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1 NO Ivrt S-bst Copproxmto -ormd Spcs Slw Slm bd Dprtmt of Mthmtcs Collg of ducto For Pur scc, Ib l-hthm, Uvrsty of Bghdd l Musddk Dlph Dprtmt of Mthmtcs,Collg of Bsc ducto, Uvrsty of Ms BSTRCT Th purpos of ths ppr s to dscuss vrt S-bst copproxmto cocpt for o-mpty compct strshpd subst of lr -ormd spcs, wth rspct to two lr cotrctv commutg oprtors. Kywords: -ormd spc, S- bst copproxmto, commo fxd pot, vrt pproxmto. ثبات افضل اقت ارب مشترك S المعيارية الثنائية في الفضاءات جامعة بغداد, أ.م.د سلوى سلمان عبد كلية التربية للعلوم الصرفة ابن الهيثم, قسم الرياضيات م.م علي مصدق دلفي قسم الرياضيات جامعة ميسان, كلية التربية االساسية, المستخلص ثابت مشترك S في المجموعات الجزئية الهدف من هذا البحث مناقشة مفهوم افضل اقت ارب الخطية االنكماشية المؤث ارت الى المرصوصة النجمية للفضاءات المعيارية الثنائية, بالنسبة المتبادلة.

2 NO INTRODUCTION Th cocpts of lr -ormd spcs wr tlly troducd by Wht[]. Th th subjct hs got grt ttto of rsrchd my rsrchrs s [-3]. Ths spc hs subsqutly b studd by showg th xstc of fxd pot of cotrctv, oxpsv, symptotclly oxpsv mppgs. s othr spcs, th fxd pot thory hs b dvlopd such spc lso. Fxd pot thorms hv b usd t my plcs pproxmto thory, th ltrtur, most studs o fxd pot thorm to pproxmto thory cosdr wth ormd lr spcs lk [ ]. O th othr hd, som mthmtcs ppld th d of bst pproxmto -ormd spcs such s [8-9-0-]. Th cocpts of bst copproxmto r othr kd of pproxmto thory ws frst troducd by Frchtt d Fur [], to study som chrctrstc proprts of rl Hlbrt spcs. Subsqutly, Vjyrgv[3], dvlopd ths problm d dlt wth som fudmtl proprts of th st of strogly uqu bst copproxmto lr -ormd spcs. Dlph [4] rctly, troducd d study w cocpt mly, S-bst copproxmto lr -ormd spcs, whr troducd th otos S-bst copproxmto d S-orthogolty - ormd spcs d th, som chrctrzto d mportt thorm bout xstc of S-bst copproxmto covx subst of -ormd lr spcs wr provd. For ths ppr, w gv two rsults bout vrt S-bst copproxmto lr -ormd spcs, for ths purpos w rcll som dfto d fcts s follows: Dfto. [9]: Lt X b lr spc ovr rl umbr wth dmso d, whr d d lt.,. : b o-gtv rl vlud fucto o X : X X stsfyg th followg proprts for ll, b, c - x, y 0 x, y r lrly dpdt - x, y y, x 3- x, y x, y whr R 4- x, y z x, y x, z Th.,. s clld -orm d th pr X,.,. lr spc X s clld lr -ormd spc. stdrd xmpl of -ormd spc s R quppd wth th followg -orm, x, y : th r of th trgl hvg vrtcs o, x d y.

3 NO Obsrv tht y -ormd spc X,.,. w hv x, y 0 d x, y x x, y for ll x, y X d R. lso, f x, y, z r lrly dpdt ths hpps for stc, wh d th x, y z x, y x, z [6]. Evry subspc of - ormd spcs X s covx. I prtculr vry -ormd spcs X s covx. Sc, f s subspc of X d, th, for ll sclrs,, thus prtculr f w put d, for ll [0, ], th w hv b, d so s covx. Dfto.[3]: Lt b subst of rl lr -ormd spc X d x X,th s sd to b bst copproxmto to x X from th lmt of, f for vry, zx, z, z X / V x,, whr V s th subspc grtd by x d. Th st of ll lmts of bst copproxmto to x X from wth rspct to z s dotd by R whr R st. { G, z x, z } d t s clld coproxml Suppos tht X,.,. s -ormd spc, wth dmso d, whr d, d z,..., z } b ts bss. w strt wth : { d Dfto.3[4]: lt b o-mpty subst of lr -ormd spcs X. lmt s sd to b S-bst copproxmto of d x X from f R z. Th st of ll lmts of S-bst G copproxmto of X from s dotd by SR, ths ms SR d R z. lso f ch x X hs t lst rspctvly xctly o S-bst copproxmto, th SR s clld S- bst coproxmlrspctvly S-coChbyshv st. Rmrk.4 [4]: If ch x X hs t lst rspctvly xctly o S- bst copproxmto, th SR s clld S-bst coproxmlrspctvly S-coChbyshv st. 3

4 NO Exmpl.5[4]: Suppos X R wth usul bss, } d th { x x orm x, z xz xz,lt {, 0} z z b subst of X, to prov tht 0, R,,, SR,, w hv,, x, 0,,, for y x 0 x 0 0 0,,,,,,, 0 0 d so 0, R,,, w hv th sm rsult f rplc by,, g x, 0 x x 0 0 0,,,,,, d so 0, R,,,thrfor, 0, R,,,, d so 0, SR,,,, d so SR s S-coChbyshv st. Dfto.6[8]: lt X b o-mpty st d T : X X slf mp. W sy tht x X s fxd pot of T, f T x x d dot by FT th st of ll fxd pot of T. Dfto.7[5]: lt X b o-mpty st d o X. W sy tht T x S x x. 0 0, T, S r slf mppg x X s commo fxd pot of T d S, f Dfto.8[8]: lt X b lr -ormd spcs th th mppg T : X X s sd to b cotrctv f T stsfs T x T z x y, z for ll x, y, z X. 4

5 NO Dfto.9 [5]: lt X b o-mpty st d T, S r slf mppg o X. W sy tht T, S commutg mppg o X, f TS x ST x for ch x X. - MIN RESULTS I ths scto, w troduc som fxd pot thorms d ts pplcto to vrt S-bst copproxmto -ormd spcs. Thorm.:lt T b lr cotrctv oprtor o -ormd lr spc X, lt b T-vrt subst of X d T-vrt pot. If th st of S-bst copproxmto to x X s o-mpty, compct, d strshpd, th F T. SR Proof: lt Q b th st of S-bst copproxmto to th th for ll,..., d Q, x X, th T : Q Q, sc f, z T T, z, z x z, th T Q T, Tk q Q such tht q h Q for ll h Q d 0, lt k, 0 k, b rl umbr such tht k s. Th df T, : Q Q by T k T k q for ll h Q, sc T mps Q to Q, lso T mps Q to Q for ch., lso for ll,..., d, w hv T T z k T T z k h y, z h y, z for ll h, y Q, h y. Th, sc Q s compct d uqu fxd pot, sy h for ch. thus, h hs covrgt subsquc T cotrctv, th T hs T h h. sc Q s compct, h covrgg to h. W clm tht k q k T h, by th followg T h. Now sc h T h qulty T h, z T T h, z h h, z d by tkg lmt s, k, w hv T h h th T h T st s h cotrctv thus h s T-vrt. To llustrt thorm., w gv th followg xmpl: Exmpl.: lt T b lr cotus cotrctv cotrctv oprtor o X R wth usul bss, } d th orm x { x x d, } b, z xz xz z z { 5

6 NO T-vrt subst of X d T-vrt pot, th wth smpl,, clculus c b show tht,, 0, R,, SR,,, d so SR s o-mpty d compct, lt Q SR,,, th T : Q Q, sc 0, SR,,,,, th for, w hv, T 0,,, T0, T,, 0,,,,,, th for ll [0, ], th Q s T 0, Q, sc 0,, Q strshpd, lt k, 0 k, b rl umbr such tht k s Th df T. : Q Q by T 0, k T0, k,, sc T mps Q to Q, lso T mps Q to Q for ch., lso for,, w hv T 0, T,, k T0, T,, k 0,,, 0,,,, d so T cotrctv oprtor o Q d Q s compct, th T hs uqu fxd pot, thus, T 0, 0, for ch. Now, by th qulty T 0, 0,, z T0, T 0,, T 0, 0,,, d by tkg lmt s, k, w hv T 0, 0,, thus 0, s T-vrt. Now, w xtd Thorm. for pr of cotrctv oprtor: Thorm.3:lt T, I two lr cotrctv commutg slf oprtors o -ormd lr spc X, lt b subst of X such tht T :, d F T F I. Furthr, T d I stsfy T x T z I x I z for ll, d lt I b lr, cotuous, o, d x, y SR I T x T I y for ll x, y SR to SR, f th st of S-bst copproxmto x X s o-mpty, compct, d strshpd, wth rspct to pot q FI d f I SR SR, th SR F T F I. Proof: lt Q b th st of S-bst copproxmto to sc f From Q x X, th T : Q Q d hc I Q Q. Furthr, y sc T T x T z I x I z t follows tht. T, z T T, z I I, z x, z, th T Q,thus T mps Q to Q. lt k } { b squc of rl umbrs such tht 0 k, b rl umbr such tht k s. Df T : Q Q by T k T k q for ll h Q, sc T mps Q to Q, lso T mps Q to Q for ch., lso for ll,..., d, w hv T T z k T T z k h y, z h y, z for ll, 6

7 NO h, y Q, h y. Th, sc Q s compct d uqu fxd pot, sy h for ch. thus, h hs covrgt subsquc T cotrctv, th T hs T h h. sc Q s compct, h covrgg to h. W clm tht k q k T h, by th followg T h. Now sc h T h qulty T h, z T T h, z h h, z d by tkg lmt s, k, w hv T h h th T h T st s h cotrctv thus h Q FT. Now, sc I s lr d commuts wth T o Q, w hv tht T I k T I k I q k I T k I q I kt k q I T for ll h Q. Thus I commuts wth T o Q for ch, Q Q I Q. Furthr, th for ll,..., d, w hv T T T z k T T z k I I z I I z whvr I I sc Q s compct d I s cotuous, w dduc tht F T F I { h } for ch. Furthr, th cotuty of I mpls tht I Ilm h lm I h lm h h, Q F T F I..., h Q FI, d so To llustrt thorm.3, w gv th followg xmpl: Exmpl.4: lt T b lr cotrctv oprtor o X R wth usul x x bss {, } d th orm x, z xz xz d z z, } b T-vrt subst of X d T- { vrt pot, th wth smpl clculus c b show tht,, 0, R, 3,3, SR, 3,3,, d so SR s ompty d compct, lt Q SR 3,3,, th T Q Q sc 0, SR 3,3,, th for, w hv,,, :, I : Q Q T 0, T,, T0,,, I0, I,, 0,,,, 3,3,, th T 0, Q, sc 0,, Q for ll [0,], th Q s strshpd, lt, 0 k, k b rl umbr such tht k s. Th df : Q Q by T T 0, kt0, k و, sc T mps Q to Q, lso T mps Q 7

8 NO to Q for ch., lso for,, w hv T 0, T, و k T0, T,, k 0,,, 0,,, d so T cotrctv oprtor o Q d Q s compct, th T hs uqu fxd pot, thus, T 0, 0, for ch. Now, T 0, k T0, k,, tkg lmt s, k, w hv T 0, T0,, sc T s cotrctv w hv, T 0, 0,, T0, T 0,, T 0, 0,, s, k w hv T 0, 0, th Q F T. Now, sc I s lr d commuts wth T o Q, w hv T I0, k T I0, k I, = k I T0, k I, I T 0, Thus I commuts wth T for ch. o Q. d so for,, w hv T 0, T,, k T0, T,, k I0, I,, I0, I,,, th, sc Q s compct, d I s cotuous, th T I0, I T 0, 0, by tkg lmt s, k, w hv F T F I. Th cotuty of I o Q mpls tht I 0, 0,, I0, T 0,, T 0, 0,, by tkg lmt s, k,w hv Q F I, d so Q F T F I., 3- Cocluso I ths ppr two rsults hv b provd bout S-bst copproxmto s fxd pot -ormd spcs. Ths ppr c b xtdd to othr sttg, such s S-bst copproxmto s commo fxd pot -ormd spcs d S-bst copproxmto s fxd pot -ormd spcs wth dffrt kds of oprtors. 8

9 NO Rfrcs [] Wht,., " -Bch Spcs", Mthmtsch Nchrcht 4, [] Dm, S., Ghlr, S., d Wht,., "Strctly Covx -ormd Spc", Mthmtsch Nchrcht 59, [3] Frs, R.W., d Cho,Y.J., "Gomtry of Lr -ormd spcs", Nov Scc Puplshrs, Ic., Nw York 00. [4] Sch, S.P, " pplcto of fxd pot thorm to pproxmto thory", Jourl of pproxmto thory 5, [5] Mukhrj, R.N. d Som, T., " ot o pplcto of fxd pot thorm pproxmto thory ", Id J. Pur ppl. Mth. 63, [6] Shb, S.. d Kh, M. S., " Rsult Bst pproxmto thory", Jourl of pproxmto thory 55, [7] Hbk, L., " fxd pot thorm d vrt pproxmtos", Jourl of pproxmto thory 56, [8] Kr, M. d Kzltuc, H., " O cotrcto mppg d fxd pot thorms -ormd spcs", mthmtcs, Sop trsctos o ppld [9]. Khors d M. brshm Moghddm,, '' Bst pproxmto I Probblstc -Normd Spcs '', Nov Sd J. Mth.40, [0]Y. Domc d M. Mrud, '' Bst pproxmto Uformly Covx -Normd Spc '', It. Jourl of Mth. lyss 6, [] Rfh Sjd bd l, "Fxd Pots d Bst pproxmtos - Normd Spcs", M.Sc., Collg of ducto For Pur scc,ib l- Hthm, Uvrsty of Bghdd, Irq03. 9

10 NO [] C. Frchtt, M. Fur," Som chrctrstc proprts of rl Hlbrt spcs", Rv. Roum Mth Purs ppl. 7, [3] Vjyrgv, R., "Strogly Uqu Bst Copproxmto Lr -ormd Spcs", Itrtol Jourl of Egrg Rsrch d Tchology, [4] Dlph,.M, "Som Rsults o S-bst Copproxmto Lr - Normd Spcs", l-mustsry Jourl of Scc 7, [5] Chdok, S. d Nrg, T., "Commo Fxd Pots d Ivrt pproxmtos for Cq-commutg Grlzd oxpsv mppgs ", Ir Jourl of Mthmtcl Scc d Iformtcs 7,