Unbounded convergence in the convergence vector lattices: a survey

A convergence [s-convergence^ c for nets [resp.. for sequences] in a set X is defined by Ute following two conditions:

• ( a )    x_a = x ^ x_a —fo x [resp.. x n = x ^ x n —fox]:

• ( b )    x_a —fo x ^ x e —fo x for every subnet x e оf x_a [resp.. x n —fo x ^ x nk —fo x for every subsequence x nk c)f x n].

A convergence set is a pair (X, c) where c is a con vergence in a set X. A mapping f from a. convergence set (X1,(71) into a. convergence set (X₂,( d₂) is said to be continuous, if x_a —fo x implies f (x_a) —fo f (x). s-Continuity of f is defined by replacing nets with sequences.

A subset A оf ( X, ( ;) is called: c-dosed if A 3 x_a —^ x ^ x G A: с-compact if any net a_a in A possesses a subnet ae such that ae —> a for some a G A. sc-Closedness and sc- compactness are defined by using sequences. If the set {x} is c-closed for every x G X then c is called Tv convergence. It is iminediaie I,о see that c G T i if every constanl net x_a = 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. Under convergence in a vector lattice X we always understand a convergence c in the set X, which agrees with the linear and lattice operations in the following way:

X 3 xa   / x, X 3 ye A y, R 3 rY A r imply rY * xa + Ув    ' Г • x + У                                  (1)

and rY • xa Л ye —^ r • x Л y.                               (2)

In other words, the linear and lattice operations in X are continuous with respect to the c-convergence in X and I,о the usual convergence in IR. In this case, we say that, X = (X, (::) is a convergence vector lattice, s-Convergence vector lattices are defined by using in (1) and (2) sequences instead of nets.

A net, x_a [rosp.. a sequence xn] in (X,(::) is called a. c-Cauchy. whenever

(x_a — x e ) —^0 [resp.. (x_m — x n ) —^0 (m, n ^ to)].               (3)

A convergence vector lattice (X,c) is said to be с-complete [resp., sc-complete], if every c-Cauchy net [resp., c-Cauchy sequence] in X is c-convergent.

A ^-convergence C1 in a vector lattice X is said to be minimal \s-minimal], if for any other ^-convergeпсе c in X satisfying x_a —^ 0 ^ x_a —^ 0 for al 1 nets x_a in X [resp.. xn —^ 0 ^ xn —^ 0 for all sequences xn in X]. it folk»vs that c = (A-

A convergence c in a. vector lattice X is said I,о be Lebesgue- [resp.. S-Lebesgue], if for every riel, xa [resp.. for er"cry sequence xn] iri X xa —°G 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, r) with its r-convergence [3]: a. space of Lebesgue measurable functions on [0,1] with the almost everywhere convergence, that is a sc-Lebesgue convergence; a vector lattice X with the 0- convergence [ru- convergence] [4]; a lattice normed vector lattice (X,p, E) with the R- 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 т such that the т-convergence coincided with the corresponding c-convergence.

Example 1. Let X be a vector la nice. Clearly. (X, —a) is a Tl-eom-ergenee vector lattice. Furthermore. (X, — -a) is a convergence vector lattice, where ”— a” is Ti iff X is Archimedean WP-4.&9.101).           '                               ..... o

In a Lebesgue and complete metrizable locally solid vector lattice, x_a —a x iff x_a —a x [20, Proposition 3]. It was also shown in [20, Proposition 4] that, in R^Q, “—A” is equivalent l,o ” — a'" for riets iff’ Q is countable. Furthermore, it was proved that I,lie о-convergence in X is topological iff dim(X) <  to [11, Theorem 1], and that the RU-convergence is topological iff X lias a strong order unit [20. Theorem 5]. It is worth I,о notice that the so-eom"ergeriee in a, Banach lattice X of countable type coincides with I,lie norm convergence iff X is lattice isomorphic l,o co [21. Theorem 1].

Example 2. Let M = {m^ }_?G5 be a family of Riesz semi norms on a vector lattice X. If, for any 0 = x E X, there is m^ E M such that m^(x) >  0, ( X, M ) is said to be a multinormed lattice (cf. [10, Definition 5.1.6]), abbreviated by MNL, with the Riesz multi-norm M. Convergence in a Riesz multi-norm (и-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, fmiction r : X A R ₊ is called a, Riesz pseudoseminorm (cf. [3, Definition 2.27]), whenever:

• ( a )    r(x + y) 6 r(x) + r(y) for all x,y E X ;

• ( b )    limn^^ r(a_nx) = 0 for all x E X arid for all R 3 an A 0;

• ( c )    | y | > | x | irrq>lies r(y) > r(x).

If r(x) = 0 for any 0 = x E X. r is called a, Riesz pseudonorm arid ( X, r) is said I,о Ire a pseudonormed lattice (abbreviated by PNL\

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 = {r_€}_{€ e}= of Riesz pseudosominorms on X is a Riesz multi-pseudonorm, if for any 0 = x E X, there is r^ E R with r^ (x) > 0. In this case, ( X, R ) is said l,o l>e a multi-pseudonormed lattice (abbreviated Irv MPNL\

Notice that, by the Fremlin theorem (cf. [3, Theorem 2.28]), MPNLs are exactly the locally solid vector lattices.

The Riesz multi-pseudonorm convergence (w-convergence) in (X,R), xa —A x AA (V r^ E R) r^(x — xa) A 0,                   (6)

coincides with т-convergence. where т is the corresponding кicallv solid topology in (X, R ).

Example 5. Given vector lattices X and E. a, function p : X A E₊ is called an E-mlued Riesz seminorm (cf. [4, 9]), whenever:

• ( a )    p ( x + y ) 6 p ( x ) + p ( y ) for all x, y E X ;

• ( b )    p ( ax ) = | a | • p ( x ) for all x E X. a E R:

• ( c )    | y | > | x | imi:dies 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 la Hico (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 A X о у p(x — Xa)  - 0 .                              (7)

Notice that, the р-convergence in ( X, | • | ,X ) coincides with the o-com-ergence in X winch is not topological if dim( X ) = to.

Example 6. A vector lattice X = ( X, M,E) equipped with a. separating family M = {p^}g_e= оf E-valued Riesz seminorms is said to be a lattice multi-normed lattice (abbreviated by LMNL). The corresponding convergence:

Xa —~— X о у (V p^ G ^M) p^ (x — Xa )  A 0                     (8)

is called the w-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-valued Riesz pseudonorm, whenever:

• ( a )    p ( x + y ) 6 p ( x ) + p ( y ) for all x,y G X ;

• ( b )    p(a_n x ) —A 0 for all x G X am f R 3 an ^ 0;

• ( c )    | y | > | x | i^mfdies p ( y ) >  p ( x );

• ( d )    x = 0 imp lies p ( x ) = 0.

If condition (d) is dropped, p is said to be an E-valued Riesz pseudoseminorm.

A vector lattice X equipped with an E-valuod Riesz pseudonorm p is called a, lattice pseudonormed lattice (abbreviated by LPNL and denoted by (X, p, E)). The corresponding convergence:

Xa    X о у p(x — Xa )  A 0                            (9)

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^ }_{?G 5} оf E-valued Riesz psoudosominorms is said to Ire separating whenever, for any 0 = x G X, there is p^ G R such that p^ (x) > 0. If R is separating, we call it an E-valued Riesz multi-pseudonorm.

Xa ' A X о у (V p^ G R ) p^ (x — X a)    0                     (10)

is called the L№-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 un-convergence with respect to an ideal I of a normed lattice (X, k • k) [14].

DEFINITION 1. The unbounded c-convergence w.r. to I (shortly, DjC-convergence) is defined by xa —to x if' |xa — x| Л u —to x for all u E I+. (11)

It follows directly from (1) and (2). that (X, (JI() is a convergence vector lattice, where иIc: E T1 <> c E T1 aiid I is order dense.

Furthermore. the и Ic-coiivergeiice is coarser than c and uIuIc: = tJIc. Th us. if I is order dense and c is Ti and miiiinial. then U/C = c. If c is topeLogical. then U/C is topological as well (cf. [20, 30]). Unbounded ©-convergence w.r. to I = X is denoted by uc.

The ©©-convergence was studied recently in [11, 13, 16, 18, 27, 28, 31]. The ш-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 им-convergence; and to [20, 29, 30, 31, 33] for the и т-convergence.

It may happened that a c-convergence is not topological, yet the ©c-convergence is topological. For example, if X is an atomic order continuous Banach lattice, then the mo-convergence in X is topological [12, Theorem 5.3], whereas the ©-convergence in X is not topological except dim(X) <  to [11. Theorem 1].

The following proposition is a ©©-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 T1-convergence in a vector lattice X and Y a sublattice of X. Y is uc-closed iff it is c-closed.

It was shown in [30, Theorem 6.4] that in a Hausdorff locally solid vector lattice (X, т) the т-convergence minimal iff it is Lebesgue and ©т = т. The question, whether or not any T i-com'ergence in a vector lattice is miriimal iff it is Lebesgue and uc = c. remains open.

Two further questions arise in the case of topological ©©-convergence (i. e. uc is a т-convergence for some locally solid т in X). Under which conditions the topology т is locally convex? Metrizable? In the case of ©-convergence (norm convergence) in a Banach lattice X, it was proved that: (1) ©©-topology is metrizable iff X has a quasi-interior point [15, Theorem 3.2]; (2) if X is order continuous, then (X, и и) is locally convex iff X is atomic [15, Theorem 5.2]. In the general case, no investigation was conducted yet.

• 3.2.    ©©-Convergence and ©R©-convergence. The ©©-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 locally solid vector lattice X, x_a —to x toto x_a —to x for every net x_a. In [20, Proposition 4], j w» *™ lh,L In X = Rⁿ -to" fe «цт* .... to" for n_HS iff ^fi is сипЫ,1е. Furthermore, it was proved in [11], that the ©-convergence is topological iff dim(X) <  to [11, Theorem 1], and that the RU-convergence is topological iff X has a strong order unit [11, Theorem 5].

• 3.3.    ©©-Convergence and и т-convergence. Recently, ©©-Convergence was studied in [7], whereas и т-convergence in [7, 29, 30, 33]. Among other things, it was shown that in a metrizable м-complete MNL (X, M ) the им-convergence is metrizable iff X has a quasiinterior point [7, Proposition 4]. In [20, Proposition 5] it was shown that in a complete metrizable locally solid vector lattice (X, т) with a countable topological orthogonal system, the и т-convergence is metrizable.

• 3.4.    Unbounded P-, lm-, and LMF-convergences. The up-convergence was introduced and investigated in [5]. As in (12) above, it can be seen that the up-convergence in X is the LMP-convergence in the LMPNL (X, P , E), where P = {n_u}_uGx+ is given by

Notice that, in the case of the м-convergence in an MNL (X, M) with the Riesz multinorm M = {m^}^G=. the им-convergence in X is the MP-convergenee in the MPNL (X,R). where R = {m^u}^Es,uEX+ ^ given by m^,u(x) = my(|x| A u) (£ G S, u 6 X+).                        (12)

In Hie ease of a locally solid vector lattice ( X,t ). in order to we consider a Riesz multi-pseudonorm on X, say P = {p^}^_G=, generating topology т (such a Riesz multi-pseudonorm exists by the Fremlin theorem). Now, the vт-convergence in X is the MF-convergence in the MPNL (X, R), where R = {p^,_u}^_Gs,u_Gx₊ is given by:

P^,u(x) = P^(|x| A u) (£ G S, u G X+).(13)

nu(x) = p(|x| A u) (u G X+).(14)

In Idic case of an LMNL X = (X, M ,E) with the E-valucd Riesz multi-norm M = {p ^ }_?G5. the ULM-convergence in X is the LMP-com"ergenee in the LMPNL (X, P , E). where P = ^{ny,u}yG=,uGX+ consists of E-valued Riesz psoudoseminorms n^ defined by

• ngu(x) = pg(|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-pseudonorm R = {p^}^_G=. Then the ulmp -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 £ G S,u G X₊.

For more results on up-convergence we refer to [5, 6, 24, 25].