Research Article | Open Access

A. Alotaibi, B. Hazarika, S. A. Mohiuddine, "On the Ideal Convergence of Double Sequences in Locally Solid Riesz Spaces", *Abstract and Applied Analysis*, vol. 2014, Article ID 396254, 6 pages, 2014. https://doi.org/10.1155/2014/396254

# On the Ideal Convergence of Double Sequences in Locally Solid Riesz Spaces

**Academic Editor:**Adem Kilicman

#### Abstract

The aim of this paper is to define the notions of ideal convergence, -bounded for double sequences in setting of locally solid Riesz spaces and study some results related to these notions. We also define the notion of -convergence for double sequences in locally solid Riesz spaces and establish its relationship with ideal convergence.

#### 1. Introduction and Preliminaries

In 1951, Fast [1] and Steinhaus [2] introduced the concept of statistical convergence for single sequences, independently. Some basic and important properties of this concept were studied by Buck [3], Šalát [4], Schoenberg [5], and Fridy [6]. Later, the notion of statistical convergence for single sequences was further defined in various spaces; see Çakalli and Khan [7–9], Di Maio et al. [10, 11], Hazarika [12–14], Maddox [15], Mohiuddine et al. [16–19], and so forth. Some application of statistical summability methods is presented in [20, 21]. In 2003, the notion of statistical convergence for single sequences has been extended to double sequences by Mursaleen and Edely [22]. Recently, the statistical convergence and statistical Cauchy for double sequences have been defined in the framework fuzzy and intuitionistic normed spaces by Mohiuddine et al. [23] and Mursaleen and Mohiuddine [24], respectively, and established some interesting results related to the concept of statistical convergence and statistical Cauchy double sequences. Recently, it was defined and studied by Mohiuddine et al. [25] in the setting of locally solid Riesz spaces while for single sequences this concept was first studied by Albayrak and Pehlivan [26] (also see [27–29]). An application of locally solid Riesz spaces in economics can be found in [30].

The notion of ideal convergence for single sequences, which is a generalization of the concept of statistical convergence, was first defined and studied by Kostyrko et al. [31]. Let us recall the notion of ideal convergence and related concepts by Kostyrko et al. [31] as follows. Let be a nonempty set. Then a family of sets (power set of ) is said to be an ideal if is additive; that is, and . A family of sets (power sets of ) is called an* ideal* if and only if, for each , we have and, for each and each , we have . A nonempty family of sets is a* filter* on if and only if ; for each , we have and for each and each , we have . An ideal is called nontrivial ideal if and . Clearly is a nontrivial ideal if and only if is a filter on . A nontrivial ideal is called* admissible* if and only if . A nontrivial ideal is* maximal* if there cannot exist any nontrivial ideal containing as a subset.

We remark that if , then the corresponding convergence coincides with the usual convergence. Also, if , then the corresponding convergence coincides with the statistical convergence (where denotes the natural density of the set ). In the above cases, both and are nontrivial admissible ideals of .

Kumar [32] defined the notions of and -convergence of double sequence and studied some properties of these notions. Recently, Das et al. [33] introduced the concepts of and -convergence of double sequences in the setting of metric space and established some relationship between these types of convergence. Quite recently, Mursaleen and Mohiuddine defined and studied the notion of -convergence, -convergence, -limit points, and -cluster points for single and double sequences, in [34, 35], respectively, in probabilistic normed spaces. Şahiner et al. [36] and Gürdal and Açik [37] introduced the notion of ideal convergence and -Cauchy sequence in 2-normed spaces, respectively. Mursaleen and Alotaibi [38] introduced the notion of ideal convergence in random 2-normed spaces and later on it was extended by Mohiuddine et al. [39] from single to double sequences. For more details on these concepts, one can be referred to [40–52].

Now we recall the definition of locally solid Riesz spaces and some related concepts as follows. Let be a real vector space and let be a partial order on this space. is said to be an* ordered vector space* if it satisfies the following properties:(1)if and , then for each ;(2)if and , then for each .If, in addition, is a lattice with respect to the partial ordering, then is said to be a* Riesz space* (or a* vector lattice*) (see [53]).

For an element of a Riesz space , the positive part of is defined by , the negative part of by , and the absolute value of by , where is the zero element of .

A subset of is said to be* solid* if and implies .

A topology on a real vector space that makes the addition and scalar multiplication continuous is said to be a linear topology, that is, when the mappings
are continuous, where is the usual topology on . In this case the pair is called a* topological vector space*.

Every linear topology on a vector space has a base for the neighborhoods of satisfying the following properties.(1)Each is a* balanced set*; that is, holds for all and every with .(2)Each is an* absorbing set*; that is, for every , there exists such that .(3)For each there exists some with .

A linear topology on a Riesz space is said to be* locally solid* (see [54]) if has a base at zero consisting of solid sets. A* locally solid Riesz space * is a Riesz space equipped with a locally solid topology . For more details on these concepts, one can be referred to [55–57].

Throughout the paper, the symbol will stand for a base at zero consisting of solid sets and satisfying conditions (1), (2), and (3) in a locally solid topology. Also we assume that is a nontrivial admissible ideal of .

#### 2. Ideal Convergence of Double Sequences in LSR-Spaces

Throughout the paper will denote the Hausdorff locally solid Riesz space, which satisfies the first axiom of countability. For our convenience, here and in what follows, we will write an LSR-space instead of a locally solid Riesz space.

The notion of convergence for double sequence was first introduced by Pringsheim [58] as follows. We say that a double sequence of reals is convergent to in Pringsheim’s sense (briefly, -convergent) provided that given there exists a positive integer such that whenever .

Let and denotes the number of in such that and (see [22]). Then the lower natural density of is defined by . In this case, the sequence has a limit in Pringsheim’s sense; then we say that has a* double natural density* and is defined by .

In the recent past, Mohiuddine et al. [25] introduced the notion of statistical convergence of double sequences in LSR-space as follows. Let be a LSR-space. A double sequence of points in is said to be -*convergent* to an element if for each -neighborhood of zero
Now we introduce the notions of -convergence and -bounded double sequences in LSR-spaces.

*Definition 1. *Let be a LSR-space. A double sequence of points in is said to be -*convergent* to an element of if for each -neighborhood of zero
That is,
In this case, one writes or .

*Definition 2. *Let be a LSR-space. Then, a double sequence of points in is said to be -*bounded* in if, for each -neighborhood of zero, there is some ,

*Definition 3. *Let be a LSR-space. One says that a double sequence is -*Cauchy* in if, for each -neighborhood of zero, there exist such that, for all and ,

*Definition 4. *Let be a LSR-space. Then, a double sequence in is said to be -convergent to if there is a set , , with such that . In this case, one writes -.

Theorem 5. *Let be a LSR-space. Every -convergent sequence in has only one limit.*

*Proof. *Suppose that is a double sequence in such that - and -. Let be any -neighborhood of zero. Also for each -neighborhood of zero there is a set such that . Let in be such that . We define the sets and as follows:
Since and , we get . Now, let . Then we have
As we know, intersection of all -neighborhoods of zero is the singleton set because is Hausdorff. Hence ; that is, .

Theorem 6. *Let be a LSR-space and let and be two double sequences of points in . Then,*(i)*if - and -, then -;*(ii)*if -, then - for .*

*Proof. *Assume that - and -. Suppose that is an arbitrary -neighborhood of zero. Then there exists such that . Let such that . Thus, we can write
Then we have .

Let . Hence we have and
Therefore
Since is arbitrary, we have .

(ii) Suppose that - and also suppose that is an arbitrary -neighborhood of zero. Then there exists such that , so we have
Since is balanced, holds for all and for every with . Therefore
Thus, we have
for each -neighborhood of zero. Now let and be the smallest integer greater than or equal to . Then there exists such that . From our assumption that -, we obtain that
Therefore
Since is solid, . It follows that . Thus,
for each -neighborhood of zero. We conclude that -.

Theorem 7. *Let be a LSR-space. If a double sequence in is -convergent, then it is -bounded.*

*Proof. *Assume that -. Suppose is an arbitrary -neighborhood of zero. Then, there exists such that . Let such that . Using our assumption, we obtain that
Since is absorbing, there exists such that . Let be such that and . Since is solid and , we have . Also, since is balanced, implies . Then we have
Thus
Hence is -bounded.

Theorem 8. *Let be a LSR-space and let , , and be three double sequences of points in such that*(i)* , for all ,*(ii)

*--.*

*Then*-*.**Proof. *Suppose that the given conditions (i) and (ii) hold for the double sequences , , and . Suppose is an arbitrary -neighborhood of zero. Then, there exists such that . Let such that . It follows from (ii) that , where
Also from the given condition (i), we have
Since is solid, we have . Thus,
for each -neighborhood of zero. Thus -.

Theorem 9. *Let be a LSR-space. A double sequence is -convergent to in if and only if for each -neighborhood of zero there exists a subsequence of such that and
*

*Proof. *Suppose that . Also, suppose that is an arbitrary -neighborhood of zero. Let be a sequence of nested base of -neighborhoods of zero. For each , put
Then, and . Let and be such that and , respectively. Then . For such that and , choose ; that is, . In general, choose and such that and hold. Then . Therefore for all which satisfy and , choose ; that is, . Hence, it follows that .

Since is an arbitrary -neighborhood of zero, there exists such that . Let such that . Now
Also and imply that
Next suppose for an arbitrary -neighborhood of zero that there exists a subsequence of such that and
Since is any -neighborhood of zero, we choose such that . Then we have
That is,
Therefore

Theorem 10. *If and , then .*

*Proof. *Let be any -neighborhood of 0. Then there exists such that . Let such that . Since , then there exist integers such that implies that . Hence
By the assumption , . Thus
That is,
This implies that .

Theorem 11. *Let be a LSR-space and let be a double sequence in . If there is a -convergent sequence in such that then is also -convergent.*

*Proof. *Suppose that and . Then for an arbitrary -neighborhood of zero, we have
Now,
Therefore, we have

Theorem 12. *Let be a LSR-space. If a double sequence is -convergent to , then it is -convergent to .*

*Proof. *Suppose that -. Let be an arbitrary -neighborhood of zero. Since -, there is a set , with such that , and implies . Then
Therefore
Hence is -convergent to .

Theorem 13. *The sequential method is regular.*

Proof of the theorem is straightforward, so it is omitted.

From Theorem 12, we can easily obtain the following useful result.

Theorem 14. *The sequential method is subsequential.*

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.

#### References

- H. Fast, “Sur la convergence statistique,”
*Colloquium Mathematicum*, vol. 2, pp. 241–244, 1951. View at: Google Scholar | Zentralblatt MATH | MathSciNet - H. Steinhaus, “Sur la convergence ordinate et la convergence asymptotique,”
*Colloquium Mathematicum*, vol. 2, pp. 73–84, 1951. View at: Google Scholar - R. C. Buck, “Generalized asymptotic density,”
*American Journal of Mathematics*, vol. 75, pp. 335–346, 1953. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. Šalát, “On statistical convergence of real numbers,”
*Mathematica Slovaca*, vol. 30, pp. 139–150, 1980. View at: Google Scholar - I. J. Schoenberg, “The integrability of certain functions and related summability methods,”
*The American Mathematical Monthly*, vol. 66, pp. 361–375, 1959. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. A. Fridy, “On statistical convergence,”
*Analysis*, vol. 5, no. 4, pp. 301–313, 1985. View at: Google Scholar | Zentralblatt MATH | MathSciNet - H. Çakalli, “Lacunary statistical convergence in topological groups,”
*Indian Journal of Pure and Applied Mathematics*, vol. 26, no. 2, pp. 113–119, 1995. View at: Google Scholar | Zentralblatt MATH | MathSciNet - H. Çakalli, “On statistical convergence in topological groups,”
*Pure and Applied Mathematika Sciences*, vol. 43, no. 1-2, pp. 27–31, 1996. View at: Google Scholar | Zentralblatt MATH | MathSciNet - H. Çakalli and M. K. Khan, “Summability in topological spaces,”
*Applied Mathematics Letters*, vol. 24, no. 3, pp. 348–352, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Di Maio and L. D. R. Kočinac, “Statistical convergence in topology,”
*Topology and Its Applications*, vol. 156, no. 1, pp. 28–45, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Caserta, G. Di Maio, and L. D. R. Kočinac, “Statistical convergence in function spaces,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 420419, 11 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. Hazarika, “On ideal convergence in topological groups,”
*Scientia Magna*, vol. 7, no. 4, pp. 80–86, 2011. View at: Google Scholar - B. Hazarika, “On generalized difference ideal convergence in random 2-normed spaces,”
*Filomat*, vol. 26, no. 6, pp. 1273–1282, 2012. View at: Google Scholar | Zentralblatt MATH | MathSciNet - B. Hazarika, “Lacunary generalized difference statistical convergence in random 2-normed spaces,”
*Proyecciones. Journal of Mathematics*, vol. 31, no. 4, pp. 373–390, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - I. J. Maddox, “Statistical convergence in a locally convex space,”
*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 104, no. 1, pp. 141–145, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. A. Mohiuddine and M. Aiyub, “Lacunary statistical convergence in random 2-normed spaces,”
*Applied Mathematics & Information Sciences*, vol. 6, no. 3, pp. 581–585, 2012. View at: Google Scholar | MathSciNet - S. A. Mohiuddine, H. Şevli, and M. Cancan, “Statistical convergence in fuzzy 2-normed space,”
*Journal of Computational Analysis and Applications*, vol. 12, no. 4, pp. 787–798, 2010. View at: Google Scholar | Zentralblatt MATH | MathSciNet - S. A. Mohiuddine, A. Alotaibi, and M. Mursaleen, “A new variant of statistical convergence,”
*Journal of Inequalities and Applications*, vol. 2013, article 309, 8 pages, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. Savaş and S. A. Mohiuddine, “$\overline{\lambda}$-statistically convergent double sequences in probabilistic normed spaces,”
*Mathematica Slovaca*, vol. 62, no. 1, pp. 99–108, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Belen and S. A. Mohiuddine, “Generalized weighted statistical convergence and application,”
*Applied Mathematics and Computation*, vol. 219, no. 18, pp. 9821–9826, 2013. View at: Publisher Site | Google Scholar | MathSciNet - N. L. Braha, H. M. Srivastava, and S. A. Mohiuddine, “A Korovkin's type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallée Poussin mean,”
*Applied Mathematics and Computation*, vol. 228, pp. 162–169, 2014. View at: Publisher Site | Google Scholar | MathSciNet - Mursaleen and O. H. H. Edely, “Statistical convergence of double sequences,”
*Journal of Mathematical Analysis and Applications*, vol. 288, no. 1, pp. 223–231, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. A. Mohiuddine, H. Şevli, and M. Cancan, “Statistical convergence of double sequences in fuzzy normed spaces,”
*Filomat*, vol. 26, no. 4, pp. 673–681, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Mursaleen and S. A. Mohiuddine, “Statistical convergence of double sequences in intuitionistic fuzzy normed spaces,”
*Chaos, Solitons & Fractals*, vol. 41, no. 5, pp. 2414–2421, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. A. Mohiuddine, A. Alotaibi, and M. Mursaleen, “Statistical convergence of double sequences in locally solid Riesz spaces,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 719729, 9 pages, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Albayrak and S. Pehlivan, “Statistical convergence and statistical continuity on locally solid Riesz spaces,”
*Topology and Its Applications*, vol. 159, no. 7, pp. 1887–1893, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. A. Mohiuddine, B. Hazarika, and A. Alotaibi, “Double lacunary density and some inclusion results in locally solid Riesz spaces,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 507962, 8 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - S. A. Mohiuddine and A. Alotaibi, “Statistical summability of double sequences through de la Vallée-Poussin mean in probabilistic normed spaces,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 215612, 5 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - S. A. Mohiuddine and M. A. Alghamdi, “Statistical summability through a lacunary sequence in locally solid Riesz spaces,”
*Journal of Inequalities and Applications*, vol. 2012, article 225, 9 pages, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. D. Aliprantis and O. Burkinshaw,
*Locally solid Riesz spaces with Applications to Economics*, vol. 105 of*Mathematical Surveys and Monographs*, American Mathematical Society, Providence, RI, USA, 2nd edition, 2003. View at: MathSciNet - P. Kostyrko, T. Šalát, and W. Wilczyński, “$\mathcal{I}$-convergence,”
*Real Analysis Exchange*, vol. 26, no. 2, pp. 669–685, 2000-2001. View at: Google Scholar | MathSciNet - V. Kumar, “On $I$ and ${I}^{*}$-convergence of double sequences,”
*Mathematical Communications*, vol. 12, no. 2, pp. 171–181, 2007. View at: Google Scholar | MathSciNet - P. Das, P. Kostyrko, W. Wilczyński, and P. Malik, “$I$ and ${I}^{*}$-convergence of double sequences,”
*Mathematica Slovaca*, vol. 58, no. 5, pp. 605–620, 2008. View at: Publisher Site | Google Scholar | MathSciNet - M. Mursaleen and S. A. Mohiuddine, “On ideal convergence in probabilistic normed spaces,”
*Mathematica Slovaca*, vol. 62, no. 1, pp. 49–62, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Mursaleen and S. A. Mohiuddine, “On ideal convergence of double sequences in probabilistic normed spaces,”
*Mathematical Reports*, vol. 12(62), no. 4, pp. 359–371, 2010. View at: Google Scholar | Zentralblatt MATH | MathSciNet - A. Şahiner, M. Gürdal, S. Saltan, and H. Gunawan, “Ideal convergence in 2-normed spaces,”
*Taiwanese Journal of Mathematics*, vol. 11, no. 5, pp. 1477–1484, 2007. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. Gürdal and I. Açık, “On $I$-Cauchy sequences in 2-normed spaces,”
*Mathematical Inequalities & Applications*, vol. 11, no. 2, pp. 349–354, 2008. View at: Publisher Site | Google Scholar | MathSciNet - M. Mursaleen and A. Alotaibi, “On $I$-convergence in random 2-normed spaces,”
*Mathematica Slovaca*, vol. 61, no. 6, pp. 933–940, 2011. View at: Publisher Site | Google Scholar | MathSciNet - S. A. Mohiuddine, A. Alotaibi, and S. M. Alsulami, “Ideal convergence of double sequences in random 2-normed spaces,”
*Advances in Difference Equations*, vol. 2012, article 149, 8 pages, 2012. View at: Publisher Site | Google Scholar | MathSciNet - H. Çakalli and B. Hazarika, “Ideal quasi-Cauchy sequences,”
*Journal of Inequalities and Applications*, vol. 2012, article 234, 11 pages, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. Dündar, Ö. Talo, and F. Başar, “Regularly (${\mathcal{I}}_{2}$, $\mathcal{I}$)-convergence and regularly (${\mathcal{I}}_{2}$, $\mathcal{I}$)-Cauchy double sequences of fuzzy numbers,”
*International Journal of Analysis*, vol. 2013, Article ID 749684, 7 pages, 2013. View at: Publisher Site | Google Scholar - B. Hazarika and S. A. Mohiuddine, “Ideal convergence of random variables,”
*Journal of Function Spaces and Applications*, vol. 2013, Article ID 148249, 7 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - B. K. Lahiri and P. Das, “$I$ and ${I}^{*}$-convergence in topological spaces,”
*Mathematica Bohemica*, vol. 130, no. 2, pp. 153–160, 2005. View at: Google Scholar | MathSciNet - H. I. Miller, “A measure theoretical subsequence characterization of statistical convergence,”
*Transactions of the American Mathematical Society*, vol. 347, no. 5, pp. 1811–1819, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Mursaleen, S. A. Mohiuddine, and O. H. H. Edely, “On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces,”
*Computers & Mathematics with Applications*, vol. 59, no. 2, pp. 603–611, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Nuray and W. H. Ruckle, “Generalized statistical convergence and convergence free spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 245, no. 2, pp. 513–527, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. Šalát, B. C. Tripathy, and M. Ziman, “On some properties of $I$-convergence,”
*Tatra Mountains Mathematical Publications*, vol. 28, pp. 279–286, 2004. View at: Google Scholar | MathSciNet - T. Šalát, B. C. Tripathy, and M. Ziman, “On $I$-convergence field,”
*Italian Journal of Pure and Applied Mathematics*, no. 17, pp. 45–54, 2005. View at: Google Scholar | MathSciNet - B. C. Tripathy and B. Hazarika, “$I$-monotonic and $I$-convergent sequences,”
*Kyungpook Mathematical Journal*, vol. 51, no. 2, pp. 233–239, 2011. View at: Publisher Site | Google Scholar | MathSciNet - B. C. Tripathy and B. Hazarika, “$I$-convergent sequence spaces associated with multiplier sequences,”
*Mathematical Inequalities & Applications*, vol. 11, no. 3, pp. 543–548, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. Tripathy and B. C. Tripathy, “On $I$-convergent double sequences,”
*Soochow Journal of Mathematics*, vol. 31, no. 4, pp. 549–560, 2005. View at: Google Scholar | MathSciNet - U. Yamancı and M. Gürdal, “${\mathcal{I}}^{\mathcal{K}}$-convergence in the topology induced by random 2-normed spaces,”
*International Journal of Analysis*, vol. 2013, Article ID 451762, 7 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - A. C. Zaanen,
*Introduction to Operator Theory in Riesz Spaces*, Springer, Berlin, Germany, 1997. View at: Publisher Site | MathSciNet - G. T. Roberts, “Topologies in vector lattices,”
*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 48, pp. 533–546, 1952. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. V. Kantorovich, “Lineare halbgeordnete Raume,”
*Recreational Mathematics*, vol. 2, pp. 121–168, 1937. View at: Google Scholar - W. A. J. Luxemburg and A. C. Zaanen,
*Riesz Spaces I*, North-Holland, Amsterdam, The Netherlands, 1971. - F. Riesz, “Sur la décomposition des opérations fonctionelles linéaires,” in
*Atti Del Congr. Internaz. Dei Mat., 3, Bologna, 1928*, pp. 143–148, Zanichelli, 1930. View at: Google Scholar - A. Pringsheim, “Zur Theorie der zweifach unendlichen Zahlenfolgen,”
*Mathematische Annalen*, vol. 53, no. 3, pp. 289–321, 1900. View at: Publisher Site | Google Scholar | MathSciNet

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