More Properties of Semi-Linear Uniform Spaces

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1 pplied Mathematis Published Olie Jue 25 i SiRes Moe Popeties of Semi-Liea Uifom Spaes Depatmet of Mathematis l-balqa pplied Uivesity lsalt Joda suadhihi@baueduo suadalhihi@yahooom Reeived 22 pil 25; aepted 7 Jue 25; published Jue 25 Copyight 25 by autho ad Sietifi Reseah Publishig I This wok is liesed ude the Ceative Commos ttibutio Iteatioal Liese (CC BY) bstat I this pape we shall geealize the defiitio give i [] fo Lipshitz oditio ad otatios fo futios o a o-metizable spae besides we shall give moe popeties of semi-liea uifom spaes Keywods Lipshitz Coditio Cotatios Not Metizable Spaes Semi-Liea Spaes Uifom Spaes Fixed Poit Itodutio The otio of uifomity has bee ivestigated by seveal mathematiia as Weil [2]-[4] Cohe [5] [6] ad Gaves [7] The theoy of uifom spaes was give by Bubaki i [8] lso Wiels i his booklet [4] defied uifomly otiuous mappig Fo moe ifomatio about Uifom spaes oe my efe to [9] I 29 Tallafha ad Khalil R [] defied a ew type of uifom spaes amely semi-liea uifom spaes ad they gave example of semi-liea spae whih was ot metizable lso they defied a set valued map ρ o by whih they studied some ases of best appoximatio i suh spaes Moe peisely they gave the followig Let ( ) be a semi-liea uifom spae; E is poximial if fo ay x thee exists e E suh that ρ( xe ) = ρ( xe ) They asked that must evey ompat is poximial they gave the aswe fo the ases i) E is fiiate; ii) If x oveges to x the { xx x2 } is poximial I [] Tallafha defied aothe set valued map δ o ad gave some popeties of semi-liea uifom spaes usig the maps ρ ad δ lso i [] [2] Tallafha defied Lipshitz oditio ad otatios fo futios o semi-liea uifom spaes whih eabled us to study fixed poit fo suh futios How to ite this pape: lhihi S (25) Moe Popeties of Semi-Liea Uifom Spaes pplied Mathematis

2 Lipshitz oditio ad otatios ae usually disussed i meti ad omed spaes ad eve bee studied i othe weake spaes We believe that the stutue of semi-liea uifom spaes is vey ih ad all the kow esults o fixed poit theoy a be geealized The obet of this pape is to geealize the defiitio of Lipshitz oditio ad otatio mappig o semi-liea uifom spaes give by Tallafha [2] lso we shall give a ew topopologial popeties ad moe popeties of semi-liea uifom spaes 2 Semi-Liea Uifom Spae Let be a set ad D be a olletio of subsets of suh that eah elemet V of D otais the diagoal = {( xx ) : x } ad V= V = {( yx ) : V} fo all V D (symmeti) D is alled the family of all etouages of the diagoal Defiitio [] Let be a sub olletio of D the pai ( ) is alled a semi-liea uifom spae if i) is a hai ii) Fo evey V thee exists U suh that U U V iii) V = iv) V = Defiitio 2 [] Let ( ) be a semi-liea uifom spae fo let { V : ( ) V} is defied by ρ ( ) = The the set valued map ρ o Clealy fo all we have ρ = ρ( yx ) ad ρ { } Let = V \ = = : V fom ow o we shall deote \ by Defiitio 3 [] Let ( ) be a semi-liea uifom spae The the set valued map δ o V if x y defied by δ ( ) = φ if x = y The followig esults ae give i [2] be a semi-liea uifom spae ad Λ is a sub olletio of the Popositio Let if a oly if thee exist U suh that U V Coollay Let ( ) be a semi-liea uifom spae If ρ ) Thee exist U suh that U ρ 2) U δ the is V Let V D the family of all etouages of the diagoal the fo all by V we mea V V V -times ad V = so fo all V D V V ad V D fo all be a semi-liea uifom spae If x y δ x y = V Popositio 2 Let Popositio 3 Let V = V Questio Does the be a semi-liea uifom spae If Λ is a sub olletio of the V ρ x y? 3 Topologial Popeties of Semi-Liea Uifom Spaes Defiitio 4 [3] Fo x ad V The ope ball of ete x ad adius V is defied by B xv y: x y V B xv = y: ρ x y V = { } equivaletly { } 7

3 Clealy if y B( xv ) the thee is a W suh that B( yw ) B( xv ) So is a base fo some topology o This topology is deoted by τ τ = G : fo all x G thee exist suh that B xv G Moe pesily { } { B xv x } I [] it is show that τ is Hausdof so if is fiite the we have the diseet topology theefoe iteestig examples ae whe is ifiite lso if is ifiite the should be ifiite othe wise whih implies also that the topology is the disete topology Popositio 4 Let ( ) be a semi-liea uifom spae ad a subolletio of satisfies i) Fo evey U thee is a V suh that V U ii) V = iii) V = The ( ) is a semi-liea uifom spae ad τ = τ Poof Sie is a subolletio of the is a subolletio of D ad i) iii) iv) i defiitio () ae satisfied Now fo U V the exist U suh that U ( U V ) so thee is a W suh that W U ( U V ) So ( ) is a semi-liea uifom spae Now τ τ is lea Let O be a oempty ope set i τ the if x O the exist U suh that B( xu ) O Let V suh that V U so B( xv ) O hee τ τ Theoem Let ( ) be a semi-liea uifom spae ad τ the topology o idused by the i) a be oside as asubset of τ τ ii) Foe all δ τ τ = U : U whee U is the iteio of U with espet to the topology τ τ o Poof i) Let { } is a semi-liea uifom spae idusig the topology τ o Sie we deal with the toplogy τ we may eplae with so fo all V V = U fo some U hee V τ τ ii) Is lea by defiitio of δ I [] Tallafha gave some impotat popeties of semi-liea uifom spaes usig the set valued map ρ ad δ Now we shall give moe popeties of semi-liea uifom spaes By Popositio 22 4 Moe Popeties of Semi-Liea Uifom Spaes Let ( ) be a semi-liea uifom spae the is a hai so thee exist a well ode set that = { U R } Fo x y the the exist U suh that U R suh that U This implies if U the U U Hee S = R : U So δ ( ) U ρ( ) R suh That is thee exist { } is bouded above by By Zo s Lemma S has a maximal elemet This opleet the poof of the followig lemma Lemma Let ( ) be a semi-liea uifom spae the = { U R } whee R is a well ode set the fo x y the exist R suh that δ U ρ Remembe that fo all V D the family of all etouages of the diagoal V satisfies the followig ie popeties V V ad V D fo all Let Λ δ ad ρ be a subolletios of D defied by δ = { δ : x y} ρ = { ρ : x y} ad Λ = δ ρ Fo all Λ by Co- ollay 6 thee exist U suh that U So we a defie fo a elemet Λ by Defiitio 5 Fo ad Λ Defie by = { U : U } Clealy U = ad D fo all Λ But eed ot be a elemet i evi if 7

4 But we have Lemma 2 Let ( ) be a semi-liea uifom spae = { U R } R is a well ode set ad Λ The thee exist R suh that U ad if R satisfied the U \ φ Poof Fo Λ thee exist R suh that U so fo evey U satisfies U we have U U Hee T = { R : U } is bouded above by Hee be the maximal elemet of T Theoem 2 Let Λ ad σ a suboletio of Λ Fo we have i) ii) If B satisfies B the iii) iv) v) = vi) σ σ σ σ B Poof i) Let ( s s ) the thee exist s s suh that ( s s ) = So thee exist U U2 U i suh that ( s s) U ad U = Sie is a hai thee exist { } suh that U U fo all = So ( s ) s U ii) ad iv) ae lea by defiitio of iii) { U :( ) U } { U : U } V = = v) fo all σ so σ U U Covesly Let ( ) ( ) σ σ s s the fo all σ thee exist s s suh that σ s s = So thee exist U U2 U s s U ad U = Sie is a hai thee exist { } suh that U U fo all = So s s U Let ad α D α Λ eplaig by o by α we have i suh that α Coollay 2 Fo whee ( ) is a semi-liea uifom spaes we have i) δ ii) ρ V if x y = V = φ if x = V V Coollay 3 Let ad α D α Λ the i) α α y 72

5 ii) If B V satisfies B { α : α Λ} iii) iv) α α α α v) If V D satisfies B the the B { α α Λ} α B α : vi) α α lso we have the follwig oollay Coollay 4 Fo whee ( ) is a semi-liea uifom spaes the ρ ρ Coollay 5 Fo V if V the V is the lagest elemet i satisfies V V lso by Defiitio 36 fo x y whee ( ) is a semi-liea uifom spaes we have i) ρ = V: V V : V if x y ii) δ = φ if x = y Popositio 5 Let x y be ay distit poits i semi-liea uifom spaes ( ) The ( ) δ δ δ Poof Let ( s s ) δ the ( s s ) V : V the thee exist s s suh that ( s s ) V : V = So thee exist 2 U U U suh that ( s s) U but is a hai implies the existee of { } suh that U U fo all = So ( s s) U U δ O the othe had by popositio 7 δ ( x y) = V So fo all V V δ ( x y) whih implies V ( δ ) Defiitio 6 Fo α α Λ ad defie i) = = m ad the geatest ommo deviso of m is m m : = : ii) { α } { α } iii) { α : } = { α : } Defiitio 7 Let x y be ay poits i semi-liea uifom spaes Fo δ = δ m whee = m m ad the geatest ommo deviso of m is Usig the a bove defiitio ad Popositio 7 we have Popositio 6 Let V ad 2 < 2 the V V 2 73

6 Popositio 7 Let x y If 2 2 the δ 2δ Defiitio 8 Let be a ieasig sequee of positive atioals If the ( ) lim δ be ay poits i semi-liea uifom spaes is defied by δ ( ) = Popositio 8 Let ( ) δ ( ) δ ( ) lim ad a ieasi sequee of positive atioals If the Poof It is a immediate osequee of Popositio (7) ad Defiitio (8) 5 Cotatios I [] the defiitios of oveges ad Cauhy ae give Now we shall disuss some topologial popeties of semi-liea uifom spaes Sie the semi-liea uifom spae is a topologial spae the the otiuity of a futio is as i topology The oept of uifom otiuity is give by Wiels [4] so we have: Defiitio 9 [4] Let f :( ) ( Y Y ) the f is uifomly otiuous if U Y V suh V f x f y U that if the Clealy usig ou otatio we have: Popositio 9 Let f :( ) ( Y Y ) The f is uifomly otiuous if ad oly if U Y V suh that fo all if ρ V the ρy ( f ( x) f ( y) ) U The followig popositio shows that we may eplae ρ by δ i Popositio 22 Popositio [2] Let f :( ) ( Y Y ) The f is uifomly otiuous if ad oly if U Y V suh that fo all if δ V the δy ( f ( x) f ( y) ) U I [] Tallafha gave a example of a spae whih was the semi-liea uifom spae but ot metizable Till ow to defie a futio f that satisfies Lipshitz oditio o to be a otatio it should be defied o a meti spae to aothe meti spae The mai idea of this pape is to defie suh oepts without meti spaes ad we ust eed a semi-liea uifom spae whih is weake as we metioed befoe Defiitio [] Let f :( ) ( ) the f satisfied Lipshitz oditio if thee exist m suh mδ f x f y δ Moeove if m> the we all f a otatio that ( ) Now we shall give a ew defiitio of Lipshitz oditio ad otatio alled -Lipshitz oditio ad -otatio Defiitio Let f :( ) ( ) the f satisfied -Lipshitz oditio if thee exists suh δ f x f y δ Moeove if < the we all f a -otatio that ( ) Questio Let ( ) be semi-liea uifom spaes ad f ( ) ( ) : what is the elatio betwee Lipshitz oditio ad -Lipshitz oditio otatio ad -otatio Remak [2] Let ( ) be semi-liea uifom spaes the if the the topology idused by ( ) is the desete topology whih is metizable Theefoe we a asumme that Questio [] [2] Let ( ) be a omplete semi-liea uifom spae d f :( ) ( ) be a otatio Does f has a uique fixed poit? Questio Let ( ) be a omplete semi-liea uifom spae d f :( ) ( ) be -otatio Does f has a uique fixed poit? Refeees [] Tallafha (23) Ope Poblems i Semi-Liea Uifom Spaes Joual of pplied Futioal alysis [2] Weil (936) Les eouvemets des espaes topologiques: Espaes omplete espaes biompat Comptes Redus de l adémie des Siees [3] Weil (937) Su les espaes a stutue uifome et su la topologie geeale tuaial Siee Idia 55 Pais [4] Weil (938) Su les espaes à stutue uifome et su la topologi gééale Hema Pais [5] Cohe LW (937) Uifomity Popeties i a Topologial Spae Satisfyig the Fist Deumeability Postulate Duke Mathematial Joual

7 [6] Cohe LW (939) O Imbeddig a Spae i a Complete Spae Duke Mathematial Joual [7] Gaves LM (937) O the Completig of a Housdoff Spae als of Mathematis [8] Boubaki N (94) Topologie Gééale (Geeal Topology) Pais [9] James IM (987) Topologial ad Uifom Spaes Udegaduate Texts i Mathematis Spige-Velag Beli [] Tallafha ad Khalil R (29) Best ppoximatio i Uifomity Type Spaes Euopea Joual of Pue ad pplied Mathematis [] Tallafha (2) Some Popeties of Semi-Liea Uifom Spaes Boleti da Soiedade Paaaese de Matematia [2] Tallafha (25) Fixed Poit i a No-Metizable Spaes To ppea [3] Egelkig R (968) Outlie of Geeal Topology Noth-Holad mstedam 75