequences of the definition of the -duals .
Remark 18. Let . Then the following statements are valid:(a); in particular, is a sequence space over ;(b)if , then ;(c) is an index set, if are sequence spaces, and if , then , where the notation stands for the span of linear subspace over ;(d).
Proof. The case (b) obviously is true, and (a) follows from . We only show the cases (c) and (d) taking . The rest of the parts can be obtained in a similar way.(c)Now, as an immediate consequence the following and hold by (b). On the other hand, if , that is, , then for all and therefore .(d)We can deduce . Let ; then for all ; thus and by (a).
Here as we get from Theorem 17 (a) in the case of and . We have . This remark gives rise to the following definition.
Definition 19 (-space, Köthe space). Let and let be a sequence space over the field . is called -space if . Further, an -space is also called a Köthe space or perfect sequence space.
From Remark 18 (d) and (b) we obtain immediately the following remark.
Remark 20. If is a sequence space over the field and , then is a -space; that is, .
Now we look for sufficient conditions for . This gives rise to the notion of solidity.
Definition 21 (solid sequence space). Let be a sequence space. Then is called solid if
Theorem 22. Let be a sequence space over the field . Then is solid if and only if
Proof. Let and with be given. Further, let , where . Obviously the condition holds because . Let , where . We obtain so since for all . We may choose a real number with and consider defined by . Then ; thus and . From Definition 7 we get . Determining such that it follows similarly that and , which imply . Hence . This step completes the proof.
Theorem 23. Consider that is any sequence space over the field ; then the following statements hold. (a)If is a Köthe space, then is solid.(b)If is solid, then .(c)If is a Köthe space, then is a -space.
Proof. Let be a sequence space over the field .(a)If is a Köthe space and , then if and only if the condition holds for all . Besides this we obtain for and and the statement holds for each . Therefore . Hence and is solid over the field .(b)Consider that is a solid sequence space over . To show , it is sufficient condition to verify in Remark 18 (a). So, let ; that is, By taking into account solidness of , for , the condition holds and there exists a sequence for all . Therefore by combining this with the inclusion (43) we deduce that the condition holds and . Hence and .(c)This is an obvious consequence of Remark 20 and the cases (a)-(b) in Theorem 23.
Theorem 24. The following statements hold. (a)The sets , , , , and of sequences are solid.(b)The sets and of sequences are not solid; therefore none of them is a Köthe space.(c)For each , then and ; and .(d)If and , then and . In particular, , and each of , is not a -space.
Proof. That the specified spaces in cases (a-b) are solid is an immediate consequence of their definition. Additionally, the cases (i-ii) of (c) can be obtained Theorem 17 and Remark 16 (d). Since and are solid, we know that . So the statements in (d) are obtained from Remark 18 (b).
Next, we determine the -duals of the spaces ,