Some critical remarks on the paper ”a note on the metrizability of tvs-cone metric spaces” Текст научной статьи по специальности «Математика»

SOME CRITICAL REMARKS ON THE PAPER "A NOTE ON THE METRIZABILITY OF TVS-CONE METRIC SPACES"

Abstract:

This short and concise note provides a detailed exposition of the approach and results established by (Lin et al, 2015, pp.271-279). We show that the obtained results are not particularly surprising and new. Namely, using an old result due to K. Deimling it is indicated that tvs-cone metric spaces over

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Suzana M. Aleksica, Ljiljana R. Paunovicb, Stojan N. Radenovicc, ~ Francesca Vetrod 0

University of Kragujevac, Faculty of Science, Department of Mathematics and Informatics, Kragujevac, Republic of Serbia, e-mail: [email protected], ORCID iD: http://orcid.org/0000-0002-2176-5091, University of Pristina - Kosovska Mitrovica, Faculty of Education in Prizren - Leposavic, Leposavic, Republic of Serbia, o

e-mail: [email protected], ORCID iD: http://orcid.org/0000-0002-5449-9367.

: University of Belgrade, Faculty of Mechanical Engineering, Belgrade, Republic of Serbia, cp

e-mail: [email protected],

ORCID iD: http://orcid.org/0000-0001-8254-6688

1 University of Palermo, Department of Energy, Information Engineering and Mathematical Models (DEIM), Palermo, Italy, e-mail: [email protected], ORCID iD: http://orcid.org/0000-0001-7448-5299

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http://dx.doi.org/10.5937/vojtehg66-15128 J

FIELD: Mathematics ^

ARTICLE TYPE: Original Scientific Paper E

ARTICLE LANGUAGE: English ot

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ACKNOWLEDGMENT: The first author is grateful for the financial support from the Ministry of Education, Science and Technological Development of the Republic of Serbia (Grant number #174024). The second author is grateful for the financial support from the Ministry of Education, Science and Technological Development of the Republic of Serbia (Grant number #174002).

solid cones are actually cone metric spaces overnormal solid cones. Hence, there are only cone metric spaces over normal solid cones or over normal non-solid cones. One question still unanswered is whether an ® ordered topological vector space with a non-normal non-solid cone exists.

Vo Key words: tvs-cone metric space, metrizable, solid, normal, non-normal.

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Introduction, preliminaries and results

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g We will denote by E a topological vector space with the zero vector

o 9 . A subset C of E is called a cone in E if and only if: o (a) C is nonempty and closed in E,

x (b) X,y e C and a, b e R+ imply ax + by e C,

uj (c) x,-x e C implies x = 9.

£ For more details we refer the reader to (Ansari et al, 2016),

* (Deimling, 1985), (Filipovic et al, 2011), (Jankovic et al, 2011), (Kothe, 1969), (Shaefer, 1971) and (Wong & Ng, 1973).

For a given cone C, a partial ordering < with respect to C is introduced in the following way: x < y if and only if y - x e C. We write

fj x < y to indicate that x < y, but x ^ y. If y - x e int C, we write x « y . It

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^ is clear that < and << are not relations of partial order. y The pair (E, C) is called an ordered topological vector space.

Throughout the paper, C stands for a solid cone, i.e. intC In most o papers it turns out that this assumption is essential (Amini-Harandi & § Fakhar, 2010), (Cakalli et al, 2012), (Du, 2010), (Proinov, 2013), (Khani & > Pourmahdian, 2011), (Lin et al, 2015), (Simic, 2011), (Vandergraft, 1967) and (Zabrejko, 1997).

The following result from (Wong & Ng, 1973) is important in the framework of ordered topological vector spaces.

Proposition 1.1. Let (E, C) be an ordered topological vector space. Then e is an interior point of C, i.e., e e intC, if and only if

[- e,e]= {x e E : -e < x < e}= (e - C)n(C - e)

is the neighborhood of 9.

Under the assumption E ^ {9}, we have the following corollary. Corollary 1.2. Let (E, C) be an ordered topological vector space with int C Then

(a) Oz int C,

(b) Aint C = int C if A> 0 and C+intC = intC,

(c) (E, C) is Hausdorff ordered topological vector space.

Let us note that the property (c) is an immediate consequence of Theorem 2.1 (2) from (Jachymski & Klima, 2016). |

For each c e int C there exists a norm ||c on the vector space E

where INI is the Minkowski functional on E = Ec = u n[-c,c], i.e., jS

||x||c = inf {A > 0: x e A[- c, c]}.

It is obvious that all norms ||c, c e int C, are equivalent. Further, we get that (e ,| |-|| c, C) is an ordered normed space with the normal-solid

cone C. For more details see (Deimling, 1985, pp.230, Proposition 19.9). Therefore, int C under the topology on E is a subset of int C under the norm topology, i.e., (intC)E c (intC)|.| . However, the converse is also

valid. Hence, (i nt C)E = (i nt CV» .

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From what has already been proved, it is clear that the notion of tvs-cone metric space in the sense of Du, see (Du, 2010), is the same as that of (Lin et al, 2015, Definition 2.3).

According to (Deimling, 1985) and (Kadelburg et al, 2016), we are | thus led to the following strengthening conclusions.

Proposition 1.3. Each tvs-cone metric space over a solid cone in the g sense of the both definitions (Du, 2010) and (Lin et al, 2015) is in fact a cone metric space over a normal solid cone. Therefore, the well known ® results, (Amini-Harandi & Fakhar, 2010), (Du, 2010), (Kadelburg et al, ° 2011), (Khani & Pourmahdian, 2011) give a positive answer to Question 1.1 that arises in (Lin et al, 2015).

Proposition 1.4. The claims from (Lin, 2015) are the immediate consequences of the corresponding results in the framework of cone metric spaces over normal solid cones, see (Huang & Zhang, 2007), <! (Jankovic et al, 2011), (Kadelburg et al, 2011), (SimiC, 2011).

Proposition 1.5. If C is a solid cone in some ordered topological vector space E, then it is a normal solid cone under a suitable defined norm. For all details see (Deimling, 1985, Proposition 19.9).

Remark 1.6. It is worth pointing out that in (Du, 2010), (Fierro, 2016), (Kadelburg et al, 2016), (Lin et al, 2015) and (Simic, 2011) the assumption:

(e, C) is Hausdorff, is superfluous. Applying Corollary 1.2 (c) we conclude that the condition int C implies that the ordered topological vector co space (E, C) is Hausdorff.

In the theory of abstract metric spaces, quite a few proofs are based on c-sequences introduced in (Alnafei et al, 2011). Namely, for a sequence {xn} in the solid cone C of a real ordered topological vector space E, we

Si say that it is a c-sequence if for every interior point c of the cone C there =5 exists n0 eN such that xn << c whenever n > n0. If C is normal

^ in E, then \xn} is a c-sequence if and only if xn ^9 in E. However, if

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o the cone C is a non-normal solid cone, then the previous is not true, i.e.,

x a c-sequence does not always converge to 9 as n ^ m. We refer the

UJ readers to (Radenovic et al, 2017), (Xu & Radenovic, 2014), (Xu et al,

£ 2016), (Bordevic et al, 2011) and (Kadelburg et al, 2011) for more details

< on c-sequences.

Theorem 1.7. Let (e,||-||) be an ordered (real) Banach space with

the underlying solid cone C and {xn} be a sequence in C. Then {xn} is a

w c-sequence if and only if ||xn|L ^ 9 as n where

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ml = in f {A > 0 : x e a[- e, e\ e e intc},

i.e., ■ is the Minkowski functional.

o Proof. First, ■ is a norm in E. If C is a normal solid cone, then this

norm is equivalent to the given one, and xn ^9 if and only if xn ^9.

If C is not normal, then C is a normal and solid cone under the norm ||e for each e e int C. Since || and |Je give the same interior points, then

by the first case {xn} is a c-sequence if and only if ||xn|^9. Consequently, in each normal cone, the conditions xn ^ 9 and {xn} is a c-sequence are equivalent.

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Conclusion

Note that we have actually proved that there exist cone metric spaces over normal solid cones or over normal non-solid cones. Therefore, the

results obtained in (Lin et al, 2015) are not new but rather the consequences of known facts (Huang & Zhang, 2007), (Jankovic et al, 2011), (Kadelburg et al, 2011) and (Simic, 2011). We can conclude that the existence of ordered topological vector spaces with a non-normal nonsolid cone is still an open question.

References

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НЕКОТОРЫЕ КРИТИЧЕСКИЕ ЗАМЕЧАНИЯ О РАБОТЕ «ЗАМЕТКИ О МЕТРИЗУЕМОСТИ ТВП-КОНИЧЕСКИХ ПРОСТРАНСТВ»

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Сузана М. Алексича, Лиляна Р. Паунович6, Стоян Н. Раденовичв, го

Франческа Ветрог

Университет в г. Крагуевац, Естественно-математический факультет, Институт математики и информатики, г. Крагуевац, Республика Сербия,

б Университет в Приштине-Косовской Митровице, Педагогический

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г Университет в г. Палермо, Департамент энергетики, информационного о

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В представленных, в данной статье, заметках приведен подробный обзор методов и полученных результатов исследования группы авторов, во главе с Шой Линь (Lin et al, 2015, pp.271-279). Мы в свою очередь доказываем, что их й результаты не являются инновационными. В частности, при применении известных результатов К. Деймлинга установлено, что ТВП-коничеческие метрические пространства с конусами с непустой внутренностью фактически являются метрическими

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пространствами с нормальной коническои и непустой ¡=

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Ключевые слова: твп-коническое метрическое пространство, & метризуемость, конус с непустой внутренностью, нормальный,

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НЕКЕ КРИТИЧКЕ НАПОМЕНЕ О РАДУ „БЕЛЕШКА О МЕТРИЗАБИЛНОСТИ ТВП-КОНУСНИХ МЕТРИЧКИХ ПРОСТОРА"

Сузана М. Алексий3, Ъиъана Р. Паунови^, Сто^ан Н. Раденови^8, Франческа Ветрог

а Универзитет у Крагу]евцу, Природно-математички факултет, Институт за математику и информатику, Крагуевац, Република Срби]а,

б Универзитет у Приштини - Косовско] Митровици, Учите^ски факултет

у Призрену - ЛепосавиЬу, ЛепосавиЬ, Република Срби]а Универзитет у Београду, Машински факултет, Београд, <g Република Срби]а

г Универзитет у Палерму, Одсек за енергетику, информационо

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о ОБЛАСТ: математика

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2 Къучне речи: твп-конусни метрички простор, метризабилан,

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О Paper received on / Дата получения работы / Датум приема чланка: 26.09.2017. -з Manuscript corrections submitted on / Дата получения исправленной версии работы /

> Датум достав^ана исправки рукописа: 10.10.2017 Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихватана чланка за об]ав^иване: 12.10.2017.

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