UNIFORM CONVERGENCE OF MULTIPLIER CONVERGENT SERIES

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1 royecciones Vo. 26, N o 1, pp , May Universidad Catóica de Norte Antofagasta - Chie UNIFORM CONVERGENCE OF MULTILIER CONVERGENT SERIES CHARLES SWARTZ NEW MEXICO STATE UNIVERSITY Received : December Accepted : February 2007 Abstract If λ is a sequence K-space and x j is a series in a topoogica vector space X, the series is said to be λ-mutipier convergent if the series j=1 t jx j converges in X for every t = {t j } λ. We show that if λ satisfies a giding hump condition, caed the signed strong giding hump condition, then the series j=1 t jx j converge uniformy for t = {t j } beonging to bounded subsets of λ. A simiar uniform convergence resut is estabished for a mutipier convergent series version of the Hahn-Schur Theorem.

2 28 Chares Swartz Let X be a Hausdorff topoogica vector space and x j a(forma) series in X. The series x j is said to be bounded mutipier convergent if the series j=1 t j x j converges in X for every t = {t j } ([D]). If X is either a ocay convex space or a metric inear space and x j is bounded mutipier convergent, then the series j=1 t j x j actuay converge uniformy for t = {t j }, ktk 1 ([Sw1], [Sw2]8.2.7 ). A series xj in X is subseries convergent if the subseries x nj converges in X for every subsequence {n j }. If σ N, et C σ denote the characteristic function of σ and if x is any sequence, et C σ x denote the coordinatewise product of C σ and x. Thus, the series x j is subseries convergent iff C σ (j)x j = j σ x j converges for every σ N. A simiar uniform convergence resut hods for subseries convergent series. Namey, if x j is subseries convergent, then the series j=1 C σ (j)x j converge uniformy for σ N ([Sw2]8.1.2). Another uniform convergence resut hods for subseries and bounded mutipier convergent series in the subseries and bounded mutipier convergent versions of the Hahn-Schur Theorem ([Sw1],[Sw2]8.1 and 8.2). We consider conditions under which the same concusions hod if the mutipier space is repaced by more genera sequence spaces. Let λ be a vector space of scaar sequences which contains c 00, the space of a sequences which are eventuay 0, and which is equipped with a vector Hausdorff topoogy under which the coordinate functionas t = {t j } t j are continuous for every j N (i.e., λ is a K space ([B]7.2.2)). Let Λ λ. The series x j in X is Λ mutipier convergent if the series j=1 t j x j converges in X for every t = {t j } Λ ([F],[Sw2]8.3); thus, a series is -mutipier convergent iff the series is bounded mutipier convergent and a series is subseries convergent iff the series is m 0 mutipier convergent, where m 0 = span{c σ : σ N} is the sequence space of a sequences with finite range. It is natura to ask if a uniform convergence resut as above for subseries and bounded mutipier convergent series hods for λ-mutipier convergent series ;i.e., if x j is λ-mutipier convergent, do the series j=1 t j x j converge uniformy for t = {t j } beongingtocertain famiies of bounded subsets of λ? Exampe 5 shows that such a resut does not hod in genera, but we show in Theorem 3 that such a resut does hod if certain subsets of the mutipier space λ satisfy a giding hump property caed the signed strong giding hump property. We aso show that a simiar uniform convergence resut hods for series in a version of the Hahn-Schur Theorem for λ-mutipier convergent series when certain subsets of the mutipier space λ satisfy the signed strong giding hump property.

3 Uniform Convergence of Mutipier Convergent Series 29 We begin by defining the giding hump property which we wi empoy. An interva in N is a set of the form [m, n] ={k N : m k n} where m n; a sequence of intervas {I k } is increasing if sup I k < inf I k+1.asign is a variabe assuming the vaues {±1}. LetΛ λ.the subset Λ has the it signed strong giding it hump property (signed-sgh) if for every bounded sequence {x k } Λ and for every increasing sequence of intervas {I k }, there exists a subsequence {n k } and a sequence of signs {s k } such that x = k=1 s k C Ink xn k [coordinate sum] beongs to Λ. The subset Λ has the strong giding hump property (SGH) if the signs above can a be chosen to equa to1 ; the SGH has been empoyed on numerous occasions ([N ],[Sw2],[SS]). The idea of mutipying the humps C I x by signs was introduced by Stuart ([St1],[St2]) and used to treat weak sequentia competeness of β duas. For exampe, the space has the SGH and the methods of [BSS] can be used to construct other spaces with SGH. Let M 0 be the subset of m 0 consisting of the sequences of 0 s and 1 s, M 0 = {C σ : σ N}. Then the subset M 0 has SGH but the space m 0 does not have SGH. We now show that the space of bounded series bs ([B]) has the signed-sgh but fais SGH; Stuart showed that bs has the signed weak giding hump property but not the weak giding hump property and we essentiay just use his proof ([St1],[St2]). Reca that bs is the space of a sequences t = {t j } such that ktk =sup nj=1 n t j < equipped with this norm ([B]1.2); an equivaent norm to kk is given by ktk 0 =sup{ k I t k : Ianinterva}. If x = {x j } and y = {y j } are sequences, we write x y = j=1 x j y j for the forma dot product of x and y when the series converges. Exampe 1. bs has signed-sgh. Actuay, bs hasanevenstronger property than signed-sgh; it is not necessary to pass to a subsequence in the definition of signed-sgh. Let {I k } be an increasing sequence of intervas and {t k } bs be bounded. ut M =sup{ j I t k j : k N,I an interva in N} <. Define signs inductivey by setting s 1 = signc I1 t 1 and s n+1 = [sign n k=1 s k C Ik t k ][signc In+1 t n+1 ]. ut y = k=1 s k C Ik t k ; we show kyk 2M. We first show by induction that max n j=1 y j M for every n. For n =1, max 1 j=1 y j = j I 1 s 1 t 1 j M. Suppose the inequaity hods for n. Then n+1 max j=1 y j = max In j=1 y j + j I n+1 y j M since j I n+1 y j = j I n+1 s n+1 t n+1 j M and max n j=1 y j M and both of these terms have opposite signs. Now for arbitrary n et k = k n be the argest integer such that max I k n. Then n j=1 y j =

4 30 Chares Swartz max k j=1 y j + n j=max Ik +1 y j max n j=1 y j + nj=min Ik+1 s k+1 t k+1 j 2M so kyk 2M as desired. Note that bs does not have SGH [consider t = {1, 1, 1, 1,...} and I k = {2k 1}; thenc Ik t = {1, 0, 1, 0,...} / bs and ikewise the same hods for any subsequence of {I k }]. Additiona spaces with the signed-sgh can be constructed empoying the methods of [BSS]. We next consider the uniform convergence of mutipier convergent series when subsets of the space of mutipiers has signed-sgh. We first estabish a emma. Lemma 2. Let Λ λ. Let x j be Λ-mutipier convergent. If the series j=1 t j x j do not converge uniformy for t B Λ, then there exist a symmetric neighborhood of 0,V, t k B and an increasing sequence of intervas {I k } such that j I k t k j x j / V. roof : If the series j=1 t j x j do not converge uniformy for t B, there exist a symmetric neighborhood of 0, U, such that for every k there exist t k B, m k k such that j=mk t k j x j / U. For k = 1, et m 1,t 1 B satisfy this condition so j=m1 t 1 j x j / U. ick a symmetric neighborhood of o, V, such that V +V U. There exists n 1 >m 1 such that j=n1 +1 t1 j x j V. Then n 1 j=m 1 t 1 j x j = j=m1 t 1 j x j j=n1 +1 t1 j x j / V. ut I 1 =[m 1, n 1 ]. Now just continue the construction. Theorem 3. LetΛ λ have signed-sgh and et x j be Λ-mutipier convergent. Then the series j=1 t j x j converge uniformy for t beonging to bounded subsets of Λ. roof: If B Λ is bounded and j=1 t j x j fais to converge uniformy for t B, et the notation be as in Lemma 2. Let n k,s k be as in the definition of signed-sgh above and t = j=1 s j C Inj t n j Λ. Then t j x j does not converge in X since j I nk t j x j = s k j I nk t n k j x j / V,i.e., t j x j doesn t satisfy the Cauchy condition. Remark 4. IfΛ =, then Theorem 3 impies that any bounded mutipier convergent series is such that the series j=1 t j x j converge uniformy for ktk 1. This statement improves the resut for bounded mutipier convergent series given in of [Sw2], removing the ocay convex and sequentia competeness assumptions. A (vector) version of Theorem 3 is estabished in [SS] Lemma 22 under the assumption that the mutipier space has SGH. A resut with the same concusion as Theorem 3 is given

5 Uniform Convergence of Mutipier Convergent Series 31 in [WLC], Theorem 7, but the assumptions there are quite different, being topoogica in nature, and difficut to compare. Remark 5. Ifλ = m 0 and Λ = M 0,then a series is subseries convergent iff the series is m 0 or M 0 mutipier convergent (spanm 0 = m 0 ). Then Theorem 3 impies that any subseries convergent series x j is such that the series j=1 C σ (j)x j converge uniformy for σ N ([Sw2]8.1.2). Thus, Theorem 3 gives a generaization of the known resuts for uniform convergence of subseries and bounded mutipier convergent series. Exampe 6. Without some type of assumption on the mutipier space λ, the concusion of Theorem 3 can fai even when the mutipier space satisfies giding hump conditions ike the weak giding hump property or the zero giding hump property ([Sw2]). Let e j be the sequence with a 1 in the j th coordinate and 0 in the other coordinates. Then e j is p mutipier covergent in ( p, kk p ) for any 1 p<, but the series j=1 t j e j do not converge uniformy for ktk p 1[Taket k = e k, j=1 t k j ej = e k.]. We next consider a mutipier version of the Hahn-Schur Theorem. The cassica scaar version of the Hahn-Schur Theorem asserts that if the scaar series j x ij is absoutey convergent for every i and im i j σ x ij exists for every σ N and if x j =im i x ij,then x j is absoutey convergent and x ij x j 0 or, equivaenty, im i j σ (x ij x j ) = 0 uniformy for σ N ([Sw2]5.4). By empoying the ast statement of the concusion of the cassica Hahn-Schur Theorem,versions of the Hahn-Schur Theorem for vector-vaued subseries and bounded mutipier convergent series have been given in [Sw2]8.1,8.2, and an abstract version of the theorem which incudes certain mutipier convergent series is given in [Sw2]9.3. We now present a version of the Hahn-Schur Theorem for mutipier convergent series when the subset Λ of the mutipier space λ satisfies the signed-sgh. We first estabish a specia case of the theorem. Lemma 7. Suppose that Λ λ has signed-sgh, j x ij is Λ-mutipier convergent for every i and im i t j x ij =0foreveryt Λ. IfB Λ is bounded, then im i t j x ij = 0 uniformy for t B. roof : It suffices to show that im i t i j x ij = 0 for any sequence {t i } B. LetU be a neighborhood of 0 in X and pick a symmetric neighborhood, V, of 0 such that V +V +V U. Setn 1 =1andpickN 1 such that

6 32 Chares Swartz j=n1 t n 1 j x n 1 j V.Sinceim i x ij =0(c 00 λ) for every j and {t i j : i N} is bounded from the K space assumption on λ, im i t i j x ij =0foreveryj ([Sw2]8.2.4) so there exist n 2 >n 1 such that N 1 1 j=1 t i j x ij V for i n 2. ick N 2 >N 1 such that j=n2 t n 2 x n2 j V. Continuing this construction produces increasing sequences {n k }, {N k } such that =Nj t n j x nj V and N j 1 =1 t i x i V for i n j. Set I j = { : N j 1 <N j }. Define the matrix M = [m ij ] = [ I j t n j x ni ]. We show that M is a signed K-matrix in the terminoogy of [St1],[St2],[Sw2] First the coumns of M go to 0 since im i x i =0forevery. Given an increasing sequence {p j }, by the signed-sgh assumption there is a subsequence {q j } of {p j } and signs {s j } such that t = {t j } = j=1 s j C Iqj t q j Λ. Then j=1 s j m iqj = j=1 s j I qj t q j x ni = j=1 t j x ni j 0byhypothesis. Therefore, M is a signed K-matrix and by the signed version of the Antosik-Mikusinski Matrix Theorem the diagona of M goes to 0 ([St1],[St2],[Sw2]2.2.4 ). Thus, there exists N such that m ii V for i N. If i N, then =1 t n i x ni = N i 1 1 =1 t n i x ni + I i t n i x ni + =Ni t n i x ni V + V + V U so im =1 i t n i x ni = 0. Since the same argument can be appied to any subsequence, it foows that im j=1 i t i j x ij =0 Theorem 8. Suppose that Λ λ has signed-sgh, j x ij is Λ- mutipier convergent for every i and im i t j x ij exists for every t Λ. Let x j = im i x ij for every j. If B Λ is bounded, then (i) x j is Λ mutipier convergent, (ii) im i t j x ij = j=1 t j x j uniformy for t B, (iii) the series j=1 t j x ij converge uniformy for t B. roof : Let t Λ. Since the space Λ has the signed weak giding hump property, it foows from Stuart s weak sequentia competeness resut that t j x j converges and im i t j x ij = j=1 t j x j.([sw2] ;see aso [St1] 3.5); Stuart s resut is for the case when Λ is a vector space with signed-wgh, but his proof is vaid for a subset with signed-wgh. Since im i t j (x ij x j )=0foreveryt Λ, Lemma 7 appies and gives (ii). Suppose that (iii) fais to hod. Then there exists a cosed symmetric neighborhood of 0, U, inx such that for every i there exist k i >i,a finite interva I i with min I i >i,t i B such that k I i t i k x k i k / U. uti 1 =1. By the above, there exist k 1 > 1, I 1 with min I 1 >i 1, t 1 B such that k I 1 t 1 k x k 1 k / U. By Theorem 3 there exists j 1 such that k=j t k x ik U for every t B,1 i k 1,j j 1. Set i 2 =max{i 1 +1,j 1 }. Again,

7 Uniform Convergence of Mutipier Convergent Series 33 by the above there exist k 2 >i 2,I 2 with min I 2 >i 2,t 2 B such that k I 2 t 2 k x k 2 k / U. Notethatk 2 >k 1 by the definition of i 2. Continuing this construction produces an increasing sequence {k i },an increasing sequence of intervas {I i } and t i B such that k I i t i k x k i k / U. (1) Define a matrix M =[m ij ]=[ k I j t i k x k i k]. We caim that M is a signed K-matrix. First, the coumns of M converge by hypothesis. Next, given any increasing sequence {p j } there is a subsequence {q j } of {p j } and signs {s j } such that t = {t j } = j=1 s j C Iqj t q j Λ. Then the sequence s j m iqj = j=1 s j I qj t q j x ki = j=1 t j x ki j converges by hypothesis. Hence, M is a signed K-matrix and the diagona of M converges to 0 by the signed version of the Antosik-Mikusinski Matrix Theorem ([Sw2]2.2.4). But, this contadicts (1). A subseries version (M 0 = Λ m 0 = λ) of the Hahn-Schur Theorem is given in [Sw2]8.1 and a bounded mutipier version of the Hahn-Schur Theorem is given in [Sw2]8.2. Both versions foow from Theorem 3. A (vector) version of Theorem 7 for spaces with SGH is given in Theorem 25 and Coroary 27 of [SS]. Without some assumption on the mutipier space λ, the concusion of Theorem 8 can fai. Exampe 9. Let x ij = e j if 1 j i and x ij =0ifi<j. Then j x ij is p -mutipier convergent in p for 1 p< for every i. Ift p, t j x ij j=1 t j e j in p. However, the convergence is not uniform for ktk p 1[Taket k = e k,so j=1 t k j x ij = i j=1 t k j ej = e k if i k.]. There is an abstract version of the Hahn-Schur Theorem which covers certain mutipier convergent series given in [Sw2]9.3; however, the vector version given there uses essentiay a strong giding hump type hypothesis. A (vector) version of Theorem 7 for spaces with SGH is given in [SS] Theorem 25. Usefu vector forms of the signed-sgh seem to be difficut to formuate. There is a version of Theorem 7 given in [WCC],Theorem 7, and in [A], Theorem 3.1, but the assumptions there are of a topoogica nature unike the agebraic signed-sgh.

8 34 Chares Swartz References [A] Aizpuru, A. and erez-fernandez, J., Spaces of S-bounded mutipier convergent series, Acta. Math. Hungar., 87, pp , (2000). [B] Boos, J., Cassica and modern methods in summabiity, Oxford University ress, Oxford, (2000). [BSS] Boos, J.;Stuart, C.;Swartz,C.,Giding hump properties of matrix domains,anaysis Math., 30, pp , (2004). [D] Day, M., Normed Linear Spaces, Springer-Verag, Berin, (1962). [F] Forencio, M. and au,., A note on λ-mutipier convergent series, Casopis ro est. Mat., 113, pp , (1988). [N] No, D., Sequentia competeness and spaces with the giding hump property,manuscripta Math., 66, pp , (1960). [St1] Stuart, C., Weak Sequentia Competeness in Sequence Spaces, h.d. Dissertation, New Mexico State University, (1993). [St2] Stuart, C., Weak Sequentia Competeness of β-duas, Rocky Mountain Math. J., 26, pp , (1996). [SS] Stuart, C. and Swartz, C., Uniform convergence in the dua of a vectorvaued sequence space,taiwanese J. Math., 7, pp , (2003). [Sw1] Swartz, C., The Schur emma for bounded mutipier convergent series, Math. Ann., 263, pp , (1983). [Sw2] Swartz, C., Infinite Matrices and the Giding Hump, Word Sci. ress, Singapore, (1996). [WCC] [WCC]Wu,J.;Li,L.;Cui,C.,Spaces of λ-mutipier convergent series, Rocky Mountain Math. J., 35, pp , (2005).

9 Uniform Convergence of Mutipier Convergent Series 35 Chares Swartz Mathematics Department New Mexico State University Las Cruces NM USA e-mai : cswartz@nmsu.edu