The space of Pettis integrable functions is barrelled

p oceedings o he ame ican ma hema ical socie y Volume 114, Numbe 3, MARCH 1992 THE SPACE OF PETTIS INTEGRABLE FUNCTIONS IS BARRELLED LECH DREWNOWSKI, MIGUEL FLORENCIO, AND PEDRO J. PAÚL (Communica ed by William J. Da is) Abs ac . I is well known ha he no med space o Pe is in eg able unc- ions om a ini e measu e space o a Banach space is no comple e in gene al. He e we p o e ha his space is always ba elled; his ells us ha we may ap- ply wo impo an esul s o his space, namely, he Banach-S einhaus uni o m boundedness p inciple and he closed g aph heo em. The p oo is based on a heo em s a ing ha a quasi-ba elled space ha ing a con enien Boolean alge- b a o p ojec ions is ba elled. We also use his heo em o gi e simila esul s o he spaces o Bochne in eg able unc ions. 1. Ba elledness o spaces wi h Boolean algeb as o p ojec ions This pape deals wi h some condi ions, ela ed o Boolean algeb as o p ojec- ions modelled o e a ini e measu e space, unde which quasi-ba elled spaces a e indeed ba elled. Ba elled spaces we e in oduced as he "good" class o locally con ex spaces, as domain spaces o ope a o s, o which uni o m bound- edness p inciples and closed g aph heo ems hold (see [6, §39.5(1), §34.6(9)] o [10, 4.1.3, 4.1.10, 7.1.12]). Al hough we e e he eade o he monog aphs [2, 6, 10] o he e ms used in his pape , we wan o ecall b ie ly he mos ele an de ini ions. Le F be a locally con ex space wi h dual E'. Deno e, espec i ely, by c (F/ , E) and /?(F', E) he co esponding weak and s ong opologies o he duali y on E'. A subse F o F is said o be a ba el i i is absolu ely con- ex, closed, and abso ben . I a ba el T abso bs e e y bounded subse o E hen i is called bo ni o ous. Ba els a e he pola s o ß(E', F)-bounded subse s o E'. A locally con ex space E is said o be ba elled ( esp. quasi- ba elled) i e e y ba el ( esp. bo ni o ous ba el) is a ze o-neighbo hood. Equi alen ly, E is ba elled ( esp. quasi-ba elled) i and only i e e y subse o E' ha is o(E', F)-bounded, i.e., poin wise bounded on E ( esp. ß(E',E)- bounded, i.e., uni o mly bounded on he bounded subse s o E ) is equicon- inuous. Me izable locally con ex spaces a e quasi-ba elled and Banach (and Recei ed by he edi o s July 6, 1990 and, in e ised o m, Augus 30, 1990. 1991 Ma hema ics Subjec Classi ica ion. P ima y 46A08, 46E40; Seconda y 46G10. Key wo ds and ph ases. Ba elled spaces, Pe is in eg al, Bochne in eg al, Boolean algeb as o p ojec ions. © 1992 Ame ican Ma hema ical Socie y 0002-9939/92 $1.00+ $.25 pe page 687 License o copy igh es ic ions may apply o edis ibu ion; see h ps://www.ams.o g/jou nal- e ms-o -use 688 LECH DREWNOWSKI, MIGUEL FLORENCIO, AND P. J. PAÚL F éche ) spaces a e ba elled; on he o he hand, he e a e no med spaces ha a e no ba elled [10, 4.1.8]. In wha ollows, ( i, X, p) s ands o a measu e space, whe e p is a ini e, posi i e, coun ably addi i e measu e de ined on a a-algeb a X o subse s o i. We a e going o use a a ian , s a ed in he nex de ini ion, o he concep o Boolean algeb a o p ojec ions in oduced by W. G. Bade (see [13] o a good lis o e e ences abou his opic). De ini ion. We say ha a locally con ex space E admi s an ( i, X, p)-Boolean algeb a o p ojec ions i he e exis s a se {Pa : A £ X} o linea p ojec ions in E such ha : (1) Pçi is he iden i y on E, PAnB = Pa- Pb o all A, B e X, and PAub = PA + PB o all disjoin A, B £ X. (2) PA is con inuous o e e y A £ X. (3) Fo e e y x £ E, he ec o measu e Fx : X —> E de ined by FxiA) := PaÍx) is p-con inuous, meaning limß(A)->o Pa(x) = 0. (No e ha his condi ion implies ha Fx is coun ably addi i e [2, 1.2.4] and, in pa icula , bounded.) The s anda d examples o Boolean algeb as o p ojec ions a ise in he cases when F is a space o measu able unc ions (o classes o unc ions) de ined on i and he p ojec ions a e de ined by Pa(x) := Xa • x, whe e xa is he cha ac e is ic unc ion o A e X. In hese examples, condi ions (l)-(3) a e usually easy o e i y o well known. Ou main esul , Theo em 1 below, will be applied only o me izable spaces in he nex sec ions o his pape . Howe e , we hink ha i is o independen in e es and may be applied o mo e gene al si ua ions; hence, we s a e and p o e Theo em 1 o quasi-ba elled spaces. We begin by ecalling a use ul ac abou ba els ha will be used in he sequel (see [10, 3.1.3]). Fac . Le E be a locally con ex space wi h dual E'. I a ba el T in E abso bs all null sequences in E hen T is bo ni o ous. Dually, i M is a a(E', E)- bounded subse o E' and M is uni o mly bounded on e e y null sequence in E, hen M is ß(E', E bounded. Lemma 1. Le E be a quasi-ba elled locally con ex space admi ing an ( i, X, p)-Boolean algeb a o p ojec ions {PA : A £ X} ; hen {PA : A £ X} is equicon inuous. P oo . Le U be an absolu ely con ex and closed ze o-neighbo hood in E. Conside he se T := {x £ E : Pa(x) £ U o e e y A £ X} . I is clea ha T is absolu ely con ex and closed. T is also abso ben because o e e y x £ E he ange o he ec o measu e Fx is bounded in E. Hence F is a ba el. Now, le ix„) be a null sequence in E. Since e e y Pa is con inuous, we ha e lim FXn iA) = lim PA(x„) = 0 o all A € S. n n This implies ha {FXniA) : n = 1, 2, ...} is bounded o e e y A £ X, so ha we may apply he Nikodym Boundedness Theo em [2, 1.3.1] o deduce ha {Pa : A e X} is uni o mly bounded on (x„) and, he e o e, ha T abso bs ix„). The ac quo ed abo e ells us ha F is a bo ni o ous ba el so ha , since License o copy igh es ic ions may apply o edis ibu ion; see h ps://www.ams.o g/jou nal- e ms-o -use THE SPACE OF PETTIS INTEGRABLE FUNCTIONS IS BARRELLED 689 E is quasi-ba elled, F is a ze o-neighbo hood. This p o es he equicon inui y o {PA : A € 2}. Q.E.D. Theo em 1. Le E be a quasi-ba elled locally con ex space admi ing an ( i, X, p)-Boolean algeb a o p ojec ions {Pa : A £ X} such ha he ollow- ing condi ion is sa is ied: (4) Whene e iA„) is a disjoin sequence in X and (x„) ¿s a null sequence in E such ha Pak(xn) = xn , « = 1,2,..., hen he e exis s x £ E and a sequence n(l) < n(2) < ■ ■ ■ in N such ha pAn(k)ix) = x„(k), k = 1,2, ... . iEqui alen ly, he e exis s a sequence «(1) < «(2) < • • ■ in N such ha J2xn(k) con e ges.) I ( i, X, p) is a omless, hen E is ba elled. P oo . Acco ding o he Fac abo e, we ha e o p o e ha i M c E' is o(E', F)-bounded and (xm) is a null sequence in E, hen sup{|(«, xm) : m £ N, u £ M} < +oo. Suppose, on he con a y, ha his sup emum is +oo . We s a by making a sliding-hump ype induc i e cons uc ion. S ep 1. Call io := i ; on accoun o ou assump ion we ha e sup{|(w, F i0(.xm))| : m £ N, u £ M} = - -co, (*) and, he e o e, we can ind ux e M and m(l)eN such ha 7 ■= {ui,PoúixmW)) > 1. Applying condi ion (3) o xm(X), he e exis s ô > 0 such ha whene e p(C) < Ô, (ux, Pc(xm{x))) <7i- 1. Since he measu e space is a omless, io can be w i en as a ini e union o measu able se s o measu e a mos ô . By (*), on one o hese se s, say ii, he co esponding sup emum mus be in ini e; hence sup{|(M, Pa Xm))] : m > m(l), u £ M} - +oo. (**) Se Ax := io ii ; hen, because p( ii) < ô, we ha e (ux, PAl(xm{X))) = |(»i, Fc^x^i,)) - (wi, Fn,(xm(,)))| > |("1 . Aio(*m(l))}| - l("l , lal *m(l))>l >?1 -i? -1) = L S ep 2. We s a om (**) inding u2 £ M and w(2) > w(l) such ha (u2, Pai(xm{2))) >2. P oceeding as in S ep 1, we can ind a se i2 C ii such ha sup{|(w, Pa2(xm)) : m > m(2), u e M} = +oo and |(«2, PA2(xm{2))) > 2, whe e A2 := i, i2 . License o copy igh es ic ions may apply o edis ibu ion; see h ps://www.ams.o g/jou nal- e ms-o -use 690 LECH DREWNOWSKI, MIGUEL FLORENCIO, AND P. J. PAÚL In his way, we a e able o ind a sequence (m„) in M, a disjoin sequence iA„) in X, and a subsequence (xm(nA o (x„) such ha (un, PAn(Xm(n))) > n, «=1,2,.... Since xm(n) -* 0 and, by he lemma abo e , {PA : A £ X} is equicon inuous, we ha e limn PAn(xm(nA — 0 so ha we may apply condi ion (4): he e exis s y £ E and a subsequence «(1) < «(2) < • • • such ha PAnW(y) = PAnW(xm{n{k))), k= 1,2, ... . Fo he sake o simplici y, le us w i e k :— u„(k) and Bk := An^ ; hen ( k,PBk(y)) >n(k)>k, k=l,2,.... Since M is o(E', F)-bounded, he scala measu es mk: A ^ mk(A) := ( k,PA(y)), k= 1,2,... o m a se wise bounded se , i.e., supijm c^)! : k = 1, 2, ...}.< +oo o all A £ X. The e o e, we may apply he Nikodym Boundedness Theo em [2, 1.3.1] o de- duce ha sup{|w c(^)| :A£Z,k=l,2,...}< +00, in con adic ion wi h he ac ha mk(Bk) = ( k,PBk(y)) >k, k=l,2,.... Q.E.D. Co olla y. Le E be a me izable locally con ex space admi ing an ( i, X, p)- Boolean algeb a o p ojec ions {Pa : A £ X} and sa is ying condi ion (4) o Theo em 1. I F is a closed subspace o E such ha Pa(F) c F o all A e X and ( i, X, p) is a omless, hen F is ba elled. Rema k. A locally con ex space E is said o ha e p ope y (K) i e e y null sequence (x„) in E has a subsequence (xn(kAk such ha ,xn[-k) is con e gen . This no ion was in oduced by S. Mazu and W. O licz and edisco e ed by P. An osik. P ope y (K) has been used o gi e nonca ego ical p oo s o se e al classical heo ems, as well as some new esul s, abou open mappings, closed g aphs, and ba elledness (see [1] and e e ences he ein o [10, 1.2.15, 1.4]). I is clea ha p ope y (K) implies condi ion (4) o spaces admi ing an ( i, X, p)-Boolean algeb a o p ojec ions. On he o he hand, we shall gi e in his pape wo examples, in Rema k 3 o §2 and Rema k 2 o §3, o spaces sa is ying condi ion (4) bu no p ope y (K). No e, howe e , ha he heo em o [1] s a es ha e e y me izable space ha ing p ope y (K) is ba elled, no needing he hypo hesis o he exis ence o an ( i, X, p)-Boolean algeb a o p ojec ions. 2. Applica ion o spaces o Pe is in eg able unc ions Le X be a Banach space. A weakly p-measu able unc ion /: i -» X is said o be Pe is in eg able i he composi ion -* (x*, ( )) is a unc ion in Lx(p) o e e y x* £ X' and i o e e y measu able se A he e is an elemen in X called he Pe is in eg al o / o e A and deno ed by JA dp such ha (x*, j dp = j (x*, ( )) dpi ) o all x* £ X'. License o copy igh es ic ions may apply o edis ibu ion; see h ps://www.ams.o g/jou nal- e ms-o -use THE SPACE OF PETTIS INTEGRABLE FUNCTIONS IS BARRELLED 691 We shall deno e by S®(p, X) he space o (classes o ) Pe is in eg able unc- ions /: i —> X, endowed wi h i s na u al no m gi en by he o mula := sup {¡ {x i )) dpi ) :x*£X', ||x*||<l}. No e ha jA dp < o e e y / e 0>(ji, X) and A £ X. Fo mo e de ails, we e e he eade o [2, II.3; 4, 3.7] and he o iginal pape by Pe is [11, S3]. Fo he case when p is he Lebesgue measu e in [0, 1 ], Pe is showed ha he no med space 3a(p, L2(p)) is noncomple e [11, 9.4]. La e , i was p o ed by Thomas [14, p. 131] and Janicka and Kal on [5] ha he same holds i we eplace L2 wi h an a bi a y in ini e-dimensional Banach space; he o me au ho ema ks ha he esul is alid o any a omless ini e measu e space. We shall p o e ha &>(p,X) is ba elled o e e y ini e measu e space. As we poin ed ou in he p e ious sec ion, his esul will make i possible o apply he Banach-S einhaus and Closed G aph Theo ems o ¿Pip, X), in spi e o he ac ha his space is noncomple e, in gene al. Theo em 2. Le X be any Banach space; hen he no med space £Pip,X) o Pe is in eg able unc ions is ba elled. P oo . Assume i s ha ( i, X, p) is a omless. We shall apply Theo em 1. Fo A £ X and / £ â°ip, X), de ine PAi ) '■= Xa* ■ I is clea ha PA is a linea p ojec ion in &(p, X) and, since ||P¿(/)|| = %a ' /II < 11/11 » i ollows ha {Pa : A £ X} is equicon inuous. Condi ion (3) holds by [11, 2.51] (see also [4, 3.7.2]). Condi ion (4): Le (An) be a disjoin sequence in X and i ) a null se- quence in S^(p, X) wi h XA„' n = n ■ We show ha any absolu ely summable subsequence o ( n) ul ills he equi emen s in (4). Wi hou loss o gene ali y, assume ha a := ¿„ n is ini e. Since he /„ a e disjoin ly suppo ed, he poin wise sum ( ) := ¿„/»(0 exis s o all £ i; xa„ - = n o e e y « = 1,2,...; and o e e y x* £ X' we ha e / (x*, i )) dpi ) = [JT (x*, n( )) dp( ) < a x* , Ja Jsin=x so ha ix*, /(•)) is in Fi(p). Mo eo e , o e e y A £ X, we ha e Y,n II JA ndp < a, hence he se ies £„ JA n dp o he co esponding Pe is in eg als con e ges o some elemen xA in he Banach space X. Now, o e e y x* £ X' and A £ X we ha e (X* , XA) - V (x* , [ ndp) = jï2[ (X* , ( )) dßi ) = (x* , ( )) dp( ). n= V JA I n=x JA An JA Hence / is Pe is in eg able. This inishes he p oo o he case when ( i, X, p) is a omless. Now suppose ha ( i, X, p) con ains a oms. W i e i as ÇlnA U i^ , whe e p is a omless on i„^ and pu ely a omic on i^ . I is clea ha 3s (p, X) is he opological di ec sum o he spaces o Pe is in eg able unc ions on Q„a and CIa > espec i ely. O hese wo spaces, he one co esponding o Q.„a is ba elled as we ha e seen. On he o he hand, i he e is a ini e numbe m License o copy igh es ic ions may apply o edis ibu ion; see h ps://www.ams.o g/jou nal- e ms-o -use 692 LECH DREWNOWSKI, MIGUEL FLORENCIO, AND P. J. PAÚL o a oms, he space co esponding o i^ is isomo phic o Xm ; and, on he con a y, i he numbe o a oms is coun able hen i can be iden i ied wi h he Banach space o uncondi ionally con e gen se ies in X. Thus in any case, ¿®(p, X) is he opological di ec sum o a ba elled space plus a Banach space, hence i is ba elled. Q.E.D. Rema ks. 1. The p oo o Theo em 2 can be al e ed o s udy he subspace &>„ip, X) o 3°ip, X) o med by hose unc ions /: i — X wi h (x*, i-)) in Lg(p) o e e y x* £ X' [11]; when endowed wi h he no m U/H, :=sup{||(x*,/(. X* £X', <!}, ¿?q(p, X) is a noncomple e (in gene al) no med space. I can be p o ed, using he same echnique as abo e, ha his space is ba elled as well. 2. The space ¿®(p, X) is also ba elled when he measu e space is c - ini e. The p oo ha we ha e ound o his esul , howe e , does no ollow di ec ly om Theo em 2; i is a consequence o a gene aliza ion o he sliding-hump echnique o Theo em 1. This, and ela ed esul s, will be included in a o h- coming pape [3], whe e we e e he in e es ed eade . 3. We shall p o e now ha ¿®(p, L2(p)) (whe e p is he Lebesgue mea- su e on he uni in e al) does no ha e p ope y (K). We use Pe is's ex- ample [11, 9.4] wi h a sligh change o no a ion. Le (y/j) be a comple e o - hono mal sequence in L2(p), (A,) a sequence o consecu i e segmen s in N such ha A, has 2' elemen s, and (F7) a sequence o in e als such ha o ieN, he amily (F, : / e A,) is he pa i ion o [0, 1) in o subin e als [(/-1)/2', 1/2'), I = 1, 2, ... , 2''. Conside he simple unc ions i| : [0, 1] -► Lain) de ined by M )-=YlVj-Xji ), whe e Xj deno es he cha ac e is ic unc ion o Ej. The sequence (/) con- e ges o ze o in ^(p,L2(p)) [11, 9.4]. Suppose ha a subse ies J^,k i(k) con e ges in S®ip, L2ip)) o a unc ion /. Then, o each ixed index ko and e e y ;' £ Ai{ko), we ha e 0lim n—>oo/ ■Ao.i] «0,¿¿w (0-/(0 k= dß( )= j Xj( )-(Wj, ( )) dp( ), J[0, ] because, by o hono mali y, ZU =i(V0'»/'(&)(0) — Xj(() o n>ko and £ [0, 1]. I ollows ha he e is a p-null se ./V, c Ej such ha (xpj, ( )) = 1 o all £ Ej Nj . Le N be he union o all Nj (j £ Ai{k), k = 1,2, ...). Now, i £ [0, 1], hen o e e y k £ N he e exis s j(k) £ A,(y ) such ha £ F;( c). Hence, i addi ionally £ N, hen (xpj(k), ( )) = 1 o e e y k £ N. In consequence, o almos all is [0,1], ( ) is a unc ion in L2 ( emembe /: [0, ]—>L2) ha has an in ini e numbe o Fou ie coe icien s equal o 1, which is clea ly impossible. 3. Applica ion o spaces o Bochne in eg able unc ions Le L(q) be a solid Banach la ice o measu able scala unc ions p: i —► R. (As usual, we iden i y unc ions ha a e equal p-a.e.) Thus i j> e L and ip is a measu able unc ion such ha <p(-) < |< H-)l p-a.e., hen xp £ L and License o copy igh es ic ions may apply o edis ibu ion; see h ps://www.ams.o g/jou nal- e ms-o -use THE SPACE OF PETTIS INTEGRABLE FUNCTIONS IS BARRELLED 693 q( ) < Q(4>) ■ Examples o L a e he classical Lpip) spaces (1 < p < oo), O licz spaces, and Kö he no med spaces. These and mo e examples can be ound in, e.g., [7, 9, 15], o [16, Chap e 15]. Recall ha L is said o be o de -con inuous i lim Xa • 4> - 0 o all 0 e F . ß{A)^0 This is sa is ied, o ins ance, when L — LPip) (1 < p < oo) o L = F<j>(p) is an O licz space, whe e O sa is ies he A2 condi ion. De ini ion. Le X be a no med space (no necessa ily comple e). We de ine he space L(X) as he space o (classes o ) s ongly measu able unc ions /: i —» X such ha he scala unc ion >(-) := ||/(-)|| is in L. We shall conside on L(X) he opology de ined by he no m -* ?(||/(-)l|). In gene al, L(X) will be noncomple e. When L = Lp(p) and A' is a dense subspace o a Banach space Y, L(X) = Lp(p, X) is a dense subspace o he space Lp(p, Y) o T- alued, Bochne p-in eg able unc ions [2, II.2]. O he examples can be ound in [16, Ch. 15]. Theo em 3. Le L be an o de -con inuous solid Banach la ice o measu able unc ions de ined o e an a omless, ini e measu e space ( i, X, p), and le X be a no med space. Then he space L(X), endowed wi h he co esponding no med opology, is ba elled. In pa icula , LP(X) is ba elled o 1 < p < oo. P oo . We shall apply Theo em 1. Fo A £ X and / £ L(X), de ine Pa(P) '■= Xa • "• Then i is clea , by he assump ions on L and he de ini ion o he opology in L(X), ha {Pa : A £ X} is an ( i, X, p)-Boolean algeb a o p ojec ions in L(X). To e i y condi ion (4), le (A„) be a disjoin sequence in X and ( „) a null sequence in L(X) such ha PA„( n) = n- The ac ha /„ —> 0 in L(X) implies ha ||/i(-)ll -» 0 as unc ions in L. Since F is a Banach space, he e exis s a subsequence «(1) < «(2) < • • • such ha he se ies £ ||/,( c)(*)ll con e ges in L, and i is clea ha he sum o his se ies is, p ecisely, he poin wise sum (in R). Now, he unc ion /(•) := E /«(£)(•) (poin wise sum in X) is s ongly measu able and ||/(-)|| = E ll/«(/ )(OII € F, so ha / £ L(X) and / • XA„m = n(k) o all k = 1, 2, ... . Q.E.D. Rema ks. 1. This heo em is a bi s iking since we do no equi e X o be ba elled. I is indeed he case ha e e y sepa able Banach space Y con ains a dense subspace X ha is no ba elled and, ne e heless, we ha e ha Lx (p, X) is a dense ba elled subspace o Lx (p, Y) i ( i, X, p) is a omless. 2. Take ( i, X, p) as he uni in e al wi h Lebesgue measu e and X a no med space no sa is ying p ope y (K) [10, 1.2.16(b)]. Then LX(X) sa is ies condi ion (4) as we saw in he p oo o Theo em 3, bu Lx (X) does no ha e p ope y (K) : simply no e ha he cons an unc ions o m a closed subspace o LX(X). 3. I is easy o see, using he same me hod o p oo , ha Theo em 3 can be ex ended o he case when L is an o de -con inuous F éche la ice (some examples may be ound in [8, 9, 12]). Mo eo e , using a diagonal p ocedu e in he e i ica ion o condi ion (4), one can show ha Theo em 3 also holds when X is me ely me izable (L(X) being de ined in an analogous way). License o copy igh es ic ions may apply o edis ibu ion; see h ps://www.ams.o g/jou nal- e ms-o -use 694 LECH DREWNOWSKI, MIGUEL FLORENCIO, AND P. J. PAUL No e. M. Flo encio, P. J. Paúl and C. Sáez ga e a p oo o Theo em 3 o he case o he space LX(X) unde es ic i e hypo heses, namely, ha ( i, X, p) is an a omless Radon measu e space and X' sa is ies he Radon-Nikodym p op- e y. This was p esen ed o he II Con e ence on Func ion Spaces held Sep em- be 1989 in Poznan, whe e he collabo a ion wi h L. D ewnowski s a ed. The au ho s would like o hank F. Bombai (Mad id), F. F eniche (Se illa), P. G eim (Cha les on, Sou h Ca olina), and he e e ee o hei help ul commen s and ema ks. Thanks also o La Conseje ía de Educación y Ciencia de la Jun a de Andalucía o pa ially suppo ing he s ays o P. J. Paúl a Poznañ and L. D ewnowski a Se ille. Re e ences 1. P. An osik and J. Bu zyk, Sequen ial condi ions o ba elledness and bo nology, Bull. Acad. Polon. Sei. Sé . Sei. Ma h. 35 (1987), 457-459. 2. J. Dies el and J. J. Uhl, Vec o measu es, Ma h Su eys and Monog aphs, ol. 15, Ame . Ma h. Soc., P o idence, RI, 1977. 3. L. D ewnowski, M. Flo encio, and P. J. Paúl, Uni o m boundedness o ope a o s and ba - elledness in spaces wi h Boolean algeb as o p ojec ions. IV Con egno di Analisi Reale e Teo ía de Misu a, Cap i, I aly, 1990, ( o appea ). 4. E. Hille and R. S. 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Reihe , Weig hed induc i e and p ojec i e limi s o no med Kö he unc ion spaces, Resul- a e Ma h. 13(1988), 147-161. 13. W. J. Ricke and H. H. Schae e , The uni o mly closed algeb a gene a ed by a comple e Boolean algeb a o p ojec ions, Ma h. Z. 201 (1989), 429-439. 14. G. E. F. Thomas, To ally summable unc ions wi h alues in locally con ex spaces, Measu e Theo y (P oc. Con . Obe wol ach, 1975), Lec u e No es in Ma h., ol. 541, Sp inge -Ve lag, Be lin, Heidelbe g, and New Yo k, 1976, pp. 117-131. 15. R. Weiland, On Kö he spaces, T ans. Ame . Ma h. Soc. 112 (1964), 267-277. 16. A. C. Zaanen, In eg a ion, No h-Holland, Ams e dam, 1967. L. D ewnowski, Ins y u Ma ema yki, Uniw. Adama Mickiewicza, ul. Ma ejki 48/49, 60-769 Poznan, Poland M. Flo encio, P. J. Paúl, E. S. Ingenie os Indus iales, A da. Reina Me cedes s/n, 41012- Se illa, Spain License o copy igh es ic ions may apply o edis ibu ion; see h ps://www.ams.o g/jou nal- e ms-o -use