Suama Uer Bou formar Oeraors Uer Bou For Mar Oeraors O Some Sequece Saces Suama Dearme of Mahemacs Gaah Maa Uersy Yogyaara 558 INDONESIA Emal: suama@ugmac masomo@yahoocom Isar D alam aer aa susa masalah ecara baas aas a orma oeraor mar Hausorff aa beberaa ruag barsa Kaa uc: orma-f fugs- mars Hausorff baas aas Absrac I hs aer e cosere he roblem of fg he uer bou a he orm of he Hausorff mar oeraor o some sequece saces Keyors: F-orm -fuco Hausorff mar uer bou Prelmares a Some Basc Noos Oeraor heory lays a mora role boh ure a ale mahemacs Therefore alays recees a lo of aeo from mahemacas from hose areas I hs aer e scuss abou he orm of a cera mar oeraor o a cera sequece sace The ey refereces are Jameso a Lasharour [] [] Lasharour [][4][5] a Pecar eal [] I hs seco e ge some basc oos As usual R a N eoe he real a aural umbers sysem resecely R + eoes he colleco of all ose real umbers The colleco of all sequeces R ll be eoe by S Le X S be a lear sace oer R A fuco : X R s calle a F-orm f sasfes for eery X y y for eery y X a f X s a sequece such ha lm for some X a a s a sequece of real umbers hch coerges o some a R he lm a a 54
Berala MIPA 3 Jauar 3 The lear sace X eque h he F-orm eoe X s calle a F- orme sace Whe he F-orm has bee elcly o e re X sea of X A F-orme sace s sa o be comlee f eery auchy sequece he sace s coerge A comlee F-orme sace s calle a Frèche sace or shorly a F-sace A fuco : R R s calle a -fuco f sasfes for eery R s creasg o R + s couous o R a lm A -fuco s sa o sasfy a -coo f here ess a real umber M such ha M for eery For ay sequece of ose umbers a -fuco ha sasfes -coo e efe { } : l S l { } S : We obsere ha l a l are comlee F-orm saces h resec o a resecely here a I case e re l sea of l Le be a ecreasg ose sequece of real umbers such ha lm a We efe * { }: here * s a ecreasg sequece hch ca be fou by rearragg I ca be sho ha s a sace of all sequeces h fely o-zero elemes Furher s a F-orme sace h resec o * 55
Suama Uer Bou formar Oeraors Mar Oeraors Le a be a sequece of real umbers h a For ay N {} e efe he oeraor as follos a a a a a a a a 34 Furher he mar H h here a h s calle he Hausorff mar Le be a robably measure o [ ] For ay N e efe he sequece a by a 3 he e ge he Hausorff mar h H h h The follogs are some of Hausorff marces: H here log H H here a 3 G H here here s ay real umber The marces H a G are calle a esaro Holer a Gamma mar resecely Le a mar oeraor A : l l A y y be sequeces of ose umbers We coser he y a 56
Berala MIPA 3 Jauar 3 The orm of A s ge by A su A : l We obsere he follog heorem Theorem Le be a ecreasg sequece of ose real umbers If he Hausorff mar oeraor H mas he sace l o self he H su Proof: For smlcy e re H su H H Tae ay l he su As a sragh cosequece e he hae he follog corollary orollary If he Hausorff mar oeraor H mas he sace l o self he H I case he -fuco s of he form equales for he Hausorff mar oeraor H he e ge Theorem 3 Le a be ecreasg sequeces of ose umbers h If he Hausorff mar oeraor H mas l o l he f Proof: We re H su H H for he smlcy Le l he 57
Suama Uer Bou formar Oeraors 58 H su su These roe he rgh ha se of he equaly Furher e are gog o roe he lef ha se of he equaly Le a I s clear ha l Sce for eery N he l Tae N a such ha a N N he N H Hece N H Furher
Berala MIPA 3 Jauar 3 59 N H H These mles H f If he Theorem 4 e ae for eery he e ge he follog corollares orollary 5 If he Hausorff mar H mas he sace l o self he H orollary 6 Le q be such ha q If he marces a G H ma he sace l o self he log G H q q f
Suama Uer Bou formar Oeraors Le be a mooo ecreasg sequece of ose real umbers such ha lm a We efe * { }: * here s a mooo ecreasg sequece fou by rearragg he sequece I ca be roe ha s a sace ha s members are all fe sequeces Furher s a F-orme sace h resec o * Lemma 7 Le A a be he oeraor o ha sasfes a a for eery a M K m a a m elemes resecely for eery subse M K N ha cosss of The for eery o egae eleme e hae * A A Proof: See Lasharour R [] Lemma 8 Le a a for eery a If for eery A a he he follog saemes are equale a y y heeer b A be a oeraor from o self such ha a r a s a sequece such ha r r for eery 6
Berala MIPA 3 Jauar 3 Proof: a b : Le be a arbrary he for some h N If e ha s a sequece h he -coorae s equal o a he ohers are he e Furher by he hyohess e hae y y a a b a : If he for some N For ay e hae y Hece a r r y y r r r r r r 3 heeer Le H be a Hausorff mar such ha a a M K m for ay subse M K N hch coss of m elemes resecely Follog Lemma 7 a Lemma 8 he for ay o egae ecreasg sequece e hae H H Furher by usg Theorem 4 e hae he follog heorems Theorem 9 Le a H be a Hausorff mar oeraor such ha M K a m a for ay subses elemes resecely The H mas o self a H M K N hch coss of m Theorem Le A a be a mar ha sasfes he coos a Lemma 7 a be coerge If s a sequece such ha a 6
Suama Uer Bou formar Oeraors here S su V S s s a a V he A s a boue lear oeraor from o a A S su V Proof: Le be sequece such ha If M A Sce he Ths mles A M S M V a A M V s S su V he Furher by leg a for eery N he e hae V a A S So A M 6
Berala MIPA 3 Jauar 3 3 oclug Remars I hs aer e hae succesfully cosruce he sequece saces l a hch s a F-sace resecely Furher s a sequece sace here all of s elemes are fe sequeces By resrcg he fuco of he form he e ca formulae he uer bou a orm of cera mar oeraor o a The ors ll be coue for mar oeraors l ac o l a 4 Acolegeme Ths aer s a ar of he 7 research gra acy fue by he Dearme of Mahemacs Gaah Maa Uersy uer a corac umber /JO8/PL6//7 Therefore he auhor oul le o ha he Dearme of Mahemacs Gaaah Maa Uersy Refereces Jameso GJO a Lasharour R Loer bous of oeraors o eghe l saces a Lorez sequece saces Glasgo Mah J 4 3 Jameso GJO a Lasharour R Norm of cera oeraors o eghe l saces a Lorez sequece saces J Iequaly Pure Al Mah 3 7 Lasharour R Oeraors o Lorez sequece sace II WSEAS Tras O Mah 6 Lasharour R 4 Weghe meas mar o eghe sequece sace WSEAS Tras O Mah 34 789 793 Lasharour R 5 Trasose of eghe meas oeraors o eghe sequece sace WSEAS Tras O Mah 44 38 385 Pecar J Perc I a Ro R O bous for eghe orms for marces a egral oeraors Lear Algebra a Al 36 35 63