D029019023

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D029019023

  1. 1. Research Inventy: International Journal Of Engineering And ScienceVol.2, Issue 9 (April 2013), Pp 19-23Issn(e): 2278-4721, Issn(p):2319-6483, Www.Researchinventy.Com19ON SOME ASPECTS OF u -IDEALS DETERMINED BY 1AND 0cJohn Wanyonyi Matuya1, Achiles Simiyu2and Shem aywa21Department of Mathematics, Maasai Mara University, P.O Box 861-20500, Narok , Kenya2Department of Mathematics, Masinde Muliro University of science and Technology,P.O BOX 190-50100, Kakamega, Kenya.Abstract: The M -ideals defined on a real Banach space are called u -ideals. The u -ideals containingisomorphic copies of 1 are not strict u -ideals. In this paper we show that u -ideals with unconditionalbasis  nx which is shrinking has no isomorphic copy of 1 and thus a strict u -ideal. Finally we show thatu -ideals with unconditional basis  nx which is boundedly complete is not homeomorphic to copies of 0cimplying that they are weakclosed in their biduals X.Key Words: u -ideals, strict u -ideals, Shrinking, boundedly complete2010 Mathematics Subject Classification: 47B10; 46B10; 46A25 This work was supported by NationalCouncil of Science and Technology (NCST), Kenya.I. INTRODUCTIONA sequence   1n ixis called a basis of a normed space X if for every x X there exists a uniqueseries1i iia x that converges to x . The basis   1n ixfor a Banach space X is an unconditional basis if foreach x X there exists a unique expansion of the form1n nnx a x  where the sum convergesunconditionally. The basis  nx is said to be boundedly complete whenever given a sequence  na of scalarsfor which  1: 1nk kka x n is bounded, then 1limnn k Kka x exists. If nx X is not boundedlycomplete then X contains an isomorphic copy of 0c [1].Let y X which belongs to X . We say that  nx is shrinking,1 1, , . , , .n ni i i ii in nu y u x y u x u            converges to ,y u for every u X .The notion of u -ideals was introduced and studied thoroughly by Godfrey et al [2]. They generalized M -ideals defined on a real Banach space. In their paper on unconditional Ideals they established that u -idealscontaining copies of 1 are not strict u -ideals. A Banach space X is said to be a strict u -ideal in its bidualwhen the canonical decomposition XXX is unconditional. In other words for X to be a strictu -ideal the u -complement of Xmust be norming, that is, the range V of the induced projection on Xis a norming subspace of X. Vegard and Asvald  3 characterized Banach spaces which are strict u -ideals in their biduals and showed that X is a strict u -ideal in a Banach space Y if it contains 0c . Matuya etal [4] using the approximation properties, hermitian conditions, isometry studied properties of u -ideals and
  2. 2. On Some Aspects Of u -Ideals Determined….20their characterization. They showed that u -ideals containing no copies of sequence space 1 are strict u -ideals. In this paper we show that Banach spaces with unconditional basis  nx that is shrinking is nothomeomorphic to copies of 1 and so it a strict u -ideal. We also show that the u -ideals with unconditionalbasis which are boundedly complete are not bicontinuous to 0c meaning their u -complement is weakclosed.II. RESULTS ON STRICT u -IDEALS2.1 PropositionLet X be u -ideal with unconditional basis  nx . The following statements are equivalent:[1] The sequence  nx is shrinking[2] X contains no copy of 1[3] X is a strict u -idealProof: i ii it is clear from the definition that if a sequence  nx is shrinking then X contains noisomorphic copy of 1 . In fact proving by contradiction it suffices to show that X contains a copy of 1 . Let nc be a sequence of coefficient functional associated with  nx . Our hypothesis applies that, for somec X , the series1, .n nnc x c does not converge in  , ,X X X   implying that1, .kn nn kc x c    cannot be Cauchy in  , ,X X X   . We therefore find a bounded set 1L   andfor each k   we define maps , :P Q L Xby  1, .n nnP c x c  and 1, .kn nn kQ c x c     which implies together with   2L P Q    that     2L Q L P      holds. P is continuous since  nx is bounded, so that P maps Lhomeomorphically into X . Now Lis dense in 1 . Since 1 is weakly sequentially complete the same is truefor the subspace  1P  of X . In particular  nc has a weak limit in X and so does  nx because of, lim , ,k n knx c x c k     , this limit has to be x and this is the desired contradiction.ii iii Considering the map P it is clear from the definition that if x X  and P x x  then x Uwhere  U p x . Now for each X we consider the set   |F x X P x x      thenthere is a net  d Xx B  converging in the weak-topology of Xto u . However, u x X   so thatdxconverges to x. Thus      lim ddu x x     so that    u x x   . Now   P x ux   so we conclude that x F . Hence Fis norming.iii i Suppose X is strict u -ideal. We proceed to show that  nx is shrinking. Let y X whichbelongs to X . Since  nx is shrinking,1 1, , . , , .n ni i i ii in nu y u x y u x u           
  3. 3. On Some Aspects Of u -Ideals Determined….21converges to ,y u for every u X . We conclude from this that1, .ni iiy u x    is bounded in X so ithas a limit sayx .1, , . , ,i iix u y u x u y u  , for all u X ,we get y x X  .2.2 Proposition Let X be u -ideal with unconditional basis  nx . Show that if X contains no copy of 0cthen[1]  nx is boundedly complete[2]  nxis weakclosed in XSuppose  nx is not boundedly complete. Then there exists  iT  with1ni ii nx    bounded but notCauchy in X . Let now e   and let    andi im n be increasing sequences in  such that1i i im n m   andii j j ema x u  , that is,   1,ie a i    . Here we set , ,.....,i i i k iM m m n . Choose0i eu U such that , 1i iu a  , i   and let f   besuch that    ,, , andMe e x f x x X M      .By construction of the sets iM , it follows thatii i i gi i j Mg a g x e             ,Whence , i giv a e0, gg v U    .There exists a constant 0c  such that, .i giv a c e0, gg v U    .Consider now T as a subspace of 0c and define: T X  by   n nna    .Then  is linear and continuous, because of  n nng g a      0lub , . .gn n gUnv a c e    , ,g T    . is also open , because given T  and j   such that j   , then we can choose   such that
  4. 4. On Some Aspects Of u -Ideals Determined….22, ,i i j i i j iu a u a   for all i   .Therefore ,j n j nnu a    ,j n n n n n nn nv a e a          f   .Thus  is bicontinuous between T and X . This implies that T is dense in 0c and since X is sequenciallycomplete,  extends a linear homeomorphic embedding 0c into X which is a contradiction.We shall show that when    an dn nx X x X     satisfy1 n nnx x   and 10n nnx s x    for  ,s F X X ,then 10n nnx x   .We may assume that0nx and1nM x    .Let 0  and choose N   such that4n Nx . Since X has the weak approximating sequence,for  ˆˆ ,LXK X Z  , there exists a net    ˆ ˆ,s F X X such that lub 1s   and         L s x k L x k L k x   .Therefore s x x   weakfor all  x L k . In particular s x x   for all x F . Since1,s  we also have 1,s L if 1x then x L k for some k K . Thus 1s x , so s x  is weakconvergent. Hence there is some ssuch that2n nx s xM    , 1,....,n NNow we have 1 1n n n nn nx x x s x        1Nn n nnx x s x      n n nn Nx L k s L k  2n nn Nx L s L k    
  5. 5. On Some Aspects Of u -Ideals Determined….232 2   Hence 10n nnx x  III. CONCLUSIONWe have shown that u -ideals can be characterized using sequence spaces. In particular we consideredthe sequence spaces 1 and 0c . The u -ideals containing no isomorphic copies of 1 are strict u -idealswhereas those that are not homeomorphic to the copies of 0c their u -complement is weak-closedIV. ACKNOWLEDGEMENTThe authors convey their appreciation to the National Council of Science and Technology (NCST),Kenya for funding this research work and Masinde Muliro University of science and Technology for theirtechnical support.REFERENCES[1] Diestel. J. “ Sequences and Series in Banach spaces’’. Springer-Verlag Newyork , 1984.[2] Godefrey, G. Kalton, J. and Saphar,P. “ Unconditional ideals in Banach spaces ’’ .studiaMathematica.104 No.1,(1993) 13-59.[3] Lima, V. Lima, A. “ Strict u -ideals in Banach spaces ’’.studiaMath.3 No.195 (2009) 275-285.[4] Matuya . J. W. et al “On characterization of u -ideals determined by sequences’’ A.J.Curr.Eng and Math.1( 2012) 247-250.

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