Completeness of spaces wth Toeplitz decompositions

Re .R.Acad.Cien.Exac .Fis.Na . (Esp) Vol. 92, N.o 1, pp 27-33, 1998 Ma emá icas COMPLETENESS OF SPACES WITH TOEPLITZ DECOMPOSITIONS1 (Decomposi ions/Comple eness/Su n nabiIi y/Bases) PEDRO J. PAÚL, CARMEN SÁEZ, JUAN M. VIRUÉS Depa amen o de Ma emá ica Aplicada JI. Escuela Supe io de Ingenie os. Uni e sidad de Se illa. P esen ado po José Bone , 3 de diciemb e de 1997. Acep ado el 14 de ene o de 1998. ABSTRACT We ex end o hese ing o Toepli z decomposi ionso alocally con ex space Ein o subspaces (Ek)a esul abou Schaude decomposi ions due o Kal on ha links he com- ple eness o E o he comple eness o bo h he decomposi- ion and he pieces (E k ). The p oo is simpli ied by he applica ion o adouble limi echnique. Then we s udy wha we call he Ga ling opology o aspace wi h a Toepli z decomposi ion wi h espec o ama ix Tin he amewo k o he .B duali y o sequence spaces. RESUMEN En es e abajo ex endemos al ma co de las descom- posiciones de Toepli z de un espacio localmen e con exo Een subespacios (E k)un esul ado sob e descomposiciones de Schaude debido aKal on que liga la comple i ud de E con la comple i ud de la decomposición ylas de las piezas (E k ); el uso de una écnica de lími e doble nos pe mi e simpli ica la p ueba o iginal. Pos e io men e es udiamos lo que llamamos opología de Ga ling de un espacio con una descomposición de Toepli z con espec o auna ma iz Ten el ma co de la .BT-dualidad de espacios de sucesiones. INTRODUCTION Up o wha poin can one subs i u e o dina y summa- bili y by ama ix summabili y me hod in he de ini ion o aSchaude decomposi ion and, s ill, ob ain nice esul s abou he locally con ex s uc u e o he space in e ms o he locally con ex s uc u e o i s pieces? Ou pu pose he e is o ex end he cha ac e iza ion o he comple eness o aSchaude decomposi ion ob ained by Kal on (9), (10) o he se ing o decomposi ions de ined in e ms o mo e gene al ma ix summabili y me hods. The p oo is simpli- ied by he applica ion o adouble limi echnique. Gi en an in ini e ma ix Tone can conside i s con e - gence ield cTand de ine, o asequence space A, i s co - esponding .BT -dual APT. This dual pai has been s udied by Bun inas (2), (3), Meye s (12) and Noll (14) as acon in- ua ion o he~-duali y heo y o Ga ling (5), (6). Re e we use he heo em abou comple eness o s udy wha we call he Ga ling opology, because i is ana u al ex ension o he <Jy- opology de ined in (5), o aspace wi h aT-decom- posi ion and, in pa icula , we gi e so ne applica ions o he dual pai (A, APT). Te minology and No a ion. Al hough ou no a ion and e minology will be mos 1y s anda d, e.g. <p is he space o ini ely nonze o sequences, Cis he space o con e gen sequences, e[k] s ands o he k- h uni sequence (we e e he eade o (16), (17), (19) and (21», le us ecall a ew ac s om summabili y heo y. Le T= Und be an in ini e ma ix o scala s om he ield K o eal o complex num- be s. The ma ix T is said o be: ow- ini e i each ow o Tis in <p, an Sp¡-ma ix i each column o Tis con e gen o 1, and e e sible i o e e y sequence yEc he in ini e sys em o linea equa ions T.x == yhas uniquesolu ion. I is well-known (21, 5.4.5-5.4.9) ha each ow- ini e and e e sible Thas aunique wo-sided in e se ma ix TI such ha each ow o 11 is in II and o each yEc he unique solu ion o T.x=yis 11• y. Le Ebe alocally con ex space. The con e gence ield o T in Eis he space cT(E) o all sequences (x k) am E such ha he p oduc T.(xk) is acon e gen sequence in E. Fo (xk)Ec (E) he limi o he sequence T.(xk) is called he T-limi o (x k)and will be deno ed by T-lim .xk, in o he wo ds IThis esea ch has been pa ially suppo ed by la Conseje ía de Educación yCiencia de la Jun a de Andalucía and by la Di ección de In es igación Cien í ica yTécnica, p ojec PB94-l460. 28 Ma emá icas: Ped o J. Paú] e al. Re .R.Acad.Cienc.Exac .Fis.Na . (Esp), 1998; 92 We simply deno e by cT he con e gence ield o T in JK. I T isa ow- ini e and e e sible ma ix hen he no m IIxll := liT· xlL makes CTaBanach space (isomo phic o c). De ini ions. Le T= [ "d be a ow- ini e in ini e ma ix o scala s. Asequence (P k) o non- i ial, mu ually o hogonal and con inuous linea p ojec ions de ined on a locally con ex space E is said o be a Toepli z decompo- si ion o Ewi h espec o he ma ix To , sho Iy, aT- decomposi ion o E, i x=T -lim J x o e e y xE E. Al e na i ely, i we de ine he sequence o ope a o s T" :xEE -7 T,,(x):= .~,>"kPkX E E, k hen (P k) is aT-decomposi ion o Ewhene e lim" T~ = x o e e y xE E. I is impo an o no e ha he ope a o s T,,'s a e no p ojec ions in gene al ( hey a e inc easing p ojec ions in he case o aSchaude decomposi ion), howe e we do ha e T"Pk=PkT" = "k Pk o all n, kEN. No e also ha he sequence o ope a o s (T,,) is p ecisely he p oduc T . (P k) hence saying ha lim" T" x=xis he same as saying ha he sequence T· (P0) con e ges o x. Call Ek:= PiE). Since Ekdoes no educes o he ze o subspace and o e e y xkEEkwe ha e xk =lim" T,h =lim" ,,0k' i ollows ha lim" nk = 1, i.e., Tis an SP ma ix. S ill ano he way o lookinga aToepli z decomposi- ion is he ollowing: E e y Ekis acomplemen ed sub- space o Eand we can iden i y e e y xEEwi h he ec- o - alued sequence (pkx) E I1 Ek,so ha Ebecomes a linea subspace o I1 Ek ha , wi h he opology ansla ed om E, has he se o all ini e sequences as adense sub- space because lim" T"x =x o e e y xEEand T is ow" ini e. AT-decomposi ion (P k) o alocally con ex space Eis said o be: ini e-dimensional i e e y Ek is ini e-dimen- sional; equicon inuous i he sequence o ope a o s (T n) is equicon inuous; and comple e i o each squence (x k) E I1 Eksuch ha he p oduc T. (Xk) is aCauchy se- quence in E he e exis s xEEsuch ha xk= P0 o e e y kENand, a o io i, T· (x k)con e ges o x. Example 1. E e y Schaude decomposi ion is a Toepli z decomposi ion wi h espec o he o dina y sum- mabili y ma ix L= IO'"d, whe e 0"k =1 i n ::; kand 0 nk :::: Oo he Wise. Example 2. ACesa o basis induces aone-dimension- al Toepli z decomposi ion wi h espec o el . L, whe e el = [c,~d is he Cesa o ma ix o o de 1de ined by C/~k :::: n- l i n ::; kand C],k =oo he wise. (The ma ix el .L is some imes called he se ies- o-sequence Cesa o ma íx.) Decomposi ions o Banach spaces wi h espec o Cesa o ma ices We e i s ly conside ed by Bu ze and his collab- o a o s in Aachen (see (19, pp. 785 and 801 o ol. I1)). Example 3. AK-space is alocally con ex sequence space }¡,::J cp such ha he k- h p ojec ion de ined by í' k((x,,)J := xke[k] is con inuous o e e y kE N. A K- spaceA is said o ha e p ope y T-AK i x=T-lim xke[k] o e e y sequence x= (x k)E A. Thus, asequence space Ahas p ope y T-AK i and only i he sequence (J k)is a (one-dimensional) T-decomposi ion o Ao , in o he wo ds, he sequence o ope a o s de ined by"" := L../"kí' k,i.e. (",,) := T. (í' k ), sa is ies x=limn ",,(Xk) o e e y sequence x= (x k)EA(see (2), (3) o (12)). (When dealing wi h scala sequences, we shall keep he no a ions (1 k)and (1: k) h oughou he pape ). In pa icula , Ahas p ope y L-AK means p ecisely ha (e 1kJ )is aSchaude basis o A. We shall be in e es ed in ma ices Tsuch ha CThas p ope y T-AK. These ma ices we e cha ac e ized by Bun- inas (3, Thms. 8-10). Bun inas's Theo em. Le T be ow- ini e and e e s- ible SPI-ma ix. Then he ollowing condi ions a e equi a- len : (1) The sequence o coo dina e p ojec ions (J k) is a T-decomposi ion o cp (2) The sequence o ope a o s ( n) is equicon inuous on cp (3) J we deno e Ti by[ /~i] hen sup {~I~ mk "k ijII : m, nE N} <oo. (4) The dual (c T )' can be iden i ied wi h he mul ipli- e space (cT -7 cT) o med by he sequences y such ha he coo dina ewise p oduc xy is in cT o e e y xEcT and, in his case, he bilinea o m o he dual pai is gi en by The i s non- i ial examples o ma ices Tsuch ha cThas p ope y T-AK a e he se ies- o-sequence Cesa o ma ices o o de a ~ O; his was p o ed by Zelle (22). The e o e, o a oid c1umsy epe i ions, a ow- ini eand e e sible Sp¡-ma ix Tsuch ha cThas p ope y T-AK will be called a Zelle -Bun inas ma ix. Fo such ama ix Twe de ine b(T) := sup" 11"" 11, whe e 11",,11 is he no m o 1: nas a bounded ope a o om he Banach space cTin o i sel . No e also ha i T is aZelle -Bun inas ma ix hen cT is a sum space in he sense o Ruckle (17). Example 4. Le Qbe an open, bounded and balanced subse o cm and le A(Q) be he space o all unc ions ha Ma emá icas: Ped o J. Paúl e al. Re .R.Acad.Cienc.Exac .Fis.Na . (Esp), 1998; 92 29 a e holomo phic on Qand can be ex ended con inuosly o he c10su e o Qendowed wi h he opology o uni o m con e gence on Q. I is men ioned wi hou p oo in (15) ha A(Q) has he bounded app oxima ion p ope y. As a ma e o ac , wha happens is ha A(Q) has an equicon- inuous and a ini e-dimensional Toepli z decomposi ion wi h espec o he se ies- o-sequence Cesa o ma ix Cl. L. This decomposi ion is he na u al one gi en by he Taylo se ies: Each E A(Q) can be uniquely w i en as O = L~=OPk(J)O (poin wise con e gence), whe e each Pi ) is ak-homogeneous polynomia1. Now, he se o aH homogeneous polynomials is dense in A(Q) ( o ap oo see (1), he me hod u ilized in Sec ion 1 o ha pape can easi1y be adap ed o show he p esen esul ). On he o he hand, he co esponding sequence o ope a o s (T n)is equi- con inuous by (1, Lemma 1.1) o (13, 5.2 P oposi ion), whe e i is shown ha ITII(J)(z)1 ~ 11 o aH zE Q. Finally, as anda d a gumen abou equicon inuous se s (11, §39.4.(1) shows ha =T-lim Pi ) o all EA(Q). COMPLETENESS OF SPACES WITH TOEPLITZ DECOMPOSITIONS Ou i s pu pose is o ex end o he se ing o Toepli z decomposi ions a esul due o Kal on (10) ha links he comple eness o E o he comple eness o bo h he decom- posi ion and he pieces Ek.We shall make use o adouble limi echnique ha lies behind he p oo gi en by Kal on o Schaude decomposi ions. The Double Limi Lemma is ce ainly well-known o double sequences bu we need a e o mula ion in e ms o adouble ne ha can be p o en analogously. Double Limi Lemma. Le Ebe alocally con ex space and {xij : (i, j) EJxJ} be adouble ne in Esuch ha o each iEJ he e exis s he limi y¡ =limj xij and o each jEJ he e exis s he limi Zj =limi Xii' J he con e - gence o (xij)j o y¡ is uni o m in J hen he h ee ne s (Xi)' (y) and (z) a e Cauchy ne s. I , in addi ion, Eis comple e hen he h ee ne s aboye a e con e gen o he same limi o Theo em 1. Le (P k) be an equicon inuous T-decom- posi ion o a locally con ex space E. Then he ollowing a e equi alen : (1) Eis comple e ( esp. quasi-comple e o sequen- ially comple e). (2) (P k)is comple e and each Ekis comple e ( esp. quasi-comple e o sequen ially comple e). P oo! I is c1ea ha (1) implies (2), so we ha e o show ha (2) implies (1). We shall deal only wi h he comple eness case because he p oo s o he h eecases a e essen ially he same. Le (z);,=/ be aCauchy ne in E. Fo each kEN he e exis s xkEEksuch ha (P k Z)¡E/ con e ges o xk because Pk is con inuous and Ek is com- ple e. Since Tis ow- ini e, o e e y nENwe ha e On he o he hand, limnT nZ¡ = Z¡ o e e y iEI. To see ha we can apply he Double Limi Lemma o he double . ne {Tnzi: (i, n) EJx N} ,le us check ha he con e - gence o (TnZ)iE/ is uni o m in N. Gi en acon inuous sem- ino m q/ on E, he e exis s acon inuous semino m q2 such ha because (P k)is an equicon inuous T-decomposi ion. Since (Z)¡E/ is aCauchy ne , i ollows ha he e exis s so ne index ioEJsuch ha q2(Zi ~ z) ::; 1whene e i, j ~ io' The e o e, ql(TnZ¡ - Tnz j) ~ 1 o all nENand i, j ~ io. Take limi s in j o ob ain This shows ha he con e gence o (TnZ)¡E/ is uni o m in N. The Double Limi Lemma ells us ha he p oduc T· (x k)= (Lk llkXk) is aCauchy sequence. Since (P k) /leN is acomple e T-decomposi ion, he e exis s XEEsuch ha xk=P ~ o e e y kE Nand T.(xk)con e ges o X. Finally, since he ne (Z)¡E/ is con e gen in he comple ion o E, he Double Limi Lemma ells us now ha (z¡hE/ mus con e ge o x as well .• Co olla y 1. Le (P k) be an equicon inuous and i- ni e-dimensional Toepli z decomposi ion o alocally con- ex space E. Then he ollowing a e equi alen : (1) Eis comple e. (2) Eis quasi-comple e. (3) Eis sequen ially comple e. (4) . (P k)is comple e. Co olla y 2. J a ba elled and sequen ially comple e locally con ex space has a ini e-dimensional Toepli z de- composi ion hen i is comple e. Rema k. An ex ended Schaude basis o alocally con ex space Eis a amily (X)iE/ wi h he p ope y ha o e e y xEE he e is aunique amily (a;(x))¡E/ o scala s such ha xcan be w i enas x= L¡a¡(x)x¡ and he unc ionals x ~ a¡{x) a e con inuous. Webb (20) p o ed ha asepa able, non-comple e, Mon el locally con ex space canno ha e any ex ended Schaude basis; ou Co - olla y 2shows ha i canno ha e any ini e-dimensional Toepli z decomposi ion nei he . THE GARLING TOPOLOGY OF A SPACE WITH A TOEPLITZ DECOMPOSITION Le Ebe alocally con ex space wi h aT-decomposi- ion (P k ). In his sec ion we will see ha he e exis s a 30 Ma emá icas: Ped o J. Paúl e al. Re .R.Acad.Cienc.Exac .Fis.Na . (Esp), 1998; 92 coa ses E'-pola opology on E o which (P k)is anequi- con inuous Toepli z decomposi ion. This opology u ns ou o be ana u al gene aliza ion o he ay- opology in o- duced by Ga ling in his deep s udy o he ~-duali y be- ween sequence spaces (5), (6) and, acco dingly, will be called he e he Ga ling opology o E. In he case o Schaude decomposi ions his opology has been s udied by Kal on (9). Using p imes o deno e adjoin ope a o s, o e e y xE Eand e e y uE E' we can w i e (x, u) = lim(T"x, u) =lim L, "k(llx, u) = 1I . II .k This shows ha (p~) is also aT-decomposi ion o El endowed wi h he weak opology a E', E). I we call Ek := PiE) and E~ := P~(E') hen he dual o Ekcan be iden- i ied wi h E~. The compu a ion aboye also shows ha he sequence (T;,u) is a(E', E)-bounded. De ini ion. Le (Pk) be aT-decomposi ion o alocal- ly con ex space E. The Ga ling opology o Eis he pola opology yy{E, El) o uni o m con e gence on he amily {(T;,u) :uE E}. Al e na i ely, yy{E, El) is gene a ed by he amily o semino ms xEE ~ supl(T"x, u)l, (u EE). " The Ga ling and he weak opology coincide on each Ekbecause o all xkEEkand uEEl we ha e we shall make use o his ac acouple o imes. The p ope ies o he Ga ling opology depend hea ily on he T-AK p ope y o he con e gence ield associa ed o he ma ix T. No e ha his is gi en o ee in he case o aSchaude decomposi ion: (e[k l)is aSchaude basis o he space cs (= cL) o all summable sequences. To see how o connec he Ga ling opology wi h he p ope ies o c1' le Fbe he ec o - alued sequence space de ined by As we no ed aboye, Ecan be iden i ied wi h asub- space o F. No e ha , using he e minology gi en in he p e ious sec ion, Eequals F i and only i (P k) is acom- ple e T-decomposi ion o E[a (E, E')]; in his case, (Pk) is said o be ~ comple e by analogy wi h he Schaude de" composi ion case (lO). Fo each uE E' we de ine he ope a o I is easy o see ha /::""sa is ies he ollowing p ope " ies (i) Fo e e y nENand (Xk) EFwe ha e ha he sequence (T"xkh is also in Fand /::""(T,,Xk)k = 'l"1l/::",,(X k)k' In pa icula , /::""(T,,PkX)k = 'l"1l/::",,(P k X)k o each xE E. (ii) J IT . (x k )]" s ands o he n- h elemen o he sequence T.(xk ), hen and, in pa icula , 11/::",,(Pkx)IIT = sup" I(T"x, u)1 o each xE Eso ha he Ga ling opology is gene a ed by he amily o semino ms x ~ 11/::,." (Pkx)IIT as uE E'. P oposi ion 1. Le T be aZelle -Bun inas ma ix and (P k) be aT-decomposi ion o alocally con ex space E. Then yy{E ,E') is he coa ses El-pola opology such ha (P k) is an equicon inuous T-decomposi ion o E. P oo! Using ha he p ojec ions (Pk)a e weakly con inuous on E, ha he Ga ling opology is s onge han he weak opology and ha he Ga ling opology induces on each subspace Eki s own weak opology 0"( Ek, E~), i ollows ha he p ojec ions (P k)a e con inuous on E o he Ga ling opology. We now show ha (Pk)is an equi- con inuous T-decomposi ion o Eendowed wi h yy{E, E'). Fo all xE E, uEE', and mENwe ha e, using (i) and (ii) aboye, 11/::,."(IlT,,,x)kIIT = 11/::,."(T,,,! X)kIIT = = 11'l"IA,(llx)kIIT :,; b(T)II/::,.,,(llx)IIT so ha (T n)is aY equicon inuous sequence. (A ema k is in o de he e: al hough yy{E, El) is gene a ed by he amily o semino s sUPn I(T"x, u)l, since he T;s a e no inc easing p ojec ions -as i is he case o aSchaude decomposi- ion- we canno conclude di ec ly ha sUPIII(T"Tmx, u)1 :,; :,; sup"I(T"x, u)I·) The ollowing compu a ion shows ha x= yy{E, E')- limnT II x o all xE E: whe e, in he la e s ep, we ha e used ha cThas p ope y T-AK. Now, le be an E'-pola opology such ha (Pk) is an equicon inuous T-decomposi ion o Eendowed wi h . Ma emá icas: Ped o J. Paú1 e al. Re .R.Acad.Cienc.Exac .Fis.Na . (Esp), 1998; 92 31 Fix uEE', hen he e is a -equicon inuous se De E' such ha sup 1(T"x, u)1 ~ sup I{x, )l· II . eD This shows ha y.JE, E') is coa se han . • Example 5. I Tis aZelle -Bun inas ma ix hen he Ga ling opology on CTcoincides wi h he IHIT - opology. To see his, conside he squence e= (1, 1, ...)E (CT)'. Then o e e y x= (X k)EcT,we ha e This shows ha he no n opology is coa se han he Ga ling opology. The con e se ollows om P oposi ion 1. (Fo he case o he Cesa o se ies o sequence summu- bili y ma ix el. 2, his ac was p o ed by Flo encio (4), using di e en echniques.) The dual o E[ T(E, E')] can be bigge han E' (see Rema k 2below), bu we can cha ac e ize i in he ollow- ing way: Gi en uE E' and aE(cT )' we may de ine alinea unc ional au on Eby I Tis aZelle -Bun inas ma ix, so ha (cT )' is also a sequence space and we w i e a=(ak) hen we ha e We deno e by (cT )' • E' he space o all linea unc ion- als hus ob ained. P oposi ion 2. Le Tbe aZelle -Bun inas ma ix and le (P k)be aT-decomposi ion 01 alocally con ex space E. Then he dual space 01 Eendowed wi h i s Ga ling opol- ogy is (CT)'· E'. In pa icula , yy(E, E') is compa ible wi h he dual pai i and only i E' =(cT )' .E'. P oo! Gi en uE E' and aE (CT)' we ha e The e o e, au is YT (E, E')-con inuous. Con e sely, le Zbe a YT (E, E')-con inuous linea unc ional on E. Then he e exis s uE E' such ha !(x, z)1 ~ IIA II (Pkx)IIT o all x E E. Iden i y Ewi h i s image in F ia he injec ion x -4 (P 0). This enables us o de ine alinea unc ional aby a: ((Pkx, u)) EA/I(E) -4 (((Pkx, u)), a) := (x, z) which, obiously, is well-de ined and!HIT -con inuous. By using ha he subspaces (E k)a e non- i ial, i is easy o see ha qJ e A" (E) so ha his is adense subspace o cT Finally, ex end a o all o cTby con inui y o ob ain and he p oo is inished. • Rema ks. (1) I BTs ands o he uni ball o (cT) " he equali y E' = (CT)' . E' is equi ulen o E' =BT.E', and i his equali y holds hen E' is said o be B in a ian , as in he sequence space case (5), (9). (2) I E' is BT-in a ian hen he Ga ling opology is compa ible wi h he dual pai (E, E') and so he sequence o unc ionals (uTn)n is .~ (E', E)-bounded in E', in which case he T-decomposi ion is said o be simple. As he e a e non-simple Schaude basis (see he ema ks ollowing De . 2.3 in (9», i ollows ha no all Ga ling opologies a e compa ible. (3) I is easy o see ha aToepli z decomposi ion is simple i and only i he weak and he Ga ling opologies ha e he same amily o bounded se s. We s udy now when is EIYT(E, E')] acomple e space. P oposi ion 3. Le Tbe aZelle -Bun inas ma ix and le (P k)be aT-decomposi ion 01 alocally con ex space E. Then E[yy(E, E')] is comple e ( esp. quasi-comple e o sequen ially comple e) i and only i (P k)is 3 comple e and each Ek[ 0"( Ek, E~)] is comple e ( esp. quasi-comple e o sequen ially comple e). P oo! As we poin ed ou abo e, he Ga ling and he weak opology coincide on each El<' Hence,acco ding o Theo em 1, we ha e o p o e ha (PJis acomple e T- decomposi ion o E[yy(E, E')] i and only i i is acomple e T-decomposi ion o Eendowed wi h i s weak opology; Le., ~T"comple e. The «i » pa ollows easily om he ac ha he weak opology is coa se han he Ga ling opolo" gy. So, assume ha (P k)is acomple e T-decomposi ion o E[YT (E, E')] and le (x k)E TI Ekbe such ha T· (x k)is a weakly-Cauchy sequence in E, i su ices o show ha T . (x k)is also a YT (E, E')-Cauchy sequence. Deno e by zn he n- h elemen o T. (x k), ha is Z/I = L.k nkxk' I is clea ha Pkz/I = /lkXk o all n, kENso, using (i) abo e, we ha e Finally, using he Y con inuous semino ns as gi en in (ii), we ob ain and his la e exp esion goes o ze o as m, n -4 00 because A" (xk)is in cTand his space has p ope y T-AK. • 32 Ma emá icas: Ped o J. Paúl e al. Re .R.Acad. Cienc. Exac . Fis. Na . (Esp), 1998; 92 'Co oIla y. Le T be aZelle -Bun inas ma ix and le (Pk) be aT-decomposi ion o aBanach space E. Then iE, E') is acomple e opology ( esp. quasi-comple e) i and only i (P k) is 3 comple e and each Ek is ini e-dimension- al ( esp. e lexi e). Example 6. Le Tbe aZelle -Bun inas ma ix and E be alocally con ex space wi h an equicon inuous T-de- composi ion (P k ). I s ands o he opology o E, hen P oposi ion I ells us ha a(E, E') :;; T(E, E') :;; . Example 5shows ha o E=cT he i s inequali y is s ic , bu he second is an equali y. On he o he hand, i Eis an in ini e dimensional Ba- nach space, hen he space co(E) o med by he null se- quences in Eis aBanach space wi h ana u al in ini e- dimensional Schaude decomposi ion (7) and i s Ga ling opology, which canno be comple e, does no coincide wi h i s no m opology so ha bo h inequali ies a e s ic . APPLICATION TO THE ~T-DUALITY OF SEQUENCE SPACES In wha ollows, Tis aZelle -Bun inas ma ix and A s ands o asequence space con aining <p. The ~T-dual o Ais he space A >T o aH sequences ysuch ha he coo di- na ewise p oduc xy is T-con e gen o e e y xE A. I Ais aK-space wi h p ope y T-AK hen i is cIea ha A' eA P . I , in addi ion, Ais sequen ially ba elled hen A' = ?J3 (see (2), (3), (12) and (14)). On he o he hand, i Ahas p ope y T-AK hen Ae (A' and, by P oposi ion 3, he equali y holds i and only i A[ T(A, A')] is sequen ially comple e. Assume now ha no opology is de ined ap io i on A. The na u al bilinea o m (x, y) ~ T-lim XkYk malees (A, A >T) asepa a ed dual pai and, cIea ly, bo h A[a(A' A P )] and AP [a(A P , A)] ha e p ope y T-AK. We may ask i he e is as onge opology on A ha in¡ p ope y T-AK and s ilI compa ible wi h he duali y (A, A ). We shall cha ac" e ize his opology by ex ending and combining esu1 s gi en by Ga ling (5) and Schae e (18) o he ~-duali y ( he duali y de ined in e ms o o dina y summabili y). Lemma. Le Tbe aZelle -Bun inas ma ix and A be a sequence space con aining ([J. Then he space A/ 3 [ T(A P , A)] is comple e and o ase Ce A P he ol- lowing condi ions a e equi alen (1) C is T(A P ,A)- ela i ely compac o (2) Cis T(A P ,A)-bounded and he con e gence o he sequence (' ny) o y in he Ga ling opology T(A P , A) is uni o m wi h espec o yE C. (3) eis T (A P , A) -bounded and o e e y x€A he con e gence o he sequence lY) o xy in he ¡HIT- opology is uni o m wi h espec o yE C. P oo! Acco ding oP oposi ion 3, o p o e ha A P [ Y (A P , A)] is comple e we ha e o show ha (1 k)is a comple e T-decomposi ion o APT a(A P , A)]. bu his ol- lows om he e y de ini ion o PT-duaI. By P oposi ion 1, (~) is an equicon inuous T-decompo- si ion o AP [ T(A P , A)j; i.e., ( n)is a T(A P ,A)-equicon- inuous sequence o ope a o s ha con e ges poin wise o he iden i y on A P hence, by using (11, §39.4(1)), we ha e ha ('en) con e ges uni o mly on T(A P ,A)-compac se s. Since ll (A P )is ini e-dimensional o e e y nE N, i ollows ha o e e y nE N, he se ll ( C) is ela i ely compac p o ided ha Cis T(A P ,A)-bounded. Then Ma- zu 's Theo em (8, Thm. 1) implies ha condi ions (1) and (2) a eequi alen . The equi alence o (2) and (3) is cIea . • Theo em 2. Le Tbe aZelle -Bun inas ma ix and A be asequen ce space con aining <p. Then he s onge o- pology on A ha has p ope y T-AK and is compa ible wi h he duali y (A, A P ) is he opology kT(A, A P ) o uni o m con e gence on he absolu ely con ex and T~AP , A) "compac subse s o A P . P oo! Tha he dual o A[kT(A, A P )] equals ?J3 ol- lows om he Mackey-A ens's Theo em and he ac ha he Ga ling opology is s onge han he weak opology. Tha AIkT(A, A PT )] has p ope y T-AK ollows om he p e ious lemma by simply no ing ha (x ~ ,,(x), Y) = (x, y ~ ,,(y)) o all xE A, yEA PT and nE N. Finally, i CeA P is an absolu ely con ex and a( A P , A) -compac se such ha [im sup i(x ~ ,,(x), y)1 =O o all xE A, Il )'EC hen, again by he lemma, Cis Y (A PT , A) -compac o This implies ha kT( A, A P )is he s onge opology sa is ying he desi ed p ope ies .• Co oIla y 1. Le Abe asequence space con aining ([J. 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