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Vladikavkaz Mathematical Journal 2018, Volume 20, Issue 2, P. 49-56
DOI 10.23671 /VNC.2018.2.14720
UNBOUNDED CONVERGENCE IN THE CONVERGENCE VECTOR LATTICES: A SURVEY
A. M. Dabboorasad1, E. Yu. Emelyanov2
1 Islamic University of Gaza; 2 Middle East Technical University
Dedicated to Professor A. G. Kusraev on the occasion of his 65th anniversary
Various convergences in vector lattices were historically a subject of deep investigation which stems from the begining of the 20th century in works of Riesz, Kantorovich, Nakano, Vulikh, Zanen, and many other mathematicians. The study of the unbounded order convergence had been initiated by Nakano in late 40th in connection with Birkhoff's ergodic theorem. The idea of Nakano was to define the almost everywhere convergence in terms of lattice operations without the direct use of measure theory. Many years later it was recognised that the unbounded order convergence is also rathe useful in probability theory. Since then, the idea of investigating of convergences by using their unbounded versions, have been exploited in several papers. For instance, unbounded convergences in vector lattices have attracted attention of many researchers in order to find new approaches to various problems of functional analysis, operator theory, variational calculus, theory of risk measures in mathematical finance, stochastic processes, etc. Some of those unbounded convergences, like unbounded norm convergence, unbounded multi-norm convergence, unbounded r-convergence are topological. Others are not topological in general, for example: the unbounded order convergence, the unbounded relative uniform convergence, various unbounded convergences in lattice-normed lattices, etc. Topological convergences are, as usual, more flexible for an investigation due to the compactness arguments, etc. The non-topological convergences are more complicated in genelal, as it can be seen on an example of the a.e-convergence. In the present paper we present recent developments in convergence vector lattices with emphasis on related unbounded convergences. Special attention is paid to the case of convergence in lattice multi pseudo normed vector lattices that generalizes most of cases which were discussed in the literature in the last 5 years.
Key words: convergence vector lattice, lattice normed lattice, unbounded convergence. Mathematical Subject Classification: 46A03, 46A40, 46B42.
1. Introduction
A convergence [s-convergence] c for nets [resp., for sequences] in a set X is defined by the following two conditions:
(a) Xa — X —^ Xa ^ X ^rGSp.^ Xn — X —^ Xn ^ xj^
(b) Xa —^ X — X^ —^ X for every subnet x^ of xa [resp., Xn —^ X — Xnk —^ X for every subsequence xnk of xn].
A convergence set is a pair (X, c) where c is a convergence in a set X. A mapping f from a convergence set (Xi,c 1) into a convergence set (X2, c2) is said to be continuous, if xa -—^x implies f (xa) -—^ f (x). s-Continuity of f is defined by replacing nets with sequences.
© Dabboorasad A. M., Emelyanov E. Yu.
A subset A of (X, c) is called: c-ciosed if A A
aa in A possesses a subnet ap such that ap a for some a e A. sc-Closedness and sc-compactness are defined by using sequences. If the set |x} is c-closed for every x e X then c is called T\-convergence. It is immediate to see that c e Ti if every constant net xa = x does not c-converge to any y = x. For further information on convergences we refer to [1, 2].
In the present paper, we investigate several special convergences in real vector lattices.
XX which agrees with the linear and lattice operations in the following way:
X 3 xa —^ x, X 3 yp —^ y, I 3 rY ^ r
imply
r7 ■ Xa + yp r ■ X + y (1)
and
r7 ■ xa A yp r ■ x A y. (2)
X
c-convergence in X and to the usual convergence in [R. In this case, we say that X = (X, c) is a convergence vector lattice, s-Convergence vector lattices are defined by using in (1) and (2) sequences instead of nets.
A net xa [resp., a sequence xn] in (X, c) is called a c-Cauchy, whenever
(xa - xp) 0 [resp., (xm - xn) 0 (m, n (3)
( X, )
X
A ^-convergence Ci in a vector lattice X is said to be minimal [s-minimal], if for any other Tl-convergence c in X satisfying xa -^ 0 ^ xa -^ 0 for all nets xa in X [resp., xn -^ 0 ^ xn -^ 0 for all seque nces xn in X], it follows th at c = c i.
X
xa xn X
xa 0 xa 0, (4)
[respectively,
xn -* 0 xn 0]. (5)
It follows from (4), (5) that every Lebesgue convergence is s-Lebesgue.
Basic examples of convergence vector lattices are: a locally solid vector lattice X = (X, t) with its t-convergence [3]; a space of Lebesgue measurable functions on [0,1] with the almost
X
vergence [ru-convergence] [4]; a lattice normed vector lattice (X, p, E) with the P-convergence [5, 6]. For more details, see [3-10].Recently, o- and uo-convergence were investigated in [7; 11-16] with some further applications in [17-19].
In the present paper, we introduce several further convergence lattices and investigate corresponding unbounded convergences.
The second author expresses deep gratitude to Prof. Anatoly Kusraev for his decisive impact on the author's choice of the functional analysis as his research area 29 years ago.
2. Examples of convergence vector lattices
In this section, we collect and shortly discuss several examples of convergence vector lattices. The convergences in Examples 2, 3, and 4 below are topological in the sense that there is locally solid topology r such that the r-convergence coincided with the corresponding c-convergence.
Example 1. Let X be a vector lattice. Clearly (X, —V) is a T1-convergence vector lattice. Furthermore, (X, —V) is a convergence vector lattice, where "—V" is T1 iff X is Archimedean (cf. [3, 4, 8, 9, 101).
In a Lebesgue and complete metrizable locally solid vector lattice, xa —V x iff xa —V x [20, Proposition 3]. It was also shown in [20, Proposition 4] that, in [Rn, "—V" is equivalent to "—V" for nets iff 0 is countable. Furthermore, it was proved that the o-convergence in X is topological iff dim(X) < to [11, Theorem 1], and that the RU-convergence is topological iff X has a strong order unit [20, Theorem 5]. It is worth to notice that the so-convergence
XX isomorphic to c0 [21, Theorem 1].
Example 2. Let M = {m^be a family of Riesz seminorms on a vector lattice X. If, for any 0 = x £ X, there is m^ £ M such that m^(x) > 0 (X, M) is said to be a multinorm ed lattice (cf. [10, Definition 5.1.6]), abbreviated by MNL7 with the Riesz multi-norm M. Convergence in a Riesz multi-norm (m-convergence) was studied recently in [7].
MNLs are also known as Hausdorff locally convex-solid vector lattices (cf. [3, p. 59]). Note that now-days the name "multi-normed space" is also used for quite different class of spaces [22].
Example 3. Given a vector lattice X, a function r : X ^ [R + is called a Riesz pseudos-eminorm (cf. [3, Definition 2.27]), whenever:
(a) r(x + y) ^ r(x) + r(y) for all X, y £ X;
(b) r(anx) = 0 for all x £ X and for all [R 9 an ^ 0;
(c) |y| ^ |x| implies r(y) ^ r(x).
If r(x) = 0 for any 0 = x £ X, r is called a Riesz pseudonorm and (X, r) is said to be
()
The convergence in a PNL is rather similar to the norm convergence in a normed lattice except of possible lack of a locally convex base for the corresponding topology
The next example presents a convergence which generalizes convergences from Examples 2 and 3.
Example 4. We say that a collection R = of Riesz pseudoseminorms on X is
a Riesz multi-pseudonorm, if for any 0 = X £ X, there is r^ £ R with r^(x) > 0. In this case, (X, R) is said to be a multi-pseudonormed lattice (abbreviated by MPNL).
Notice that, by the Fremlin theorem (cf. [3, Theorem 2.28]), MPNLs are exactly the locally solid vector lattices.
( X, R)
xa —V x (Vr^ £ R) r^(x — xa) V 0, (6)
coincides with r-convergence, where r is the corresponding locally solid topology in (X, R).
Example 5. Given vector lattices X and E, a function p : X v E+ is called an E-valued Riesz seminorm (cf. [4, 9]), whenever:
(a) p(x + y) ^ p(x) + p(y) for all X, y £ X;
(b) p(ax) = |a| ■ p(x) for all x £ X, a £
(c) |y| ^ |x| implies p(y) ^ p(x).
If, additionally, p(x) = 0 for any 0 = x G X we say that p is an E-valued Riesz norm. A vector lattice (X, p, E) equipped with an E-valued Riesz norm p is called a lattice normed lattice (abbreviated by LNL).
Several types of convergences in lattice normed lattices were studied recently in [5, 6, 23]. One of the most interesting convergences here is the P-convergence:
xa -% x ^^ p(x - Xa) 0. (7)
Notice that, the p-convergence in (X, | ■ |,X) coincides with the o-convergence in X which is not topological if dim(X) = to.
Example 6. A vector lattice X = (X, M, E) equipped with a separating family M = {p£of E-va,lued, Riesz seminorms is said to be a lattice multi-normed lattice (abbreviated by LMNL). The corresponding convergence:
xa —% x ^^ (VP£ G M) p^(x - xa) -% 0 (8)
is called the lm-convergence. Clearly, any LNL is an LMNL.
Example 7. Given two vector lattices X and E. A function p : X % E+ is called an
E
(a) p(x + y) ^ p(x) + p(y) for all x, y G X;
(b) p(anx) -% 0 for all x G X and [R 9 an ^ 0;
(c) |y| ^ |x| implies p(y) ^ p(x);
(d) x = 0 implies p(x) = 0.
If condition (d) is dropped, p is said to be an E-va,lued, Riesz pseudoseminorm.
X E p
(X, p, E)
convergence:
xa x ^^ p(x - xa) -% 0 (9)
(X, p, E)
Our last example presents a convergence which generalizes convergences from all previous examples except the ru-convergence from Example 1.
Example 8. A family R = {p^ of E-valued Riesz pseudoseminorms is said to be separating whenever, for any 0 = x G X, there is p^ G R such, that p^(x) > 0. If R is
E
(X, R, E) E R
to be a lattice multi-pseudonormed lattice (abbreviated by LMPNL). The corresponding convergence:
xa —% x ^^ (V p£ G R) p£(x - xa) 0 (10)
is called the lmp-convergence.
3. Unbounded convergences
Various unbounded convergences have been investigated recently in [5, 7, 11-16, 18, 20, 2429, 30-32]. This section is focused on the unification of approaches for unbounded convergences in different settings. After this, we discuss several types of unbounded convergences related to examples in Section 2.
3.1. General facts. Let I be an ideal in a convergence vector lattice (X, c). The following definition is motivated by the definition of unconvergence with respect to an ideal I of a normed lattice (X, || ■ ||) [14].
Definition 1. The unbounded, c-convergence w.r. to I (shortly, UiC-convergenee) is defined by
xa —V X if |xa — x| A u —V x for all u £ I+. (11)
( X, i )
UiC £ Ti c £ T1 and I is order dense.
Furthermore, the Uic-convergence is coarser than c and UiUiC = uic. Thus, if I is order dense
T1 i = i
I=X
The uo-convergence was studied recently in [11, 13, 16, 18, 27, 28, 31]. The UN-convergence was introduced and investigated in [12] (see also [14, 15, 29, 32]). We refer to [5, 6] for the UP-convergence; to [7] for the un-convergence; and to [20, 29, 30, 31, 33] for the ur-convergence.
It may happened that a c-convergence is not topological, yet the uc-convergence is topolo-X
XX dim(X) < to [11, Theorem 1].
The following proposition is a uc-version of [13, Proposition 3.15] (cf. also [5, Proposition 3.11] and [30, Proposition 2.12]). Since its proof is similar, we omit it.
Proposition 1. Let c be a Lebesgue ^-convergence in a vector lattice X and Y a sub-lattice of X. Y is uc-closed iff it is c-closed.
It was shown in [30, Theorem 6.4] that in a Hausdorflt locally solid vector lattice (X, r) the r-convergence minimal iff it is Lebesgue and ur = r. The question, whether or not any T1 =
Two further questions arise in the case of topological uc-convergence (i. e. uc is a r-convergence for some locally solid r in X). Under which conditions the topology r is locally convex?
X
X et CjlletSl- interior point [15, Theorem 3.2];
X ( X, ) X
In the general case, no investigation was conducted yet.
3.2. uo-Convergence and : : -convergence. The uo-convergence was studied deeply in many recent papers (cf. [11-14, 26-28, 31]), whereas the URU-convergence was investigated in [5, 7, 11, 20]. It was proved [20, Proposition 3] that in a Lebesgue and complete metrizable
X for every net xa. In [20, Proposition 4],
it was shown that, in X = [Rn, "—V" is equivalent to "—V" for nets iff 0 is countable. Furthermore, it was proved in [11], that the o-convergence is topological iff dim(X) < to [11,
X
Theorem 5].
3.3. ; -Convergence and ur-convergence. Recently, UM-Convergence was studied in [7], whereas ur-convergence in [7, 29, 30, 33]. Among other things, it was shown that in
(X, M) X
a quasi-
interior point [7, Proposition 4]. In [20, Proposition 5] it was shown that in a complete metrizable locally solid vector lattice (X, r) with a countable topological orthogonal system, the ur-convergence is metrizable.
(X, M)
norm M = {m?}?es, the um-convergence in X is the mp-convergence in the MPNL (X,R), where R = {m?;U}?e2;Uex+ is given by
m?,„(x) = m?(|x| A u) (f G S, u G X+). (12)
In the case of a locally solid vector lattice (X, t), in order to describe the ur-convergence, we consider a Riesz multi-pseudonorm on X, say P = {p?}?eSi generating topology r (such a Riesz multi-pseudonorm exists by the Fremlin theorem). Now, the ur-convergence in X is the mp-convergence in the MPNL (X, R), where R = {p?,u}?es,U€X+ is given by:
p?,u(x) = p?(|x| A u) (f G S, u G X+). (13)
3.4. Unbounded p-, lm-, and —convergences. The up-convergence was introduced
X
lmp-convergence in the LMPNL (X, P, E), where P = {nu}ueX+ is given by
n„(x) = p(|x| A u) (u G X+). (14)
In the case of an LMNL X = (X, M, E) with the E-valued Riesz multi-norm M = {p? }?gs,
X (X, P, E) P =
{n?)u}?e2,uex+ consists of E-valued Riesz pseudoseminorms n?,u defined by
n?,u(x) = p?(|x| A u) (x G X). (15)
Furthermore, in the most general case of the lmp-convergence from Example 8, we have the following proposition, whose straightforward proof is omitted.
Proposition 2. Let X = (X, R, E) be an LMPNL with the E-valued Riesz multi-pseu-donorm R = {p?}?gs- Then the VLf\p-convergence in X is the lmp-convergence in the LMPNL (X, P, E) where P consists of E-valued Riesz pseudoseminorms n?,u
n?,u(x) = p?(|x| A u) (x G X) (16)
for all f G S, u G X+.
For more results on up-convergence we refer to [5, 6, 24, 25].
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Received February 28, 2018
Dabboorasad Yousef Atef Mohammed Islamic University of Gaza, P.O.Box 108, Rimal, Gaza City, Palestine E-mail: yasadQiugaza.edu,ps
Emelyanov Eduard Yu.
Middle East Technical University,
Universiteler Mahallesi, Dumlupinar Bulvari No:l,
Cankaya Ankara 06800, Turkey
Sobolev Institute of Mathematics,
4 Acad. Koptyug avenue, Novosibirsk 630090, Russia
E-mail: emelanovQmath.nsc.ru
Владикавказский математический журнал 2018, Том 20, Выпуск 2, С. 49-56
НЕОГРАНИЧЕННЫЕ СХОДИМОСТИ В КОНВЕРГЕНТНЫХ ВЕКТОРНЫХ РЕШЕТКАХ
Дабурасад Ю., Емельянов Э. Ю.
Исторически, разнообразные сходимости в векторных решетках являлись предметом глубоких исследований восходящих к началу XX века. Изучение неограниченной порядковой сходимости было инициировано Накано в конце 40-х годов, в связи с эргодической теоремой Биркгофа. Идея Накано заключалась в том, чтобы определить сходимость почти всюду в терминах решеточных операций без прямого использования теории меры. Много лет спустя выяснилось, что неограниченная порядковая сходимость весьма полезна в теории вероятностей. С тех пор идея исследования различных сходимостей с помощью их неограниченных версий используется в различных контекстах. Например, неограниченные сходимости в векторных решетках привлекли внимание многих исследователей для того чтобы найти новые подходы к различным проблемам функционального анализа, теории операторов, вариационного исчисления, теории рисков в финансовой математике и т. д. Некоторые неограниченные сходимости, такие как неограниченная сходимость по норме или муль-тинорме, неограниченная т-сходимость, являются топологическими. Другие приведенные сходимости не являются топологическими в общем случае, например: неограниченная порядковая сходимость, неограниченная относительная равномерная сходимость, различные неограниченные сходимости в решеточно-нормированных решетках, и т. п. В настоящей работе представлены последние наиболее часто используемые сходимости в векторных решетках, с акцентом на соответствующих неограниченных сходимостях. Особое внимание уделяется случаю сходимости в решеточно муль-типсевдонормированных векторных решетках, обобщающих большинство случаев, обсуждавшихся в литературе за последние 5 лет.
Ключевые слова: конвергентная векторная решетка, решеточно-нормированное пространство, неограниченная сходимость.
