We present a compactness criterion of Vitali-type in a class of vector-valued real F-seminormed spaces, which satisfy the W-property

Tavernise, Marianna

Trombetta, Alessandro

Trombetta, Giulio

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

LE MATEMATICHEVol. LXVI (2011) – Fasc. I, pp. 171–179doi: 10.4418/2011.66.1.17TOTAL BOUNDEDNESS IN VECTOR-VALUEDF-SEMINORMED FUNCTION SPACESM. TAVERNISE - A. TROMBETTA - G. TROMBETTAWe present a compactness criterion of Vitali-type in a class of vector-valued real F-seminormed spaces, which satisfy the W -property.1. IntroductionThe definition of some quantitative characteristics in spaces of functions andtheir comparison with the Hausdorff measure of noncompactness has allowedmany authors to generalize some classical compactness results (see [2], [5],[12], [14], for example). In this paper we introduce a quantitative characteristic,which measures the degree of non equiabsolute continuity for subsets of spacesin a class of vector-valued real F-seminormed function spaces. We compare thisquantitative characteristic with the Hausdorff measure of noncompactness. Bythis comparison we obtain some inequalities, that give, as a special case, suffi-cient conditions for the total boundedness of a set of functions. From our resultswe derive a Vitali-type compactness criterion in spaces, of the class we haveconsidered, which satisfy the W -property (see Definition 3.7). In our context wegeneralize some of the results obtained in [12]. Moreover, the results we presentare a partial anticipation of those contained in [13], where the total boundednessEntrato in redazione: 30 ottobre 2010AMS 2010 Subject Classification: 47H08, 46E40.Keywords: Hausdorff measure of noncompactness, Measure of non equiabsolute continuity,Vector-valued F-seminormed function space.172 MARIANNA TAVERNISE - ALESSANDRO TROMBETTA - GIULIO TROMBETTAof a set is considered in a wider class of vector-valued F-seminormed functionspaces.2. Preliminaries and notationsThroughout the present paper all linear spaces are real and we adopt the conven-tion that inf /0=+∞. The notions concerning the theory of F-seminormed spaces(resp. Riesz F-seminormed spaces) can be found in [8] (resp.[10] and [16]). Wegive here only the basic ones. Let V be a linear space. An F-seminorm on V is afunction ‖ ·‖V : V −→ [0,+∞[ such that ‖u+v‖V ≤ ‖u‖V +‖v‖V , limn→∞∥∥1n u∥∥V =0, ‖λu‖V ≤ ‖u‖V , for each u,v ∈ V and for all real number λ with |λ | ≤ 1. If‖u‖V = 0 only when u = 0, then ‖ · ‖V is called an F-norm. An F-seminorm(F-norm) is called a q-seminorm (q-norm), 0 < q ≤ 1, if ‖λu‖V = |λ |q‖u‖Vfor all u ∈ V and for all real number λ . We observe that if (V,‖ · ‖V ) is an F-seminormed space and u is an element of V, then the function[0,+∞[3 t→ ϕ(t) = ‖tu‖V ∈ Ris continuous and non-decreasing.In the following we assume that (E,‖ · ‖E) is an F-normed space, Ω anonempty set, A a subalgebra of the power setP(Ω) of Ω, η :A −→ [0,+∞]a submeasure, (i.e., a monotone, subadditive function with η( /0) = 0), and η˜ :P(Ω)−→ [0,+∞] the submeasure defined by η˜(B) = inf{η(A) : B⊂A and A∈A }. Let EΩ be the linear space of all E-valued functions on Ω. For f ∈ EΩ wedenote by ‖ f‖E the function x→ ‖ f (x)‖E and we put {‖ f‖E ≥ a} = {x ∈ Ω :‖ f (x)‖E ≥ a}. Then‖ f‖0 = inf{a > 0 : η˜({‖ f‖E ≥ a})≤ a}defines a Riesz group pseudonorm on EΩ, i.e., ‖0‖0 = 0,‖ f +g‖0≤‖ f‖0+‖g‖0and ‖ f‖E ≤ ‖g‖E =⇒ ‖ f‖0 ≤ ‖g‖0 for each f ,g ∈ EΩ, (see [3], [4] for moredetails).Moreover set A0 = {A ∈A : η(A)<+∞}. We will denote byS(A0,E) = span{χAy : y ∈ E and A ∈A0}the linear space of all E-valuedA0-simple functions onΩ, where χA denotes thecharacteristic function of a set A defined on Ω. In the remainder of this paperL = (L,‖ · ‖L) stands for an F-seminormed subspace of EΩ with the followingproperties:(a) if A ∈A and f ∈ L, then χA f ∈ L;TOTAL BOUNDEDNESS IN VECTOR-VALUED . . . 173(b) S(A0,E) is a linear subspace of L dense in (L,‖ · ‖L);(c) if A ∈A0 and f ∈ L, then limη(B)→0A⊇B∈A‖χB f‖L = 0;(d) there is k ≥ 1 such that ‖ f‖E ≤ ‖g‖E implies ‖ f‖L ≤ k‖g‖L for eachf ,g ∈ L.If the condition (d) is satisfied for k = 1, L is a Riesz F-seminormed space.Throughout we denote by Bε(EΩ) = { f ∈ EΩ : ‖ f‖0 ≤ ε} and by Bε(L) = { f ∈L : ‖ f‖L ≤ ε} for ε > 0.Definition 2.1. Let M ⊆ EΩ; thenβ0(M) := inf{ε > 0 : there is a finite subset F of M such that M⊆F+Bε(EΩ)}.Let M ⊆ L; thenγL(M) := inf{ε > 0 : there is a finite subset F of L such that M ⊆ F +Bε(L)}.The set functions β0 and γL are, respectively, the inner Hausdorff measure ofnoncompactness in (EΩ,‖ ·‖0) and the classical Hausdorff measure of noncom-pactness in (L,‖ · ‖L).Clearly, M ⊆ EΩ is ‖ ·‖0-totally bounded (‖ ·‖0-tb, for short) iff β0(M) = 0, andM ⊆ L is ‖ · ‖L-totally bounded (‖ · ‖L-tb, for short) iff γL(M) = 0.3. Main resultsIn this section we introduce a quantitative characteristic which measures thedegree of non equiabsolute continuity for a subset of the space L.Definition 3.1. Let M ⊆ L. We define for A ∈A0 and δ > 0 :ΠL (M,A,δ ) = maxsupf∈M∥∥χΩ\A f∥∥L ,supf∈M supA⊇B∈Aη(B)≤δ‖χB f‖L ,ΠL (M,A) = limδ→0ΠL (M,A,δ ) ,ΠL (M) = infA∈A0ΠL (M,A) .A subset M of L is called ‖ ·‖L-equiabsolutely continuous ( ‖ ·‖L-eac, for short)if ΠL (M) = 0.174 MARIANNA TAVERNISE - ALESSANDRO TROMBETTA - GIULIO TROMBETTARemark 3.2. Let s ∈ S(A0,E) and set As = {x ∈Ω : s(x) 6= 0} . Then As ∈A0and ‖χΩ\Ass‖L = 0. Hence, by (c), ΠL({s} ,As) = 0.We are now in a position to prove the main results of this note.Theorem 3.3. Let M ⊆ L. ThenΠL(M)≤ kγL(M).Proof. The inequality is trivially true if γL(M) = supf∈L‖ f‖L. Assume thatγL(M) < supf∈L‖ f‖L. Let α > γL(M). By (b) there are s1, ...,sn ∈ S(A0,E) suchthat M ⊆n⋃i=1(si +Bα(L)). Set Ai = {x ∈Ω : si(x) 6= 0} for i = 1, ...,n. ClearlyA=n⋃i=1Ai ∈A0. Let f ∈M . Choose i∈ {1, ...,n} such that ‖ f −si‖L ≤ α . Then‖χΩ\A f‖L = ‖χΩ\A( f − si)‖L ≤ k‖ f − si‖L ≤ kα. (1)Moreover, let δ > 0. For B⊆ A,B ∈A and η(B)≤ δ , we have‖χB f‖L ≤ ‖χB( f − si)‖L+‖χBsi‖L ≤ k‖ f − si‖L+‖χBsi‖L≤ kα+ maxj=1,...,nΠL({s j},A,δ ). (2)Having in mind Remark 3.2, by (1) and (2) we getΠL(M,A)≤ kα+ maxj=1,...,nΠL({s j},A) = kα+ maxj=1,...,n‖χΩ\As j‖L = kα,hence ΠL(M)≤ kα, and therefore ΠL(M)≤ kγL(M).Theorem 3.4. Let M ⊆ L, A ∈A0 and suppose that β0(χAM)< supy∈E‖y‖E . ThenγL(M)≤ 3 limδ→β0(χAM)+ΠL(M,A,δ )+ k ‖χAy0‖L, (3)for some y0 ∈ E such that ‖y0‖E = β0(χAM). In fact, for every y ∈ E satisfying‖y‖E > β0(χAM), there is a t0 ∈ [0,1[ such that ‖t0y‖E = β0(χAM) and suchthat the inequality (3) is satisfied by y0 = t0y. In particular, if L is a Riesz F-seminormed space, the inequality (3) is satisfied by any y0 ∈E such that ‖y0‖E =β0(χAM).TOTAL BOUNDEDNESS IN VECTOR-VALUED . . . 175Proof. Fix an element y ∈ E such that ‖y‖E > δ0 = β0(χAM). Then, by thedefinition of δ0, for every δ ∈]δ0,‖y‖E ] there are a positive number σ ∈ [δ0,δ [and functions f1, ..., fn ∈M such thatχAM ⊆n∪i=1(χA fi+Bσ (EΩ)).Fix f ∈M and let i∈ {1, ...,n} such that ‖χA( f − fi)‖0 ≤ σ . Set D f ={‖χA( f −fi)‖E > σ}, then D f ⊆ A and η˜(D f ) ≤ σ . Hence, by the definition of η˜ thereexists C f ∈A such that D f ⊆C f and η(C f )≤ δ . Then B f = A∩C f ⊆ A, B f ∈A , η(B f ) ≤ δ and χB f χA fi = χB f fi. Set aδ = ΠL(M,A,δ ). By the definitionof ΠL(M,A,δ ) we have ‖χΩ\A f‖L ≤ aδ , ‖χB f f‖L ≤ aδ and ‖χB f fi‖L ≤ aδ .Therefore‖ f −χA fi‖L ≤ ‖χΩ\A f‖L+‖χA( f − fi)‖L≤ ‖χΩ\A f‖L+‖χA\B f ( f − fi)‖L+‖χB f f‖L+‖χB f fi‖L≤ 3aδ +‖χA\B f ( f − fi)‖L. (4)Since the function ϕ : [0,1] −→ [0,+∞[ defined by ϕ(t) = ‖ty‖E is continuousand ϕ([0,1]) = [0,‖y‖E ], there is tδ ∈ [0,1] such that ‖tδ y‖E = δ . Clearly thefunction χAtδ y ∈ L. Moreover, it is easy to see that‖χA\B f ( f − fi)‖E ≤ ‖χAtδ y‖E .Then‖χA\B f ( f − fi)‖L ≤ k‖χAtδ y‖L .By (4) it follows thatγL(M)≤ 3aδ + k ‖χAtδ y‖L . (5)Now, since the function ϕ is non-decreasing, we have that]δ0,‖y‖E ] 3 δ → tδ ∈ [0,1]is a strictly increasing function andlimδ→δ+0tδ = t0 = max{t ∈ [0,1] : ϕ(t) = δ0}.Thus, by the continuity of the function[0,1] 3 t→ ψ(t) = ‖χAty‖L ∈ R,we havelimδ→δ+0‖χAtδ y‖L = ‖χAy0‖L,176 MARIANNA TAVERNISE - ALESSANDRO TROMBETTA - GIULIO TROMBETTAwhere y0 = t0y. Therefore, by (5), the inequalityγL(M)≤ 3 limδ→δ+0aδ + k ‖χAy0‖L ,that is inequality (3), holds and ‖y0‖E = β0(χAM). This accomplishes the proof.Corollary 3.5. Let M ⊆ L, A ∈A0, and suppose χAM ‖ · ‖0-tb. ThenγL(M)≤ 3ΠL(M,A).The following corollary of the Theorem 3.4 gives a sufficient condition for thetotal boundedness of a subset of L.Corollary 3.6. Let M be a ‖ · ‖L-eac subset of L and suppose that χAM is‖ · ‖0-tb for all A ∈A0. Then M is ‖ · ‖L-tb.Definition 3.7. (see [15, Chapter III], [17]) A space L has the W -property iffn‖·‖L−→ 0 implies χA fn ‖·‖0−→ 0 for all sequences ( fn) of elements of L and for allA ∈A0.Proposition 3.8. If the space L has the W-property and if M is a subset of L‖ · ‖L-tb , then χAM is ‖ · ‖0-tb for all A ∈A0.Proof. Let M be ‖ · ‖L-totally bounded and let A ∈ A0. By the W -property wehave thatfor all ε > 0 there exists δ > 0 such that χABδ (L)⊂ Bε(EΩ). (6)Now fix ε > 0. By (6) there exists δ > 0 such that χABδ (L)⊂ Bε(EΩ). Since Mis ‖ · ‖L-tb there exists a finite subset F of L such that M ⊂ F +Bδ (L). HenceχAM ⊂ χAF +χABδ (L)⊂ χAF +Bε(EΩ), and therefore χAM is ‖ · ‖0-tb.Combining Theorem 3.3 and Proposition 3.8, and having in mind Corollary 3.6,we obtain the following Vitali-type total boundedness criterion.Theorem 3.9. Assume that the space L has the W-property. Then a subset M ofL is ‖ · ‖L-tb if and only if it is ‖ · ‖L-eac and χAM is ‖ · ‖0−tb for all A ∈A0.In the setting of q-seminormed spaces, 0 < q≤ 1, we have the following corol-laries of the Theorem 3.4.TOTAL BOUNDEDNESS IN VECTOR-VALUED . . . 177Corollary 3.10. Assume that E is a q-normed space, 0 < q≤ 1. Let M ⊆ L andA ∈A0, thenγL(M)≤ 3 limδ→β0(χAM)+ΠL(M,A,δ )+ k ‖β0(χAM)1q χAy0‖L,where y0 ∈ E and ‖y0‖E = 1.Corollary 3.11. Assume that L is a p-normed space, 0 < p ≤ 1, and E is aq-normed space, 0 < q≤ 1. Let M ⊆ L and A ∈A0, thenγL(M)≤ 3 limδ→β0(χAM)+ΠL(M,A,δ )+ k β0(χAM)pq ‖χAy0‖L,where y0 ∈ E and ‖y0‖E = 1.4. ExampleIn this section we give an example of a class of F-seminormed spaces of type Lsatisfying the W -property. We consider the “ Orlicz spaces ” LN introduced in[6] in the same way as Dunford and Schwartz [7, Chapter III] define the space ofintegrable functions and the integral for integrable functions. We briefly recallthe definition of the space LN . Let ‖ · ‖ : S(A0,R) −→ [0,+∞[ be a Riesz F-seminorm such that η(A) = ‖χA‖ for all A ∈ A0 and N : [0,+∞)→ [0,+∞)a continuous, strictly increasing function such that N (0) = 0 and N(s+ t) ≤n¯(N(s)+N(t)) for all s, t ≥ 0. Assume that (E,‖ · ‖E) is a complete F-normedspace and let us denote by L0 the closure of S(A0,E) in (EΩ,‖ · ‖0). For s ∈S(A0,E), ‖s‖N is defined by ‖s‖N = ‖N ◦‖s‖E‖. Then LN (see [6, p.92]) is thelinear space of all functions f ∈ L0, for which there is a ‖·‖N- Cauchy sequence(sn) in S(A0,E) converging to f with respect to ‖ · ‖0. Such a sequence (sn) ofsimple functions is said to determine f . If (sn) is a determining sequence forf ∈ LN , ‖ · ‖N is defined by ‖ f‖N = limn→+∞‖sn‖N . The function ‖ · ‖N has thefollowing properties:‖ f +g‖N ≤ 2n¯max{‖ f‖N , ‖g‖N}, limn→+∞‖1nf‖N = 0,‖λ f‖N ≤ ‖ f‖N ,for all f ,g ∈ LN and for all real number λ with |λ | ≤ 1, therefore ‖ · ‖N is a∆-seminorm on LN in the sense of [9, p.2]. Moreover, the space (LN ,‖ · ‖N)satisfies the properties (a)-(c) of Section 2 and (see [6, Proposition 2.6])‖ f‖E ≤ ‖g‖E implies ‖ f‖N ≤ ‖g‖N , for each f ,g ∈ LN .178 MARIANNA TAVERNISE - ALESSANDRO TROMBETTA - GIULIO TROMBETTASet L = LN , then by [9, Theorem 1.2], if p is choosen such that 21p = 2n¯ thenthe formula‖ f‖L = inf{ n∑i=1‖ fi‖pN :n∑i=1fi = f}defines an F-seminorm on L generating the same topology of the ∆-seminorm‖ · ‖N .Further, being 14‖ f‖pN ≤‖ f‖L≤‖ f‖pN (see [9, Lemma 1.1]), using (4) we obtainthat‖ f‖E ≤ ‖g‖E implies ‖ f‖L ≤ 4‖g‖L, for each f ,g ∈ LN . (7)Hence the space (L,‖ · ‖L) is a F-seminormed subspace of EΩ which satisfiesthe properties (a)-(d) of Section 2. Moreover, as consequence of [6, Theorem2.7], it has the W -property.5. AcknowledgementThe authors wish to thank the referee for his comments and suggestions on thesubmission.REFERENCES[1] C. D. Aliprantis - O. Burkinshaw, Locally solid Riesz spaces, Academic Press, NewYork, 1978.[2] J. Appell - E. De Pascale, Su alcuni parametri connessi con la misura di noncom-pattezza di Hausdorff in spazi di funzioni misurabili, Boll. U.M.I. 6 (3-B) (1984),497–515.[3] A. Avallone, Spaces of measurable functions I, Ric. Mat. 39 (2) (1990), 221–246.[4] A. Avallone, Spaces of measurable functions II, Ric. Mat. 40 (1) (1992), 3–19.[5] A. Avallone - G. Trombetta, Measures of noncompactness in the space L0 and ageneralization of the Arzela`-Ascoli Theorem, Boll. UMI 7 (5-B) (1991), 573–587.[6] P. De Lucia - H. Weber, Completeness of function spaces, Ric. Mat. 39 (1990),81–97.[7] N. Dunford - J. T. Schwartz, Linear Operators, Part I, Wiley-Interscience, NewYork, 1958.[8] H. Jarchow, Locally convex spaces, B. G. Teubner Stuttgart, 1981.[9] N. J. Kalton - N. T. Peck - J. W. Roberts, An F-space sampler, London Mathemati-cal Society Lecture Note Series 89, Cambridge University press 1984.TOTAL BOUNDEDNESS IN VECTOR-VALUED . . . 179[10] D. Keim, Die Ordnungstopologie und ordnungstonnelierte topologien auf vek-torverbanden, Collect. Math. 22 (1971), 117–140.[11] J. L. Kelley, General topology, D. van Nostrand, New York, 1955.[12] M. A. Lepellere - G. Trombetta, On some measures of noncompactness and themeasure of non equiabsolute continuity in function spaces, Ric. Mat. 46 (2) (1997),291–306.[13] M. Tavernise - A. Trombetta - G. Trombetta, Compactness criteria in vector-valued F-seminormed function spaces, preprint.[14] G. Trombetta - H. Weber, The Hausdorff measures of noncompactness for balls inF-normed linear spaces and for subsets of L0, Boll. UMI 6 (5-C) (1986), 213–232.[15] M. Vath, Ideal Spaces, Lecture Notes in Mathematics 1664, Springer, 1997.[16] Y. C. Wong - K. F. Ng, Partially ordered topological vector spaces, ClarendonPress, Oxford, 1973.[17] P. P. Zabrejko, Ideal Spaces of functions I (in Russian), Vestnik Jarosl. Univ.(1974), 12–52.MARIANNA TAVERNISEDepartment of MathematicsUniversity of Calabriae-mail: tavernise@mat.unical.itALESSANDRO TROMBETTADepartment of MathematicsUniversity of Calabriae-mail: aletromb@unical.itGIULIO TROMBETTADepartment of MathematicsUniversity of Calabriae-mail: trombetta@unical.it

Total boundness in vector-valued F-seminormed function spaces

We present a compactness criterion of Vitali-type in a class of vector-valued real F-seminormed spaces, which satisfy the W-property.<br /

Marianna Tavernise

Alessandro Trombetta

Giulio Trombetta

Directory of Open Access Journals

https://lematematiche.dmi.unict.it/index.php/lematematiche/article/download/866/776/

Total boundness in vector-valued F-seminormed function spaces

Abstract

Similar works

Full text

Available Versions

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

Directory of Open Access Journals