Unbounded p τ -Convergence in Lattice-Normed Locally Solid Riesz Spaces

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1 Unbounded -Convergence in Lattice-Normed Locally Solid Riesz Spaces July 6, 2018 arxiv: v3 [math.fa] 9 Jan 2018 Abdullah AYDIN Department of Mathematics, Muş Alparslan University, Muş, Turkey. Abstract Let x α ) be a net in a lattice-normed locally solid Riesz space X,p,E τ ). We say that x α ) is unbounded -convergent to x X if p x α x u) τ 0 for every u X +. This convergence has been studied recently for lattice-normed vector lattices under the name up-convergence in [3, 4, 5], under the name of uo-convergence in [13], under the name of un-convergence in [8, 15], under the name uaw-convergence in [20]. In this paper, we study general properties of the unbounded -convergence. 1 Introduction Locally solid vector lattices and lattice-valued norms on vector lattices provide a natural and efficient tools in the theory of vector lattices. We refer the reader for detail information about the theory of locally solid vector lattices and lattice-normed vector lattices; see example [1, 2, 6, 9, 14, 16, 17, 18]. In this paper, aim is to illustrate usefulness of latticevalued norms for investigation of different types of unbounded p-convergences in latticenormed vector lattice; see [3, 4, 5] and different types of unbounded convergences in vector lattices, which attracted the attention of several authors in series of recent papers; see [8, 11, 12, 13, 15, 20]. Nakano introduced uo-convergence in [17] under the name individual convergence, and Troitsky introduced un-convergence in [19] under the name d-convergence. The unbounded p-convergence was introduced in [4]. We refer the reader for an exposition on uo-convergence to [12, 13], on un-convergence to [8, 15], and up-convergence to [3, 4, 5]. For applications of uo-convergence, we refer to [12, 13]. Recall that a net x α ) α A in a vector lattice X is order convergent to x X if there exists another net y β ) β B satisfying y β 0, and for any β B, there exists α β A such o that x α x y β for all α α β. In this case, we write x α x. In a vector lattice X, a net x α ) is unbounded order convergent to x X if x α x u 0 o for every u X + ; see uo example [8, 13, 15]. In this case, we write x α x. The uo-convergence is an abstraction of a.e.-convergence in L p -spaces for 1 p <, [12, 13]. In a normed lattice X, ), a Keywords: -convergence, u -convergence, vector lattice, lattice-normed space, locally solid vector lattice, lattice-normed locally solid vector lattice, up-convergence, uo-convergence, un-convergence, 2010 AMS Mathematics Subject Classification: Primary 46A40; Secondary 46E30. a.aydin@alparslan.edu.tr 1

2 un net x α ) is unbounded norm convergent to x X, written as x α x, if x α x u 0 for every u X + ; see [8]. Clearly, if the norm is order continuous then uo-convergence implies un-convergence. For a finite measure µ, un-convergence of sequences in L p µ), 1 p <, is equivalent to convergence in measure, see [8, 19]. Zabeti introduced the following notion; a net x α ) in a Banach lattice X is said to be unbounded absolute weak convergent or uaw-convergent, for short) to x X if, for each u X +, x α x u 0 weakly; see [20]. Throughout the paper, all vector lattices are assumed to be real and Archimedean. Let X be a vector space, E be a vector lattice, and p : X E + be a vector norm i.e. px) = 0 x = 0, pλx) = λ px) for all λ R, x X, and px+y) px)+py) for all x,y X) then the triple X,p,E) is called a lattice-normed space, abbreviated as LNS; see example [14]. The lattice norm p in an LNS X,p,E) is said to be decomposable if, for all x X and e 1,e 2 E +, it follows from px) = e 1 +e 2 that there exist x 1,x 2 X such that x = x 1 +x 2 and px k ) = e k for k = 1,2. If X is a vector lattice and the vector norm p is monotone i.e. x y px) py)) then the triple X,p,E) is called a lattice-normed vector lattice, abbreviated as LNVL; see [4]. We abbreviate the convergence px α x) 0 o p as x α x, and say in this case that xα ) p-converges to x. A net x α ) α A in an LNS X,p,E) is said to be p-cauchy if the net x α x α ) α,α ) A A p-converges to 0. An LNS X,p,E) is called sequentially) p-complete if every p-cauchy sequence) net in X is p-convergent. In an LNS X,p,E), a subset A of X is called p-bounded if there exists e E such that pa) e for all a A. An LNVL o X,p,E) is called op-continuous if x α 0 implies pxα ) 0. o A net x α ) in an LNVL X,p,E) is said to be unbounded p-convergent to x X shortly, x α ) up-converges to x up or x α x), if p x α x u) 0 o for all u X + ; see [4, Def.6]. Let X,p,E) be an LNS and E, E ) be a normed lattice. The mixed norm on X is defined by p- x E = px) E for all x X. In this case, the normed space X,p- E ) is called a mixed-normed space see, for example [14, 7.1.1, p.292]). We refer the reader for more information on LNSs to [14]. In this paper, unless otherwise stated, we do not assume lattice norms to be decomposable. Let E be a vector lattice and τ be a linear topology on E that has a base at zero consisting of solid sets. Then the pair E,τ) is called locally solid vector lattice. It should be noted that all topologies considered throughout this article are assumed to be Hausdorff. It follows from [1, Thm. 2.28] that a linear topology E,τ) with vector lattice E is locally solid vector lattice iff it is generated by a family of Riesz pseudonorms {ρ j } j J. Moreover, if a family of Riesz pseudonorms generates a locally solid topology τ τ on a vector lattice E then x α x iff ρj x α x) 0 in R for each j J. In this paper, unless otherwise, the pair E,τ) refers to a locally solid vector lattice with a family of Riesz pseudonorms {ρ j } j J that generates the topology τ. We shall keep in mind the following facts. Lemma 1.1. Let x α ) and y α ) be two nets in a locally solid vector lattice E,τ). If y α x α for all α, and x α τ 0 in E then yα τ 0 in E. Proposition 1.2. Let E,τ) be a locally solid vector lattice. If a subset A of E is τ- bounded then ρ j A) is bounded in R for any j J; see [6, Prop. 1]. Let X,p,E) be an LNVL with E,τ) is a locally solid vector lattice then we call X,p,E τ ) as lattice-normed locally solid Riesz space or, lattice-normed locally solid vector lattice), abbreviated as LNLS. Dealing with LNLSs, we shall keep in mind also the following examples. Example 1.1. Let E,τ) be a locally solid vector lattice. Then E,,E τ ) is an LNLS. Example 1.2. Let X, X ) be a normed vector lattice. Then X, X, R ) is an LNLS with usual -topology and usual ordering on R. 2

3 By using the Example 4 of [4], we give the following result. Example 1.3. Let X = X, X ) be a normed vector lattice. Take the closed unit ball B X of the dual Banach lattice X. Let E = l B X ) be the vector lattice of all bounded real-valued functions on B X. Define an E-valued norm p on X by px)[f] := f x ) f B X ) for any x X. The Hahn-Banach theorem ensures that px) = 0 iff x = 0. All other properties of lattice norm are obvious for p. Consider the topology τ on E that is generated by the norm of bounded real-valued functions on X. Therefore, X,p,E τ ) is an LNLS. Notice also that the lattice norm p takes values in the space CB X ) of all continuous functions on the w -compact ball B X of X. Hence, instead of X,p,l B X )), one may also consider the LNVL X,p, CB X )). Therefore, X,p,CB X ) τ ) is also an LNLS. By considering the Example 2.4 of [16], we can get the following work. Example 1.4. Let E be the space of all Lebesgue measurable functions on R with the usual pointwise ordering, i.e., for f,g E, we define f g iff ft) gt) for every t R. Consider the map : E R defined by f = f 2 t)dt )1 2, where f E. Then the norm is a seminorm on E. It is easy to see that it is also a Riesz seminorm. Thus, the topology τ that is generated by is locally convex-solid; see [1, Thm.2.25]. Now, consider a vector lattice X and a map p : X E defined by px) = px)[f] = f x ) then X,p,E τ ) is an LNLS. The structure of this paper is as follows. In section 2, we give the definitions of -convergence, -closed set, -complete and o -continuous LNLSs. We show that - convergence coincides with un-convergence, uaw-convergence, and um-convergence; see Remark 2.1, Remark 2.3 and Remark 2.4, respectively. In Theorem 2.2, we see that the squeeze property holds for -convergence, and the lattice operations are o -continuous; see Proposition 2.3. Also, we see that -convergence implies order convergence and the Lebesgue property implies o -continuity; see Theorem 2.4 and Proposition 2.8, respectively. We have that o -continuity with -completeness implies order completeness, see Corollary In the Theorem 2.11, we have a technical result about o -continuity. In section 3, we introduce the notion of u -convergence, and give basic properties of u -convergence; see Proposition 3.3, Proposition 3.2 and Lemma 3.4. We see that a monotone u -convergence implies -convergence; see Theorem 3.6. We have that a -convergence net is -bounded; see Theorem 3.9. In the last section, We study u -convergence on sublattices. We see that u - convergence in sublattice implies u -convergence in the LNLS; see Theorem 4.1, and u -convergence in a vector lattice implies convergence in its order completion; see Corollary 4.2. We have that u -convergence of a net in an LNLS implies u -convergence in a projection band with the corresponding band projection; see Proposition 4.3. We give a technical result about order completion; see Proposition 4.4. Also, we see that u -closed coincides with -closed; see Theorem 4.5, and u -convergence with p-almost order boundedness implies -convergence; see Theorem Convergence with Topology in Locally Solid Vector Lattices Most of the notions and results in this section are direct analogs of well-known facts of the theory of normed lattices and locally solid vector lattices. We include them for the convenience of the reader. Recall that a net x α ) in an LNS X,p,E) is said p-convergence to x X if px α x) o 0 in E. We introduce the following notion on LNLS, which is motivated by the p-convergence. 3

4 Definition 2.1. Let X,p,E τ ) be an LNLS. A net x α ) in X is called -convergent to x X if px α x) τ 0 in E. We write x α x, and say that xα ) -converges to x. In the following work, we give some basic properties of - convergence. Lemma 2.1. Let x α ) and y β ) be two nets in an LNLS X,p,E τ ). Then the followings hold; i) x α x iff xα x) 0, pτ ii) if x α x then yβ x for each subnet yβ ) of x α ), iii) suppose x α x and yβ y, then axα +by β ax+by for any a,b R, iv) if x α x and xα y then x = y, v) if x α x then xα x. pτ Proof. The i) and ii) are obvious. Also, iii) follows from the Lemma 1.1. Observe the inequality px y) px x α )+px α y). Then, by Lemma 1.1, clearly iv) holds. Lastly, we prove v). By x α x x α x, we have p x α x ) px α x). Therefore, by Lemma 1.1, we get the result. -convergence coincides with some kinds of convergence, and we give them in the following remarks. As it was observed in [8], the un-convergence is topological. For every ε > 0 and non-zero u X +, put V ε,u = {x X : x u < ε}. The collection of all such sets of this form is a base of zero neighborhoods for a topology, and the convergence in this topology agrees with un-convergence. This topology is called as un-topology, see [15, p.3]. It can be seen that un-topology is locally solid. Remark 2.1. Assume E is a Banach lattice and the topology τ is un-topology on E. Consider an LNLS X,p,E τ ). For any net x α ) in X, 1) x α x in X iff pxα x) 0 un in E, 2) if E has a strong unit then x α x in X iff pxα x) 0 in E; see [15, Prop. 2.3], 3) if E has quasi-interior point then x α x in X iff pxα x) d 0 in E, where d is a metric on E which give the un-topology; see [15, Prop. 3.2]. In the following, we give another relation between -convergence and un-convergence. Remark 2.2. Let X,p,E τ ) be an LNLS with E, ) is normed vector lattice and τ is generated by a family of Riesz pseudonorms p u : E R +, for each u E + defined by p u x) = x u. Thus, for a net xα ) and x X, we have x α x iff pxα x) un 0; see [10, Thm. 2.1]. For each positive vector u in a Banach lattice E and ε > 0, and for each f E +, put V u,ε,f = {x E : f x u) < ε}. Let N be the collection of all sets of this form. Then N is a base of neighborhoods of zero for some Hausdorff linear topology; see [20, p.3-4]. Then E,τ) is locally solid vector lattice. Remark 2.3. Let X,p,E τ ) be an LNLS with τ as above topology on the Banach lattice E. Then a net x α x in X iff pxα x) uaw in E. 4

5 Let M = {m λ } λ Λ be a separating family of lattice semi-norms on a vector lattice E. A net x α ) in E is um-convergent to x if m λ x α x) u 0 for all λ Λ and u E +. Recall that N 0, the collection of all sets of the form V ε,u,λ = {x E : m λ x u) < ε} ε > 0,0 u E +,λ Λ), form a neighborhood base at zero for some Hausdorff locally solid topology in E which is called as um-topology; see [7, p.4]. Thus we can give the following result. Remark 2.4. Let X,p,E τ ) be an LNLS with E,τ) is um-topology. Then, for a net x α ) and x X, we have x α x iff pxα x) um 0 in E. Theorem 2.2. Let X,p,E τ ) be an LNLS and x α ),y α ) and z α ) be three nets in X such that x α y α z α for all α. If x α x and zα x for x X then yα x. Proof. Consider the family of Riesz pseudonorms {ρ j } j J that generates the topology τ. Since x α y α z α for all α, we have x α x y α x z α x px α x) py α x) pz α x) As x α x and zα x for x X, we have ρ j pxα x) ) ρ j pyα x) ) ρ j pzα x) ) j J). ρ j pxα x) ) 0 and ρ j pzα x) ) 0 j J). Thus, by squeeze property in R, we have ρ j pyα x) ) 0 for all j J. Therefore, we get y α x; see [1, Thm. 2.28]. In the following result, we see that the lattice operations are o -continuous. Proposition 2.3. Let x α ) α A and y β ) β B be two nets in an LNLS X,p,E τ ). If x α x and yβ y then xα y β ) α,β) A B x y. In particular, xα x implies that x α x. Proof. Let s considerthefamily ofrieszpseudonorms{ρ j } j J thatgenerates thetopology τ. Since x α x and yβ y, we have pxα x) 0 τ and py β y) 0 τ in E, or ρ j pxα x) ) 0, and also ρ j pyβ y) ) 0 in R for all j J. Then by the formula [2, Thm.1.92)], we have px α y β x y) p x α y β x α y )+p x α y x y ) p y β y )+p x α x ). Thus, wehave ρ j pxα y β x y) ) ρ j pyβ y) ) +ρ j pxα x) ) for all j J. Hence, we get ρ j pxα y β x y) ) 0 in R for allj J. Therefore, x α y β ) α,β) A B x y in X. Similarly, we have x α y β x y. In particular, xα x if and only if xα x. pτ Definition 2.2. Let X,p,E τ ) be an LNLS and A X be a subset. Then A is called -closed set in X if, for any net a α ) in A that is -convergent to a X, it holds a A. It is clear that the positive cone X + of an LNLS X is -closed. Indeed, assume x α ) is a net in X + such that it -converges x X. By the Proposition 2.3, we have x α = x + α x +, and so we get x = x +. Therefore, x X +. Remark 2.5. A band in an LNLS X,p,E τ ) is -closed. Indeed, given a band B in X and a net x α ) in B such that x α x for x X. By Proposition 2.3, we have x α b x pτ b for any b B. Thus, x b = 0 as x α b = 0. Therefore, we get x B. 5

6 Theorem 2.4. Any monotone -convergent net in an LNLS X,p,E τ ) order converges to its -limits. Proof. It is enough to show that if a net x α ) is increasing and -convergent to x X then x α x. Fix arbitrary index α. Then x β x α X + for β α. Since X + is -closed, we have x β x α β x x α X + ; see Lemma 2.1iii), and so we get x x α. Since α is arbitrary, x is an upper bound of x α. By -closeness of X +, if y x α for all α then we have y x α p y x X+. Therefore, y x and so x α x. Question 2.5. Is it true that order convergence implies -convergence? We continue with several basic notions in LNLSs, which are motivated by their analogues vector lattice theory. Definition 2.3. Let X = X,p,E τ ) be an LNLS. Then i) a net x α ) α A in X is said to be -Cauchy if the net x α x α ) α,α ) A A - converges to 0, ii) X is called -complete if every -Cauchy net in X is -convergent, iii) X is called o -continuous if x α o 0 implies pxα ) τ 0. Remark 2.6. A -closed sublattice in an o -continuous LNLS is order closed. Indeed, o suppose that Y is -closed in X, y α ) is a net in Y and x X such that y α x. Since X is o -continuous, we have y α x. Thus, since Y is pτ -closed, we get x Y. Recall that a locally solid vector lattice X,τ) is said to have the Lebesgue property if x α 0 in X implies x α τ 0; or equivalently xα o 0 implies xα τ 0; and X,τ) is said to have σ-lebesgue property if x n 0 in X implies x n τ 0. It is clear that the LNLS X,,X τ ), where X,τ) has the Lebesgue property, is o -continuous. The following result gives us a connection between -convergence and p-convergence. Proposition 2.6. Let E,τ) be locally solid vector lattice with the Lebesgue property. Then, for a net x α ) in an LNLS X,p,E τ ) and x X, x α p x implies xα x. Proof. Assume x α p x in X. Then we have pxα x) o 0, and so px α x) τ 0 since E,τ) has the Lebesgue property. Therefore, we get the result. Question 2.7. Is it true that -convergence implies p-convergence? Proposition 2.8. Let X,p,E τ ) be an LNLS, where E,τ) has the Lebesgue property. If, for any net x α ) in X, x α 0 implies px α ) 0 then X is o -continuous Proof. Suppose x α o 0 in X. Then there exists a net zβ 0 in X such that, for any β there is an index α β so that x α z β for all α α β. Hence px α ) pz β ) for all α α β. Since z β 0, by assumption, we have pz β ) 0, so px α ) o 0. By the Lebesgue property of E,τ), we get px α ) τ 0. Therefore, X is o -continuous. Proposition 2.9. For an o -continuous LNLS X,p,E τ ), if 0 x α x holds in X then x α ) is -Cauchy net in X. Proof. Let 0 x α x in X. Then there exists a net y β ) in X such that y β x α ) α,β 0; see [2, Lem.4.8]. Thus, by o -continuity, we get py β x α ) τ 0, and so we have px α x α ) α,α ) AXA) px α y β )+py β x α ) τ 0. Therefore, we get px α x α ) α,α ) AXA) 0, so the net x α ) is -Cauchy. τ 6

7 Corollary Let X,p,E τ ) be an o -continuous LNLS. If X is -complete then it is order complete. Proof. Assume 0 x α u. Then, by Proposition 2.9, x α ) is -Cauchy net. Since X is -complete, there is x X such that x α x. It follows from Theorem 2.4 that xα x, and so X is order complete. For a partial converse of Proposition 2.9, we have the following result. Theorem Let X,p,E τ ) be a -complete LNLS. If 0 x α x in X implies that x α ) α A is -Cauchy net in X then X,p,E τ ) is o -continuous. Proof. Assume that x α o 0 in X. Then there is another net yβ ) β B satisfying y β 0, and for any β B, there exists an index α β A such that x α y β for all α α β. Thus, by assumption, we can say that y β ) is -Cauchy in X. By -completeness of X, there is y X satisfying py β y) τ 0 as α. Since y β y, we get y = 0; see Theorem 2.4, so py β ) τ 0. Therefore, we get px α ) τ 0, and so X is o -continuous. 3 Unbounded -Convergence The u -convergence in LNLSs generalizes the up-convergence in lattice-normed vector lattices [4], uo-convergence in vector lattices [13, 11, 12], the un-convergence [8] and the uaw-convergence in Banach lattices [20]. In this section, we study basic properties of the u -convergence and characterize the u -convergence in certain LNLSs. For a locally solid vector lattice X,τ), a net x α ) in X is called unbounded τ-convergent to x X if, for any u X +, x α x u uτ uτ 0. This is written as x α x and say x α ) uτ-converges τ uτ to x. Obviously, x α x implies xα x. The converse hold for order bounded nets; see [6, 18]. Definition 3.1. LetX,p,E τ )beanlnls.thenanetx α )inx + issaidtobeunbounded -convergent to x shortly, x α ) u -converges to x or x α u x), if for all u X +. p x α x u) τ 0 It can be seen that under the conditions of Lemma 1.1, as x α x u x α x for all u X + and for all α, -convergence implies u -convergence. The following result is an u -version of [13, Cor.3.6]. Lemma 3.1. A disjoint sequence x n ) in a sequentially o -continuous LNLS X,p,E τ ) is sequentially u -convergent to 0. uo Proof. Assume x n ) is a disjoint sequence. Then we have x n 0 in X; see [13, Cor. 3.6]. It means x n u 0 o for all u X +. Thus, p x n u) 0 τ as X,p,E τ ) is sequentially u o -continuous. Therefore, we get x n 0. Similarly to Proposition 2.3 and Proposition 2.6, we give respectively following two results without their proof. Proposition 3.2. Let x α ) α A and y β ) β B be two nets in an LNLS X,p,E τ ). If u u u x α x and yβ y then xα y β ) α,β) A B x y. Proposition 3.3. For a net x α ) in an LNLS X,p,E τ ), where E,τ) is a locally solid up vector lattice with the Lebesgue property. Then, for x X, we have x α x implies u x α x. 7

8 Remark ) Let X,τ) be a locally solid vector lattice and x α ) be a net in X. u uτ Consider the LNLS space X,,X τ ). Then, for x X, x α x in X iff xα x in X. 2) Let X, X ) be a normed vector lattice and x α ) be a net in X. Consider the u un LNLS X, X,R ) with usual -topology on R. Then x α x in X iff xα x in X. The next result is u -version of Lemma 2.1. u u Lemma 3.4. Let x α x and yα y in an LNLS X,p,Eτ ). Then we have u i) x α x iff xα x) 0, upτ u ii) ax α +by α ax+by for any a,b R, u iii) x αβ x for any subnet xαβ ) of x α ), iv) x α upτ x. One can define u -closed subset as following; let X,p,E τ ) be an LNLS and Y be a sublattice of X. Then Y is called u -closed in X if, for any net y α ) in Y that is u -convergent to x X, we have x Y. Remark 3.2. i) Every band is u -closed. ii) A u -closed sublattice in an o -continuous LNLS is uo-closed. Then similar to Theorem 2.4, the next result can be given. Proposition 3.5. Let X,p,E τ ) be an LNLS. Any monotone and u -convergence net is order convergent its u -limit. It is known that -convergence implies u -convergence, but for converse, we generalize -version of [15, Lem.1.2ii)] in the following. Theorem 3.6. Assume x α ) is a monotone net in an LNLS X,p,E τ ) and x α ) is u - convergent to x X. Then x α x. Proof. We may assume that, without loss of generality, x α is increasing and 0 x α for all α. From Proposition 3.5, it follows that 0 x α x for some x X since x α is u -convergent. So, 0 x x α x for all α. For each u X +, we know that p x x α ) u ) τ 0. In particular, for u = x we have x xα ) x = x x α, and so we obtain that px x α ) = p x α x) x ) τ 0. Therefore, x α x. Recall that a subset A of a topological vector space E,τ) is called topological bounded or simply τ-bounded if for every τ-neighborhood V of zero there exists some λ > 0 such that λa V. Also, a subset Y of an LNVL X,p,E) is called p-bounded if there exists e E such that py) e for all y Y. Definition 3.2. Let X,p,E τ ) be an LNLS. A subset Y of X is called -bounded if py) is τ-bounded in E. Lemma 3.7. Let X,p,E τ ) be an LNLS. If a net x α ) is p-bounded then it is -bounded. Proof. Suppose that x α ) is p-bounded. Then there is e E + such that px α ) e for all α. Thus px α ) is order bounded in E. Hence, by [1, Thm.2.19i)], px α ) is τ-bounded in E. Therefore, x α is -bounded in X. For the converse of Lemma 3.7, we give the next result. Proposition 3.8. Let X,p,E τ ) be an LNLS, where E,τ) has an order bounded τ- neighborhood of zero. Then if a net x α ) is -bounded in X then it is p-bounded. 8

9 Proof. Since the net x α ) is -bounded, px α ) is τ-bounded net in E. By [16, Thm.2.2], px α ) is also order bounded in E. Therefore, x α is p-bounded in X. N sol will stand for a base at zero consisting of solid sets, and it satisfies the properties balanced, absorbing and there exists some W N with W + W V in a locally solid topology. We give the following result. Theorem 3.9. Let x α be a net in an LNLS X,p,E τ ). If a net x α ) α A is -convergent in X then it is -bounded. Proof. Assume x α x in X. Thus, pxα x) 0 τ in E. Let U be an arbitrary τ- neighborhood of zero. Choose V,W N sol such that W + W V U. Since W is absorbing, there exists a λ > 0 such that λpx) W. We can take λ 1 since W is solid. Since W is balanced, px α x) W implies λpx α x) W. Thus,by the λpx α ) λpx α x)+λpx) W +W V U, we have λpx α ) U. Therefore, px α ) is τ-bounded in E and so x α is -bounded in X. It is clear that a -convergent net in an LNLS X,p,E τ ) is -Cauchy. 4 Convergence in Sublattices The u -convergence passes obviously to sublattices of vector lattices as it was remarked in [8, p.3], in opposite to uo-convergence [13, Thm.3.2], the un-convergence does not pass even from regular any sublattices. Let Y be a sublattice of an LNLS X,p,E τ ) and u y α ) α A be a net in Y. Then we can define u -convergence in Y as following; y α 0 iff p y α u) 0 τ for all u Y +. It is clear that if a net in sublattice Y of an LNLS X,p,E τ ) is u -convergences to zero in X then it is also u -convergences to zero in Y. For the converse, we give the following work that is a -version of [15, Thm.4.3] and [4, Thm.4]. Theorem 4.1. Let Y be a sublattice of an LNLS X,p,E τ ) and y α ) be a net in Y such u u that y α 0 in Y. Thus, we have yα 0 in X if each of the following cases hold; i) Y is majorizing in X, ii) for any x X and for any 0 u px) there is y Y such that px y) u, iii) Y is a projection band in X, u Proof. Assume y α ) Y is a net such that y α 0 in Y. Take any vector 0 x X+. i) Since Y is majorizing in X, there exists y Y such that x y. It follows from 0 y α x y α y pτ 0, u thus, by Lemma 1.1, we have y α 0 in X. ii) Chooseanarbitrary 0 u px). Thenthereexists y Y such that px y) u. Since y α x y α x y + y α y, we have p y α x) p y α x y )+p y α y ) p x y )+p y α y ) u+p y α y ) Since 0 u px) is arbitrary and p y α y ) 0 τ then, by Lemma 1.1, we get y α x 0. pτ u Hence y α 0 in X. 9

10 iii) Suppose that Y is projection band in X. Thus, X = Y Y. Hence x = x 1 +x 2 with x 1 Y and x 2 Y. Since y α x 2 = 0, we have py α x) = py α x 1 +x 2 )) py α x 1 )+py α x 2 ) = py α x 1 ) τ 0 u Hence, it follows by Lemma 1.1 y α 0 in X. Recall that every Archimedean) vector lattice X is majorizing in its order completion X; see [2, p.101]. Thus, the following result arises. u Corollary 4.2. If X,p,E τ ) is an LNLS and x α ) be a net in X such that x α 0 in u X. Then x α 0 in the X,p,Eτ ). In the following work, we have a -version of [12, Lem.3.3] and [4, Lem.8]. Proposition 4.3. Let X,p,E τ ) be an LNLS, B be a projection band of X, and P B be u the corresponding band projection. If x α x in X then PB x α ) P upτ B x) in both X and B. Proof. It is known that P B is a lattice homomorphism and 0 P B I. Since P B x α ) P B x) = P B x α x x α x, we have p P B x α ) P B x) u ) p x α x u ) τ 0. Thus, by Lemma 1.1, it follows that P B x α ) upτ P B x) in both X and B. The following result is similar to [4, Lem.9]. Proposition 4.4. Let X δ,p,e τ ) be an LNLS, where X δ is the order completion of a vector lattice X, and Y be a sublattice of X. For any net y α ) in sublattice Y δ u, if y α 0 in Y δ u implies y α 0 in X, δ u u then, for any net y β ) in Y, y β 0 in Y implies yβ 0 in X. u Proof. Take a net y β ) in Y such that y β 0. Then, by Corollary 4.2, we obtain u y β 0 in Y δ. Then, by assumption, the net y β ) is u -convergent to 0 in X δ, and so in X. We work towards a version of [13, Prop.3.15]. Theorem 4.5. Let X,p,E τ ) be an o -continuous LNLS and Y be a sublattice of X. Then Y is u -closed in X iff it is -closed in X. Proof. Suppose that Y is u -closed in X and y α ) be a net in Y and x X such that u y α x. Thus, yα x, and so x Y. Conversely, suppose that Y is -closed in X. Let y α ) be a net in Y and x X such u that y α x in X. Then, by Proposition 3.2, without loss of generality, we assume yα in Y + for each α and x X +. Note that, for every z X +, y α z x z y α x z cf. the inequality 1) in the proof of [13, Prop.3.15]). Hence, py α z x z) p y α x z) 0. τ Thus, we get y α z x pτ z for every z X +. In particular, y α y x pτ y for any y Y +. Since Y is -closed, x y Y for any y Y +. For any 0 w Y and for any α, we have y α w = 0 then we have x w = y α w x w y α x w pτ 0. Therefore, x w = 0, and hence x Y. Since Y is the band generated by Y in X, there is a net z β ) in the ideal I Y generated by Y such that 0 z β x in X. Choose an element t β Y with z β t β, for each β. Then x t β x z β x = z β x in X, and so t β x x o in X. Since X is o -continuous, t β x x. pτ So, as t β x Y and Y is -closed, we get x Y. 10

11 Remark 4.1. If ρ is a Riesz pseudonorm on a vector lattice X and x X then ρ 1 n x) ρx) for all n N; see [6, p.2]. 1 n Recall that a subset A of an LNVL X,p,E) is called a p-almost order bounded if, for any w E +, there is x w X + such that p x x w ) + ) = p x x w x ) w for any x A. In the following, we see that u -convergence implies -convergence. That is a p-version of [8, Lem.2.9], and it is also similar to [12, Prop.3.7] and [4, Prop.9]. u Theorem 4.6. Let X,p,E τ ) be an LNLS. If x α ) is p-almost order bounded and x α x then x α x. Proof. Suppose that x α ) is p-almost order bounded. Then the net x α x ) α is also p-almost order bounded. So, for a arbitrary w E + and n N, there exists x w X + with p x α x x α x x w ) = p x α x x w ) +) 1 n w u Since x α x, we have p xα x x w ) 0 τ in E. Thus, for the family of Riesz pseudonorms {ρ j } j J that generates the topology τ, we have ρ j p xα x x w ) 0 for all j J. Moreover, for any α, px α x) = p x α x ) Hence, by Remark 4.1, we get p x α x x α x x w )+p x α x x w ) 1 n w +p x α x x w ) ρ j pxα x) ) ρ j 1 n w +p x α x x w ) ) j) ρ j 1 n w)+ρ j p xα x x w ) ) ρ j 1 n w) 1 n ρ jw) 0. Therefore, ρ j pxα x) ) 0 and so x α x. The following is -version of [4, Lem.7], we omit its proof. Lemma 4.7. Let X,p,E τ ) be an LNLS. If x α x and xα ) is an o-cauchy net then o uo x α x. Moreover, if xα x and xα ) is uo-cauchy then x α x. The following is a -version of [12, Prop.4.2] and is similar to [4, Prop.10]. Theorem 4.8. Given an o -continuous and -complete LNLS X,p,E τ ). Then every p-almost order bounded uo-cauchy net is uo- and -convergent to the same limit. Proof. Suppose that x α ) is p-almost order bounded and uo-cauchy. Then the net x α x α ) is p-almost order bounded. Then, by [4, Prop.10], x α x α ) is uo-convergent 0 in X. Thus, since X is o -continuous, x α x α ) 0 upτ in X and, by Proposition 4.6, we get x α x α ) 0. pτ Thus x α ) is -Cauchy, and so is -convergent since X is -complete. By Lemma 4.7, we get that x α ) is uo-convergent to its -limit. Acknowledgement: This note would not have existed without inspiring and invaluable suggestions and comments made by Eduard Emel yanov and Şafak Alpay. I would like to have a deep gratitude toward them. 11

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