BULLETIN of te MLYSİN MTHEMTİCL SCİENCES SOCİETY Bull. Malays. Mat. Sci. Soc. (2) 27 (2004), 27 223 Lacuay lost Suability i Cetai Liea Topological Spaces BÜNYMIN YDIN Cuuiyet Uivesity, Facutly of Educatio, Sivas-Tuey e-ail: baydi@cuuiyet.edu.t bstact. I tis pape, te cocept of lacuay alost suability of sequeces i locally cove spaces as bee defied ad ivestigated. It is also poved Koia-Scu ad Silvaa- Toeplitz type teoes fo lacuay alost cosevatively ad lacuay alost egulaity of te wic tasfo sequeces i a Fecet space ito a sequece i a ote Fecet space. We also stated tat te sae esults old i locally bouded liea topological spaces. 2000 Mateatics Subect Classificatio: 4005, 40H05. Itoductio ad bacgoud Lacuay suability o lacuay alost suability of sequeces of eal (o cople) ube by ifiite atices of eal (o cople) ubes wee give i [4]. I tis pape, ou pupose is to eted te cocept of lacuay alost suability to two type of liea topological spaces Fecet spaces ad locally bauded spaces, eac of te icludes Baac spaces as a special cases. By a lacuay sequeces we ea a iceasig itege sequece θ = ( ) suc tat 0 = 0 ad = as. Tougout tis pape te itevals deteied by θ will be deoted by I = (, ] [3]. We coside oly Hausdoff spaces. Tougt tis pape we use ( X, pi ) ad ( Y, q ) to deote Fecet spaces, i.e., locally cove spaces wic ae etizable ad coplete, wose topologies ae geeated espectively by te coutable clases { p i } ad { q i } of seios. If X ad Y ae liea topological spaces ad Y is locally cove (wit te topology geeated by te sei-os q i ), ta te topology of bouded covegece o te spaces L ( X, Y ) of cotiuous liea opeatos o X ito Y is a locally cove topology geeated by te seios { q M } wee fo eac ad eac bouded set M i X,
28 B. ydi q M ( u) = sup q ( u( )), u L ( X, Y ). M If ( ) is a sequece i a Fecet spaces ( X, pi ), te covegece of ( ) i X will be i te topology cosideed, i.e, ( ) is coveget to X if ad oly if pi ( ) 0 ( ) fo eac i. (X ) ad c (X ) will deote liea space of bouded ad coveget sequeces i X espectively. 2. Defiitios Defiitio. Let ( X, pi ) be a Fecet space (o locally cove space). sequece = ( ) ( X ) is said to be lacuay alost coveget to l if ad oly if li + = l uifoly i. I Let te ati = ( ),, = 0,, 2, cosist of eties eac of te is a cotiuous liea opeato fo a liea topological space X ito aote liea topological space Y. We foally defie te sequece y ), ( y = = 0, = 0,, as te -tasfo of a give sequece ( ) i X ad wite y =. Te ati is said to be cosevative if fo eac coveget sequece ( ) i X, its -tasfo ( y ) is defied ad is coveget i Y. Te ati is said to be L-egula if fo eac i X iplies, y L(), wee L is a pescibed cotiuous liea opeato fo X ito Y. I paticula, if we coside tasfoatios of sequeces i ( X, pi ) ito sequeces i te sae space te we ay defie egulaity wit espect to te idetity opeato I. I tis case is called to be egula ati. Defiitio 2. Let X ad Y be locally cove spaces. sequece ( ) i X is said to be lacuay alost -suable if te -tasfo of ( ) is lacuay alost coveget i Y. Te sequece ( ) is said to be lacuay alost -suable to y Y if te -tasfo of ( ) is lacuay alost coveget to y i Y.
Lacuay lost Suability i Cetai Liea Topological Spaces 29 Defiitio 3. Te ati is said to be lacuay alost cosevative if te -tasfo of ) is lacuay alost coveget we ( ) c( X ). ( Defiitio 4. Te ati is called to be lacuay alost L-egula if fo eac i X, its -tasfos te sequeces i X ito te sequeces i X te te idetity opeato I ca be tae istead of L ad is said to be lacuay alost egula ati. Te followig Leas will be used fequetly i te poof of te teoes. Lea. [, p. 22] If X ad Y ae locally cove spaces ad X is quasi-coplete i.e. its closed ad bouded sets of X ae coplete, te ay collectio of cotiuous liea opeatos fo X ito Y wic is siply bouded is bouded fo te topology of uifo covegece o bouded sets. Lea 2. [2, p. 55] Let ( T ) be a sequece of cotiuous liea opeatos o X ito Y wee X ad Y ae Fecet spaces. If li ( ) eists fo eac i a fudaetal set i X ad if fo eac X, ( T ) is bouded i Y te T = lit eists fo eac X ad T is a cotiuous liea opeato o X ito Y. T 3. Lacuay alost suability teoes fo Fecet spaces Teoe. Let ( X, pi ) ad ( Y, q ) be Fecet spaces. Te ati = ( ) defiig sequece to sequece tasfoatio fo X ito Y is lacuay alost cosevative if ad oly if (i) fo eac bouded set Mα i X ad fo eac fied, q = 0 v I K v +, α, fo,, = 0,, ad M, = 0,,, α (ii) fo eac X, fo eac fied ad fo eac, = 0 v I + v, eists
220 B. ydi ad li = 0 v I + v, eists, uifoly i ; (iii) fo eac X ad fo eac fied = 0,, li v +, eists, uifoly i. v I Poof. Fist, suppose tat is lacuay alost cosevative. Te fo eac = ( ) c( X ), te -tasfo of ( ) is defied ad lacuay alost coveget i Y. To sow tat (i) is ecessay, let us coside te liea space c (X ) of coveget sequeces ( ) i X. It is easily see tat P i is a sei-o o c (X ) fo eac i, wee P ( ) = sup p ( ), = ( ) c ( X ), i i ad also locally cove space c ( X ), P ) is coplete ad so quasi coplete, te fo ( i eac fied, te collectio { T :, = 0,, 2, } of liea opeatos defied by T = = 0, = ( ) c ( X ), ae cotiuous o c (X ) ito Y. By te sae easos, fo eac fied ad, te collectio { U,, : = 0,, 2, } of liea opeatos defied by U,, = v I T v +, = v = 0 I v +, = = 0 I v +,, = ( ) c ( X ) ae cotiuous o c (X ) ito Y. By te assuptio of te eistece of te -tasfo of evey = ( ) c( X ), li = U,, = U, y,
Lacuay lost Suability i Cetai Liea Topological Spaces 22 eists fo evey fied ad. Tus, by Lea 2, fo, = 0,,, eac U, is cotiuous ad liea o c (X ) ito Y. Sice is also lacuay alost cosevative ati ad = ( ) c( X ), it follows tat y, ) 0 is coveget i ( Y, q ) uifoly i ad cosequetly fo = 0,, 2,, ( (, ) 0 = ( U, ) 0 y is bouded i ( Y, q ). Tus te collectio { U, : = 0,, 2, } of cotiuous liea opeatos is poitwise (ad teefoe siply) bouded fo =,. Teefoe by Lea, tey ae bouded covegece o L ( c ( X ), Y ) fo = 0,,. Fo te desciptio of te topology of te bouded covegece we obtai sup q M ( U, ) < K M,, = 0,,, fo eac fied ad eac bouded set M i c (X ). Now coside a bouded set M α i X. Tis set cosists of poits suc tat pi ( ) < αi. Coside te sequece of te fo ( 0,,,, 0, 0, ) wee s ae i M α. ll suc sequeces ae above i te sae bouded set M α i c (X ). Hece, fo te esult we obtaied we get q = 0 v I K v +, Mα, = K α,,,, = 0,, fo Mα fo eac. Te poof of te ecessity of coditio (I) is coplete. Te ecessity of (ii) follows by cosideig te sequece (,,, ) wile tat of (iii) follows by cosideig te sequece ( ), = 0 if ad =. Fo te sufficiecy, let us coside te sequeces of te fo (,,, ) ad ( 0, 0, 0,, 0,, 0, 0,, 0, ). It is easly see tat te set of te sequeces of tis fo is a fudaetal set i c (X ). Fute, fo eac = ( ) c( X ) ad fo eac, = 0,,, te opeatos T ae liea ad cotiuous o c (X ) ito Y. Hece te opeatos U,, ae also liea ad cotiuous o c (X ) ito Y. Te by te coditio (i), fo evey fied = ( ) c( X ) ad fied ad, te sequece ( U,, ) 0 is bouded i Y. d also, by te coditio (ii), fo evey fied = ( ) belogig to te fudaetal set. li U,, = 0,,,
222 B. ydi eists fo evey fied. Hece, fo evey fied ad = 0,,, te opeatos U, defied by U, = li U,,, ae liea ad cotiuous a c (X ) ito Y by Lea 2. Fute, fo evey = 0,, ad fo fied = ( ) c( X ), te sequece ( U, ) 0 is bouded ad by (ii) ad (iii) tis sequece is coveget i Y fo evey belogig to te fudaetal set. Tus agai by Lea 2 fo evey = ( ) c( X ) ad fo = 0,, li U = U eists ad U ( = 0,, ) is liea ad cotiuous o c (X ) ito Y., Now it ay be veified tat ude te coditios of te teoe, U ca be witte i te fo of U = li, li + li v + μ v +, li μ V I = 0 μ = 0 V I μ (*) fo eac = ( ) c( X ) ad = 0,,. It is also obseved tat te igt ad side of (*) is idepedet o by te coditios (ii) ad (iiii), so tat te liit U = liu is uifo i. Tis copletes te poof of te teoe., Te followig teoe fo lacuay alost egulaity is easily poved i a siila ae to tat of te Teoe. Teoe 2. Te ati = ( ) defiig sequece to sequece tasfoatios fo te Fecet space ( X, pi ) ito te Fecet space ( Y, q ) is lacuay alost egula if ad oly if (a) of Teoe old ad (b) fo eac X fo eac fied ad fo eac = 0 v I + v, eists ad li v = o I, L, uifoly i. v + =
(c) fo evey Lacuay lost Suability i Cetai Liea Topological Spaces 223 X ad fo evey fied, li v, v I + = 0, uifoly i. Refeeces. N. Boubai, Eleets de ateatique, Live V, Espaces Vectoiels Topologiques, cap. III-V. ct. Sci. Et Id. N. 229, Pais 955. 2. N. Dufod ad J.T.Scwatz, Liea Opeatos, Pat I, New Yo, 958. 3..R Feeda, J.J. Sebe ad M. Rapacl, Soe Ceao type suability spaces, Poc. Loda Mat. Soc. 37 (978), 508 520. 4. F. Nuay, θ-alost suability sequeces, Iteatioal Joual of Mat. Sci. 4 (997), 74 744.