Köthe echelon spaces à la Dieudonné

Indag. Ma he n., N.S., 5 (l), 51-60 KG he echelon spaces CI la Dieudonnk Ma ch 28, 1994 by Miguel Flo encio, Ped o J. Paljl and Ca men Shez Depa amen o de Ma emci ica Aplicada II. E.S. Ingenie os Indus iales, A . Reina Me cedes s/n, 41012-Se illa, Spain Communica ed by P o . J. Ko e aa a he mee ing o Decembe 21,1992 ABSTRACT Le (gn) be a sequence o locally in eg able unc ions de ined on a Radon measu e space. The echelon space associa ed o (g,,) was de ined by J. Dieudonne as he Ko he-dual o (g”). i.e. he space n o all locally in eg able unc ions such ha all he in eg als s I g,l a e ini e. Deno e by Ax he Kii he-dual o A. We p o e ha A@(,$ Ax)) IS a F eche space wi h dual Ax. This e- sul gi es i s co ec sense o a w ong a i ma ion o J. Dieudonne and alida es hose ins ances whe e i has been used. As a ool o p o e his esul , we s udy he p oblem o when he s ong dual o a pe ec space coincides wi h i s Kii he-dual and gi e some necessa y and su icien con- di ions. 1. INTRODUCTION The Ko he-Toepli z heo y o pe ec sequence spaces [7, $301 has been one o he mos in luen ial in he s udy o he s uc u e o locally con ex spaces. This heo y, and in pa icula he duali y be ween echelon and co-echelon sequence spaces (see e.g. [l] and [6]), has p o ided he specialis s wi h plen y o hin s, ex- amples and coun e examples. Di e en ex ensions o his heo y a e ob ained by eplacing he C in he de ini ion o G. K&he and 0. Toepli z wi h a sui able S. One o he i s mo es in his di ec ion was made by J. Dieudonne. I is in he las sec ion o his seminal pape [3, $161 ha J. Dieudonne conside s echelon spaces o unc ions. His de ini ion is as ollows: le X be a locally compac and a-compac Hausdo opological space wi h a Radon measu e p. The eche- lon space associa ed o a sequence (g,) o locally in eg able unc ions is de ined as he Ko he-dual o he sequence: 51 n := : is locally in eg able and &n( ) = s I .g,I dp < +oo, n = 1,2,. . . x A e gi ing his de ini ion, he a i ms ha A endowed wi h he opology de ined by he semino ms ( pg,) is a F eche space and ha i s opological dual coincides wi h i s Ko he-dual Ax. This is no co ec , as was poin ed ou by J.A. Lopez Molina [8, Ex. (p. 187)] wi h he ollowing example: conside he uni in e al wi h i s Lebesgue measu e and he echelon space A associa ed o he unc ion g(x) = exp( - l/x). Then h = x[~,~I is a unc ion in ilx because he unc ions in A a e, by de ini ion, locally in eg able. Howe e , h is no con inuous o he semi- no mp, because he sequence (k xp~,k]) om n sa is ies lip pg(k. XIO,llk]) = lip s k. exp(-l/x) dx 5 lip exp(-k) = 0, 0 bu o allk= 1,2,...weha e jkqqo,,,k,(x).h(x)dx= ykdx= 1. 0 0 This mis ake led J.A. Lopez Molina o conside an al e na i e de ini ion o echelon space: equi e he unc ions in n o be only measu able ins ead o locally in eg able. The heo y o echelon spaces de ined in his way has been de eloped and gene alized by J.A. Lopez Molina [8] and [9], J.C. Diaz [2] and K. Reihe [12]. Ou pu pose in his pape is o p o e ha J. Dieudonne’s a i ma ion is essen- ially co ec in he sense ha an echelon space A is a F eche space when en- dowed wi h he s ong opology p(A, Ax) and i s opological dual equals i s Ko he-dual Ax. This esul will, in u n, alida e hose ins ances in which J. Dieudonne’s a i ma ion has been used p ecisely in i s co ec sense, as in [ll, Co . 11. In $2 we ecall J. Dieudonne’s de ini ion o pe ec spaces and gi e some necessa y and su icien condi ions o he opological dual o a pe ec space o coincide wi h i s Ko he-dual. These condi ions, which a e o independen in e es , gene alize and uni y some p e iously known esul s and will be used in $3, whe e he esul announced in he p eceding pa ag aph appea s. We e e he eade o W. Rudin’s book [13] o he esul s conce ning measu e heo y and in eg a ion, and o G. Ko he’s monog aph [7] o he heo y o locally con ex spaces. 2. WHEN DOES THE TOPOLOGICAL DUAL COINCIDE WITH THE KGTHE-DUAL? Al hough some o ou esul s in his pape can be gi en in a mo e abs ac measu e- heo e ic ame, we shall s ick o J. Dieudonne’s o iginal o mula ion [3, #lo-161. In wha ollows, X s ands o a locally compac , Hausdo opolo- gical space ha is a-compac , so ha we can w i e X = U, X,, whe e e e y X, is 52 compac and X, c in (X,+i) o all m E N. Le p be a posi i e Radon measu e on X and G be he space o all (equi alence classes o ) locally in eg able unc ions om X in o he ield [ib o eal o complex numbe s. Fo a subse A c R we deno e by A” he se o all g E $2 such ha . g is in eg able o each E A; Ax is called he K&he-dual o A. A linea subspace A o 0 is said o be a pe ec space (o Kii he space) i (Ax)” = A. In pa icula , Ax is always pe ec . Fo ins ance, L1 (,u) and L”(p) a e pe ec spaces, and each one is he Ko he-dual o he o he . The space l is also pe ec and Qx is he space @ o all (equi alence classes o ) measu able essen ially bounded unc ions wi h compac suppo . When A con- ains @, he spaces A and Ax a e pu in o duali y by means o he canonical bi- linea o m ( > g) = L (x) g(x) 44x) ( E 4 g E Ax 1. Le B s and o he uni ball o L”(p). A subse H o R is called no mal i E H o all E H and E B o , equi alen ly, i E H and g is a measu able unc ion such ha /g(x) 1 5 1 (x) 1 ,+a.e. on X, hen g is also in H. The no mal hull o a se H c 0 is de ined by { : E H, E B}. One impo an ac is ha he no mal hull o a weakly bounded se is also bounded [3, P op. 61. This means ha he s ong opology /?(A, A”) is gene a ed by he semino ms whe e H uns h ough he absolu ely con ex, no mal and o(A’, A)-bounded subse s o Ax. I A is a pe ec space, hen A(p(A, Ax)) is comple e [3, Th. 51. Deno e by A’ he opological dual o A(p(A, Ax)). The p oblem o when A’ equals Ax o , equi alen ly, when he s ong opology p(A, Ax) coincides wi h he Mackey opology ,u(A, A”), has been add essed by se e al au ho s (we shall gi e p ecise e e ences in he no es ollowing ou Theo em l), and di e en necessa y o su icien condi ions ha e been conside ed. In ou i s esul we uni y and ex- end p e iously known esul s. Theo em 1. Le A be a pe ec space and deno e by A’ he opological dual o A en- dowed wi h he s ong opology /?(A, A”). C onside he ollowing condi ions.. (i) A’ = Ax o , equi alen ly, p(A, Ax) = P(A, Ax). (ii) Fo e e y unc ion E A and e e y sequence (A,) o measu able se s such ha lim, p(A,) = 0, he sequences ( XA,) and ( - . xx,) con e ge o ze o o he s ong opology P(A, Ax). (iii) I ( n) is a dec easing sequence in A ha con e ges o ze o p-a.e., hen ( n) con e ges o ze o o he s ong opology ,@A, Ax). (i ) A(p(A, Ax)) is sepa able. ( ) The space Cc(X ) o con inuous unc ions wi h compac suppo is dense in A o he s ong opology p(A, A”). 53 Then we ha e (i ) + (i) @ (ii) @ (iii) + ( ). Besides, i he measu e space (X, p) is sepa able, hen (i ) is equi alen o (i) - (iii). Mo eo e , cyX is a me iz- able space, hen all he condi ions abo e a e equi alen . P oo . We s a by p o ing he equi alence o (i) - (iii). (i) + (ii): Take E A. T o see ha ( - XX,,) con e ges s ongly o ze o, ix a s ong semino m pH, whe e H is an absolu ely con ex, no mal and a(A’, A)- bounded subse o Ax. Condi ion (i) ells us ha H is ela i ely compac o he weak opology a(Ax , _4). Conside he mapping T:gEAx-T(g)= .gEL’(p). Since A is no mal, he unc ion . h is in A o e e y h E L”( p), and we ha e (T(g),h)(,l(,),,,(,)) = i .g.hd~ = (gJ.h)(,x,,). The e o e, T is a(A’, A) - a(L’( p), L”(p)) con inuous and consequen ly T(H) is ela i ely compac o he weak opology a(L’( p), I.,“( p)). Then, by [3, Th. 41, he e is a compac se K c X such ha sup J” IT(g)]dp:gEH J” j .gldp:gEH J’ K X K TakingmENsuch ha KcX,weha epH( - .xxm)<l o n>m. On he o he hand, i p(A,) + 0 bu ( . XA.) does no end o ze o o he Mackey opology, hen we can ind a no mal se H c Ax, absolu ely con ex and a(Ax , A)-compac , and an inc easing sequence o indices (nk) such ha bu sup Aj 1 .gldl.L: gEH >l o allk=1,2,... “k Conside he dec easing sequence (Fk) o measu able se s de ined by Fk := Ujzk A,. Then P(h) 5 xj>k P(&), h ence limk p(Fk) = 0. Fo each k E N de ine he se H/c:= gEH: j- I .gldp> 1 Fk > These se s (Hk) a e non-emp y, o(A" , A)-closed and hey ha e he ini e in e sec- ion p ope y because o kl < k2 we ha e J 2 I Wd~ I .hldp I o e e y h E Ax so ha HkZ c Hk,. Since H is o(A’, A)-compac , he e is a unc ion g E H such ha g E Hk o all k o , equi alen ly, J I . gl dp 2 1 o all k E N. Fk 54 Bu . g E L’(p). The e o e, he e is some 6 > 0 such ha i p(A) < 6 hen J, I . gl dp < 1, and his is in con adic ion wi h he inequali y abo e because limk p(Fk) = 0. (ii) + (iii): Le ( n) b e a d ec easing sequence in A ha con e ges o ze o p-a.e. and ake a no mal and a(nx, A)-bounded se H. Applying (ii) o = i E A we can ind an index N E N such ha Now, le cy =PH(xx,,,). We may assume ha cy # 0. (O he wise, e e y g E H would be ze o /l-a.e. in X, and he p oo o his implica ion would be inished.) Conside he measu able se s de ined o each II E N by: A, := {x E -=,: n(x) > 1/(4d’)}. Since p(XN) < +oc and ( n) con e ges o ze o p-a.e., we ha e ha lim, CL(&) = 0. By condi ion (ii), he e is m E N such ha ~~( i . XA,) < $. Now, o n 2 m we ha e ha I n(x)I < 1/(4a) on XN &.The e o e PH(h XX,) < PH(h . XA,) +PH( n ’ % A,) 5 PH( i XA,,,) + (l/(4a)) ‘PHh. A,) < ; + (1/(4a)). cy = 1. Hence, i n > m we ha e pH( n) 5 pH( n - n XX,) +pH( n XX,) < ; + ; = 1. (iii) =+ (i): W e a ways ha e Ax c A’, On he o he hand, ake 4 E A’. Fix an 1 index N E N. Fo each measu able se A c XN, de ine G(A) = I. Condi ion (iii) yields ha G is a a-addi i e measu e. Indeed, i (A,) is a sequence o disjoin measu able se s in X, wi h union A, apply (iii) o he unc ions de ined by , := XA - 2 X.4, E @ c A, o eachm= 1,2,... n=l Then ( m) con e ges o ze o o he s ong opology. Hence I = C,“=i ~(xA,) so ha G(A) = C,“=, G(A,). Le us see now ha G is absolu ely con inuous wi h espec o p: i we ha e a measu able se A C XN wi h p(A) = 0, hen XA = 0 p- a.e., hus G(A) = I = 0. Apply he Radon-Nikodym Theo em o deduce he exis ence, on each XN, o a unc ion gN E L’(p, X ) such ha G(A) = JgN dp o A c x . A I is clea ha i N 2 M, hen gN = gM on X, so ha he unc ion g = limN gN is well-de ined and locally in eg able. Mo eo e , o each compac K and each measu able se A c K, we ha e G(A) = s g dl.L. A 55 We p o e now ha g E Ax and 4( ) = ( , g). Suppose, wi hou loss o gene ali y, ha g > 0. We p oceed in se e al s eps. (1) Since 4 is linea , we ha e ha 4( ) = (g, ) o a simple unc ion wi h compac suppo . (2) Fo E @ ( he space o measu able and essen ially bounded unc ions wi h compac suppo ), le (&) b e a ne o simple unc ions wi h compac suppo such ha lim, Il( - u) XA llm = 0 w h e e A is he suppo o . I H is a no mal and a(Ax, A)-bounded se , hen we ha e limPH( - o?) = lim sup a J I - ol] . IhI dp: h E H > ~~11~ .PH(XA) = 0. N This p o es ha ( ol) con e ges o o he s ong opology p(A, Ax). Using his, (1) abo e and ha = lim, a in Lm(p, A), we ob ain (3) Now ake a posi i e unc ion E A. Fo each 12 E N, le n be he unc ion in @ de ined by: h(x):= {y i (x)<nandxEX, > o he wise. Then ( - n) is a dec easing sequence ha con e ges o ze o p-a.e. By condi ion (iii), n con e ges o o he opology @(A, A”). Using his and he Mono one Con e gence Theo em in L’ (p) we ha e (4) Finally, o a bi a y E A he equali y 4( ) = ( , g) ollows by linea i y. (i ) + (i): W e a wa s 1 y h a e Ax c A’. Now, ake 4 E A’. Then he e is some ab- solu ely con ex, closed and o(Ax, A)-bounded subse H o Ax such ha 4 E H”“, he bipola o H in A’. Since A(p(A, A”)) is sepa able, H”” is me izable [7, §21.3.(4)] and compac o he opology o(A’, A). The e o e, H is sequen ially dense in H”“. Bu , on he o he hand, H is o(Ax, A)-sequen ially comple e because i is closed in A’(c~(il’, A)) and his space is sequen ially comple e [3, P op. 121. Hence, c5 E H”” = H c Ax. Consequen ly, A’ c Ax. (i) + ( ): Fo e e y non-ze o g E Ax he e is a unc ion h con inuous and ha ing compac suppo such ha (h, g) # 0 [3, p. 981. Then, by he Hahn-Banach sepa a ion heo em, Cc(X) is dense in A o any opology such ha he dual o A isAx. (i) - (iii) + (i ) when he measu e space (X, p) is sepa able: Fo e e y non- ze o g E A” he e is a simple unc ion h ha ing compac suppo such ha (h,g) # 0. Condi ion (i) and he Hahn-Banach sepa a ion heo em ensu e ha he space SC(X) o simple unc ions ha ing compac suppo is dense in 56 n(,B(A, A”)). The e o e, we ha e o p o e ha S,(X) is sepa able o he s ong opology /?(A, A”). Since he measu e space (X, p) is sepa able, he e is a coun able amily C o measu able se s (we may, and do, assume ha his amily con ains all he se s o he o m Y ~ X, o Y E C and n E N) such ha o e e y measu able se A he e is a sequence ( Yj) om C wi h limi p(AnYj) = 0, whe e D s ands o he sym- me ic di e ence ope a o . I , in pa icula , A is con ained in X,,, hen bo h XA and xX, A a e in SC(X) c A. Since limj p(An Y,) = 0, we can apply condi ion (ii) o xA andxx, A = X& - XA ( ecall ha A C X,) o ob ain, o he s ong opol- ogy P(4 AX )9 O=limxAx~a ,=limxA(xA+x~-2xAx~)=li~(xA-xAxY;) .I I and 0 = lim(xx;, -xA)xAa , =lijn(x~ -XA)(XA +,YY,-~xAxY,) i = Ii,? k,~x , - x~xY,). I ollows ha limj (XA - x~~,~~,)) = 0 o he s ong opology p(il, Ax). This ensu es ha he coun able se D o all simple unc ions o he o m C c yxy, whe e each Y in he sum is om C and each QY is a ional, is dense in S,(X) o he s ong opology /?(A, /lx). ( ) =+ (i ) when X IS me izable: Acco ding o condi ion ( ), we ha e o p o e ha Cc(X) endowed wi h he es ic ion o he opology @(il, A”) is sepa able. Now, i X is me izable, hen each o he spaces C(Xn)( 1) 11,) is sepa able. Fo e e y n E FU, le D, be a coun able dense se in C(Xn)( j II,). Now, o E C(Xn) ake g E D, such ha I (x) -g(x) I 1 o all x E X,. Le H be a no mal and a(n”, A)-bounded subse o Ax. Since S and g a e sup- po ed in X,, we ha e ~~( ‘-g) = SUP . I -s/. Ihldp: h E H xn This shows ha D = (J, D, is a coun able se dense in Cc(X) o he s ong op- ology p(n, A);). q No es. J. Dieudonni: [3, $13 (p. 107)J claimed ha (i ) =+ (i). Y. K6mu a [5, Th. 1.31 showed he equi alence o (i) and (i ) o X = [w”. G.G. Lo en z 110, Th. 31 p o ed ha (iii) + (i) o he case when X is a ini e in e al in he eal line and A is a no med space. R. Welland [18, Th. 21 p o ed he equi alence o (i), (iii) and (i ) unde he hypo hesis ha he measu e space (X, ,u) is sepa able; he e we ha e shown ha his hypo hesis is no eally necessa y o p o e (i) ej (iii). Condi ion (ii) may be easie o use ha (iii) and he p oo o (ii) + (iii) abo e ollows he 57 ideas gi en by A.C. Zaanen in [19,§72]. Finally, condi ion ( ) has been conside ed by G. Sil e man o ansla ion in a ian pe ec spaces [17, Th. 2 and Th. 31. 3. THE STRONG DUAL OF AN ECHELON SPACE Le (g,) be an inc easing sequence o non-nega i e locally in eg able unc ions. The Ko he-dual o (gn) is called he echelon space associa ed o his sequence and we shall deno e i by A(g,). Thus A(a) = E Q: ~g,( ) := j- I (x)1 .8,(x) dp(x) < +cc x o alln=1,2,... When (g,) educes o a simple unc ion g, he space A(g) is commonly deno ed by Li and has many in e es ing p ope ies (see [3, #IO-121, [4] and [15]). No e ha we can w i e A(g,) = n, Li.. As we said in he in oduc ion, he main pu pose o his pape is o gi e he ollowing esul . Theo em 2. Le A = A(g,,) be he echelon space associa ed o an inc easing se- quence (g,) o non-nega i e locally in eg able unc ions. Then A is a pe ec space wi h Kii he-dual Ax = U, (Lin)‘. Mo eo e , A(p(A, Ax)) is a F che space wi h dual Ax and he s ong opology ,B(A, A”) is gene a ed by he amily o semino ms pg,: ~A+p,~( ) :=EI (x)l.g.(x)d~(x), n= 1,2,... qm: 6 A-+ qm( ) :=L I (x)Idp(x), m= I,&... P oo . Fi s , we p o e he heo em when (gn) educes o a single unc ion g E 6). Acco ding o [ 15, P op. 11, he s ong opology ,B( Li, (Li) ’ ) is gi en by he amily o semino ms { pg and qm, m = 1,2, . .} whe e pg( ) = Jx ) ) g dp. The space Lk is pe ec because i is he Ko he-dual o {g} and he e o e, i is comple e when endowed wi h he s ong opology [3, Thm. 51. Hence Lh(p(Li, (Lh)“)) is a F eche space. To p o e ha he opological dual o his space is (Li)‘, we shall apply condi ion (ii) o Theo em 1 abo e. Le E Lj. Ob iously, lim, qm( - . xx”) = 0 o e e y m E N, and since . g E L’(n), we also ha e lim pg( - XX.) = lip s 1 gl dp = 0. n x xl Now le (An) be a sequence o measu able se s wi h lim, p(A,) = 0. Since XX, is in eg able o each m, we ha e li,m qm( . x.4,) = lip j I I xx, dp = 0. A. 58 On he o he hand, using again ha g E L’(p), we ha e l$p,( . XA,) = 1,” l I I . gdp = 0. A” This inishes he p oo o LL. We u n now o he gene al case. Le T be he opology de ined by bo h se s o semino ms { ps, : II = 1,2, . . .} and {qm : m = 1,2, .}. Since n c Ljn o e e y n E N we ha e ha (Lin)’ c Ax and he e o e U, (LL”)” c Ax. Deno e by ,, he co esponding s ong opology ,B(Li., (Li”)’ ) on Lj.. Since g, < g,+l, he inclu- sion $+, (7,+1) + Lin (~~1 IS con inuous, so ha A(T) is he educed (because Cc(X) is dense in each o hese spaces by (i) + ( ) in Theo em 1) coun able p o- jec i e limi o a amily o F eche spaces. The e o e, .4(~) is a F eche space. Deno e by A’ he dual o n( ). No e ha T = @(A, A’). On he o he hand, A’ c u (L;n(Tn))’ = u (L;“y c AX. ” n The p oo will be inished i we show ha il” c A’. Take h E Ax and assume, wi hou loss o gene ali y, ha h 2 0. Call T he linea o m induced by h on il, T( ) = ( , h). Fo k = 1,2,. . de ine he unc ions h/‘(X) = i h(x), i h(x) < k and x E xk 0, o he wise, hen (hk) is an inc easing sequence ha con e ges poin wise o h. We can apply he Mono one Con e gence Theo em o deduce ha T( ) = S .hd,u=lip ~ ~hkd~=li~( ,hk). x x Now, obse e ha e e y hk E @ and ha he semino m I(., hk) 1 is domina ed by k qk. The e o e, T is he poin wise limi o he sequence ((., hk)) o -con inuous linea o ms. By he Banach-S einhaus Theo em, T is also -con inuous, i.e. hEA’. q One can see now ha he ouble in J. Dieudonne’s a i ma ion was ha he missed he amily o semino ms {qnl : m E N}. ACKNOWLEDGMENT This esea ch has been suppo ed by La Conseje ia de Educa ion y Ciencia de la Jun a de Andalucia. REFERENCES 1. Bie s ed , K.D., R. M&e and W. H. Summe s ~ Kii he se s and Kii he sequence spaces. Func- ional Analysis, Holomo phy and App oxima ion Theo y, No h-Holland Ma hema ical S udies 71, Else ie iNo h-Holland, Ams e dam, Ox o d and New Yo k, 27791 (1982). 2. Diaz, J.C. -S ic ly egula echelon Kii he spaces. A ch. Ma h. (Basel) 46,360-367 (1986). 3. Dieudonne, J. - Su les espaces de Ka he. J. Analyse Ma h. 1,81-115 (1951). 59