S. Velmurugan 1, N. Saivaraju 2, and N. Subramanian 3

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1 Le Scece Joural 03;0(3) htt://wwwlescecestecom The -Metrc Sace o Deed by Muselak S Velmuruga N Savaraju ad N Subramaa 3 Deartmet o Mathematcs Shr Agalamma college o Egeerg ad Tech Trchy Ida Deartmet o Mathematcs Sastra Uvversty Thajavur Ida ksvelmuruga09@gmalcom savaraju@yahoocom 3 smaths@yahoocom Abstract: I the reset aer we troduce the -metrc sace o multler deed by a Muselak modulus ucto We study some toologcal roertes ad rove some cluso relatos betwee these saces Ldestrauss ad Tzarr [5] used the dea o Orlcz ucto to dee the seuece sace l M whch s called a Orlcz seuece sace Aother geeralzato o Orlcz seuece saces s due to Woo [3] [S Velmuruga N Savaraju ad N Subramaa The -Metrc Sace o Deed by Muselak Le Sc J 03;0(3):30-37] (ISSN: ) htt://wwwlescecestecom 49 Key words ad hrases: aalytc seuece modulus ucto double seueces sace derece seuece sace Muselak modulus ucto metrc sace duals 00 Mathematcs Subject Classcato 40A0540C0540D05 Itroducto: Throughout w ad deote the classes o all ga ad aalytc scalar valued sgle seuece resectvely We wrte w or the set o all comlex seueces (x ) where m the set o ostve tegers The w s a lear sace uder the coordate wse addto ad scalar multlcato Some tal works o double seuece saces s oud Bromwch [] Later o they were vestgated by Hardy [3] Morcz [7] Morcz ad Rhoades [8] Basarr ad Solaka [] Trathy [] Turkmeoglu [] ad may others We rocure the ollowg sets o double seueces: M u (t): = x w :su x t < m N C (t): = x w : lm x l or somel m m t C 0 (t): = x w : lm x L u (t): = m x w t x m t : C b (t): = C t M t ad C t C t M t u 0b 0 u ; THE -METRIC SPACE OF DEFINED BY MUSIELAK Where t = (t ) s the seuece o strctly ostve real s t or all m ad lm m deotes the lmt the Prgshe s sese I the case t = or all m ; M u (t) C (t) C 0 (t) L u ( t ) C b (t) ad C 0b (t) reduce to the sets Mu C C 0 L u C b ad 30 C 0b resectvely Now we may summarze the kowledge gve some documet related to the double seuece saces Gokha ad Colak [45] have roved that M u (t) ad C (t) C b (t) are comlete araormed saces o double seueces ad gave the duals o the saces M u (t) ad C b (t) Qute recetly her PhD thess Zelter [6] has essetally studed both the theory o toologcal double seuece saces ad the theory o summablty o double seueces Mursalee ad Edely [7] ad Trathy [] have deedetly troduced that statstcal covergece ad Cauchy or double seueces ad gve the relato betwee statstcal coverget ad strogly Cesaro summable double seueces Altay ad Basar [0] have deed the saces BS BS(t) CS CS b CS r ad BV o double seueces cosstg o all double seres whose seuece o artal sums are the saces M u M u (t) C C b C r ad L u resectvely ad also examed some roertes o those seuece saces ad determed the duals o the saces BS BV CS b ad the (v) - duals o the saces CS b ad CS r o double seres Basar ad Server [] have troduced the Baach sace L o double seueces corresodg to the well-kow sace l o sgle seueces ad examed some roertes o the sace L Qute recetly Subramaa ad Msra [] have u o double seueces studed the sace M ad gave some cluso relatos The class o seueces whch are strogly Cesaro summable wth resect to a modulus was troduced by Maddox [6] as a exteso o the

2 Le Scece Joural 03;0(3) deto o strogly Cesaro summable seueces Coor [3] urther exteded ths deto to a deto o strog A summablty wth resect to a modulus where A = (a k ) s a oegatve regular matrx ad establshed some coectos betwee strog A summablty strog A summablty wth resect to a modulus ad A statstcal covergece I [4] the oto o covergece o double seueces was reseted by A Prgshem Also [5]-[6] ad [7] the our dmesoal matrx trasormato Ax a x was studed extesvely k l m kl by Robso ad Hamlto We eed the ollowg eualty the seuel o the aer For a b 0 ad 0 < < we have (a + b) a + b () The double seres x s called m coverget ay oly the double seuece (s ) s m coverget where s x j m A seuece x = (x ) s sad to be double aalytc su x / The vector sace o all double aalytc seueces wll be deoted by A seuece x = (x ) s called double ga seuece / m! x 0 a as m The double ga seueces wll be deoted by Let = {all te seueces} Cosder a double seuece x = (x ) The (m ) th secto x [m] o the seuece s deed by m m x x j 0 or all m ; where deotes the double seuece whose oly o zero term s a the ( j) th lace or each j j! A FK-sace (or a metrc sace) X s sad to have AK roerty ( ) s a Schauder bass or X Or euvaletly x [m] x A FDK-sace s a double seuece sace edowed wth a comlete metrzable; locally covex toology uder whch the coordate mags x = (x k ) (x ) (m ) are also cotuous Let M ad are mutually comlemetary modulus uctos The we have: () For all u y 0 uy M(u) + (y) (Youg s eualty) [See[3]] () () For all u 0 u(u) = M(u) + ((u)) (3) 3 htt://wwwlescecestecom () For all u 0 ad o < < M(u) M(u) (4) Ldestrauss ad Tzarr [5] used the dea o Orlcz ucto to costruct Orlcz seuece sace : xk l 0 M xw M or some k The sace l M wth the orm 0 : xk x M k becomes a Baach sace whch s called a Orlcz M t t the saces seuece sace For l M cocde wth the classcal seuece sace l A seuece = ( ) o modulus ucto s called a Muselak-modulus ucto A seuece g = (g ) deed by g (v) = su v u u : u 0 m s called the comlemetary ucto o a Muselakmodulus ucto For a gve Muselak modulus ucto the Muselak-modulus seuece sace t ad ts sub-sace h are deed as ollows / t xw : I m! x 0 as m / :! 0 h x w I m x as m where I s a covex modular deed by / m I x m! x x x t m I I a We cosder t eued wth the Luxemburg metrc / m d x y su! m u m x m I I m m Let = be a o-decreasg seuece o ostve real s tedg to ty wth = ad The geeralzed de la Vallee-Pouss meas s deed by! t (x) = / m x mi I where I A seuece x = (x ) s sad to be (V ) summable to a umber t 0 as I X s a seuece sace we gve the ollowg detos: () X = the cotuous dual o X; () X aa : a x oreachx X ; m () X a a : a x xx ; (v) M N :su m ; X a a a x oreach x X

3 (v) Le Scece Joural 03;0(3) let X bea FK sace ; the X : X ' ; / (v) aa :su a x X ; or each xx X X X are called (or Kothe Toeltz) dual o X (or geeralzed Kothe Toeltz) dual o X dual o X dual o X resectvely X s deed by Guta ad Kamta [3] It s clear that X X ad X X but X X does ot hold sce the seuece o artal sums o a double coverget seres eed ot to be bouded The oto o derece seuece saces (or sgle seueces) was troduced by Kzmaz as ollows k : k Z x x w x Z or Z = c c 0 ad l where x k = x k x k+ or all k Here c c 0 ad l deote the classes o coverget ull ad bouded scalar valued sgle seueces resectvely The derece seuece sace bv o the classcal sace l s troduced ad studed the case by Basar ad Altay ad the case 0 < < by Altay ad Basar [0] The saces c() c 0 () l () ad bv are Baach saces ormed by / x x su k x ad x bv x k k k Later o the oto was urther vestgated by may others We ow troduce the ollowg derece double seuece sace deed by Z x x w : x Z where Z = ad x = (x x + ) (x m+ x m++ ) = x x + x m+ + x m++ or all m DEFINITION AND PRELIMINARIES Let ad X be a real vector sace o dmeso w where w A real valued ucto d x x d x d x o X satsyg the ollowg our codtos: () d x d x 0 ad oly d (x ) d (x ) are learly de-edet d x d x s varat () () uder ermutato d x d x d x d x (v) d x y x y x y d X x x x dy y y y / (v) htt://wwwlescecestecom or < ; (or) dx x x x d x y x y x y : su dy y y y or x x x X y y y Y s called the roduct metrc o the Cartesa roduct o metrc saces s the orm o the -vector o the orms o the sub-saces A trval examle o roduct metrc o metrc sace s the orm sace s X = eued wth the ollowg Eucldea metrc the roduct sace s the orm: d x d x d x d x d x d x d x d x su det d x su E d x d x d x where x = (x x ) or each = I every Cauchy seuece X coverges to some L X the X s sad to be comlete wth resect to the metrc Ay comlete metrc sace s sad to be Baach metrc sace Let X be a lear metrc sace A ucto w: X s called Para ormed w(x) 0 or all x X; w( x) = w(x) or all x X; 3 w(x + y) w(x) + w(y) or all x y X; 4 I ( ) s a seuece o scalars wth as m ad (x ) s a seuece o vectors wth w(x x) 0 as m the w( x x) 0 as m A araormed w or whch w(x) = 0 mles x = 0 s called total araorm ad the ar (X w) s called a total araormed sace It s well kow that the metrc o ay lear metrc sace s gve by some total araorm (sec [3] Theorem 04 83) Let = ( ) be a Muselak-modulus ucto X d x d x d x be a - metrc sace = ( ) be bouded seuece o strctly ostve real umbers ad u = (u ) be ay seuece o strctly ostve real umbers By S( X) we deote the sace o all seueces deed over X d x d x d x a I the reset aer we dee the ollowg seuece saces: d x d x d x lm V u mi I / u m! x d x d x d x 0 3

4 Le Scece Joural 03;0(3) d x d x d x I V u su mi / m u x d x d x d x I we take (x) = x we get d x d x d x lm V u mi I / u m! x d x d x d x 0 V u d x d x d x su mi I / u x d x d x d x I we take = ( ) = or all m we get V u d x d x d x lm mi I / u m! x d x d x d x 0 d x d x d x su V u m I I / m u x d x d x d x I we take = ( ) = ad u = (u ) = or all m we get V u d x d x d x lm mi I / m! x d x d x d x 0 V u d x d x d x su mi I / x d x d x dx The ollowg eualty wll be used throughout the aer I 0 su = HK = max ( H ) the a b K a b or all m ad a b Also a H max a or all a The ma am o ths aer s to troduce some multler seuece saces deed by a Muselak-modulus ucto over -metrc saces also study some toologcal roertes ad cluso relato o above deed seuece saces 3 MAIN RESULTS 3 Theorem htt://wwwlescecestecom Let = ( ) be a Muselak-modulus ucto = ( ) be aalytc seuece o ostve real umbers ad u = (u ) be ay seuece o strctly ostve real umbers The the saces V u d x d x d x ad d x d x d x saces V u are ler Proo It s route vercato Thereore the roo s omtted 3 Theorem Let = ( ) be a Muselak-modulus uto = ( ) be aalytc seuece o ostve real umbers ad u = (u ) be ay seuece o strctly ostve real umbers The saces V u d x d x d x s a araormed sace wth resect to the araormed deed by g x mi I / H / u m! x d x d x d x where H = max ( su < ) Proo Clearly g(x) 0 or x = (x ) V u d x d x d x Sce (0) = 0 we get g(0) = 0 Coversely suose that g(x) = 0 the mi I / H / u m! x d x d x d x 0 Suose that m! x 0 or each m / Ths mles that u / m x m The It ollows that! 0 or each /! u m x d x d x d x 33

5 Le Scece Joural 03;0(3) mi I / H / u m! x d x d x d x whch s a cotradcto Thereore m x / m x /! 0 or each m ad thus! 0 or each m Let ad m / u m! x d x d x d x mi I / u m! y d x d x d x The by usg Mkowsk s eualty we have mi I u m x y d x d x d x /! / H / H / H mi I mi I u m x d x d x d x /! / u m! y d x d x d x so we have g x y m I I / H / H / u x y d x d x d x m I I / H / H / u m! x d x d x d x m I I / H / u m! y d x d x d x Thereore g(x + y) g(x) + g(y) 34 htt://wwwlescecestecom Fally to rove that the scalar multlcato s cotuous Let be ay comlex umber By deto g x m I I / H / u m! x d x d x d x where t Sce y su max we have su / H g xmax t : mi I / H / u m! x d x d x d x Ths comletes the roo 33 Theorem Let = ( ) be a Muselak-modulus ucto The the ollowg statemets are euvalet ) ) ) V u d x d x d x V u d x d x d x V u d x d x d x V u d x d x d x su u mi I xm d x d x d x / Proo () () s obvous sce V u dx dx dx V u dx dx dx () () Suose V u dx dx dx V u dx dx dx ad let () does ot hold The / su u x d x d x d x mi I m ad thereore there s a seuece ( j ) o ostve tegers such that (3) j mi I j = Dee x = (x ) by m m u d x d x d x j m! m! m! m I ; I 3; j j x x m 0 m I I j The x x V u d x d x d x V u x x d x d x d x cotradcts () Hece but whch

6 Le Scece Joural 03;0(3) () must hold () () Suose x x V u d x d x d x ad x x d x d x d x V u The (3) su / u xm d x d x d x m I I whch cotradcts () Hece () must hold 34 Theorem Let su < The the ollowg statemets are euvalet ) V u d x d x d x d x d x d x V u V u d x d x d x ) ) t > 0 d x d x d x V u / m xm u m m I I d x d x d x 0 Proo () () s obvous () () Suose V u d x d x d x d x d x d x V u ad let () does ot hold The (33) / u xm d x d x d x 0 t 0 mi I We ca choose a dex seuece ( j ) such that j mi I j j m! Dee x = (x ) by m u d x d x d x m! m I ; I 3; j j x x! 0 mi I j Thus by (33) we have x = (x ) V u d x d x d x x x V u d x d x d x whch cotradcts () Hece () must hold 35 htt://wwwlescecestecom () () Let x x V u d x d x d x That s (34) / u m! xm d x d x d x 0 mi I Suose () hold ad x x V u d x d x d x The or some umber є 0 ad dex 0 0 we have є u m x d x d x d x /! 0 m ad coseuetly (34) lm є0 0 m I I whch cotradcts () Hece V u d x d x d x V u d x d x d x Ths comletes the roo 35 Theorem Let = ( ) be a Muselak-modulus ucto Let su < The V u d x d x d x V u d x d x d x hold ad oly (35) / lm u m! xm d x d x d x m I I Proo Suose V u d x d x d x V u d x d x d x ad let (35) does ot hold There s a umber t 0 > 0 ad a dex seuece ( j ) such that (36) mi j I j N u m x d x d x d x Dee x = (x ) by x x /! m t0 mi ; I j 3; j 0 mi I j Thereore x x V u d x d x d x but x x V u d x d x d x Hece (35) must hold Coversely x x V u d x d x d x the or each s ad

7 Le Scece Joural 03;0(3) htt://wwwlescecestecom (37) mi j I j / u xm d x d x d x N uose that x x V u d x d x d x umber 0 0 є we have S The or some / є0 u xm d x d x d x ad hece or m we get 0 m I є N I or some N > 0 whch cotradcts (35) Hece V u d x d x d x V u d x d x d x Ths comletes the roo 36 Theorem Let = ( ) be a Muselak-modulus ucto Let su < The d x d x d x d x d x d x V u V u hold ad oly (38) / lm u m! xm d x d x d x 0 mi I Proo: It s smlar to above Thereore we omt the roo REFERENCES [] M Basarr ad O Solaca O some double seuece saces J Ida Acad Math () (999) [] TJI A Bromwch A troducto to the theory o te seres Macmlla ad Co Ltd New York (965) [3] GH Hardy O the covergece o certa multle seres Proc Camb Phl Soc 9 (97) [4] MA Krasoselsk ad YB Rutck Covex uctos ad Orlcz saces Gorge Nether-lads 96 [5] J Ldestrauss ad L Tzarr O Orlcz seuece saces Israel J Math 0 (97) [6] IJ Maddox Seuece saces deed by a modulus Math Proc Cambrdge Phlos Soc 00() (986) 6-66 [7] F Morcz Extetos o the saces c ad c 0 rom sgle to double seueces Acta Math Hug 57(-) (99) [8] F Morcz ad BE Rhoades Almost covergece o double seueces ad strog regularty o summablty matrces Math Proc Camb Phl Soc 04 (988) [9] H Nakao Cocave modulars J Math Soc Jaa 5 (953) 9-49 [0] WH Ruckle FK saces whch the seuece o coordate vectors s bouded Caad J Math 5 (973) [] BC Trathy O statstcally coverget double seueces Tamkag J Math 34(3) (003) 3-37 [] A Turkmeoglu Matrx trasormato betwee some classes o double seueces J Ist Math Com Sc Math Ser () (999) 3-3 [3] PK Kamtha ad M Guta Seuece saces ad seres Lecture otes Pure ad Aled Mathematcs 65 Marcel Dekker Ic New York 98 [4] A Gokha ad R Colak The double P PB seuece saces c a ad c Al Math Comut 57() (004) [5] a Gokha ad R Colak Double seuece saces l bd 60() (005) [6] M Zeltser Ivestgato o Double Seuece Saces by Sot ad Hard Aaltcal Methods Dssertatoes Mathematcae Uverstats Tartuess 5 Tartu Uversty Press Uv o Tartu Faculty o Mathematcs ad Comuter Scece Tartu 00 [7] M Mursalee ad OHH Edely Statstcal covergece o double seueces J Math Aal Al 88() (003) 3-3 [8] M Mursalee Almost strogly regular matrces ad a core theorem or double seueces J Math Aal Al 93() (004) [9] M Mursalee ad OHH Edely Almost covergece ad a core theorem or double seueces J Math Aal Al 93() (004) [0] B Altay ad F Basar Some ew saces o double seueces J Math Aal Al 309() (005) [] F Basar ad Y Sever The sace L o double seueces Math J Okayama Uv 5 (009) 49-57

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