transactions of the american mathematical society Volume 263, Number 2, February 1981
BASIC SEQUENCES AND SUBSPACES IN LORENTZ SEQUENCE SPACES WITHOUT LOCAL CONVEXITY BY NICOLAE POPA 1
Abstract. After some preliminary results (§ 1), we give in §2 another proof of the result of N. J. Kalton [5] concerning the unicity of the unconditional bases of lp, 0
1. Preliminary results. Let Ibea real linear space and 0
norm (or briefly a norm) if the following conditions are verified. 1. ||x|| = 0 if and only if x = 0. 2. ||ax|| = \a\p||;c|| for x G X and a G R. 3. H* + y\\ < \\x\\ + \\y\\ for x,y G X. Then the subsets U„ = {x G X: \\x\\ < n~x), for n G N, constitute a fundamen- tal system of neighbourhoods of zero for a metric linear topology of X. If X is complete with respect to this topology we say that X is ap-Banach space. A sequence (x„)^°_i in X is called a basis if for every x G X there is a unique sequence of scalars (a„)^x such that x = 2°1, a¡x¡. The following lemma is essentially known (see Theorem III.6.1, Theorem 6.5 of [10]): Lemma 1.1. Let (xi)f=x be a sequence in X. The following assertions are equivalent: 1. 77ie series S".| •*„(„)converges for every permutation v of the integers.
Received by the editors September 13, 1979 and, in revised form, December 4, 1979. AMS (MOS) subject classifications(1970). Primary 46A45, 46A10; Secondary 46A35. Key words and phrases. />-Banach spaces, symmetric bases, complemented subspaces. 'Supported by Alexander von Humboldt Foundation.
© 1981 American Mathematical Society 0002-9947/81/0000-O058/SO7.50 431
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2. The series 2,e/) x¡ converges for every subset A c N. 3. For every e > 0 there exists an integer n such that ||2ie//Jc,|| < efor every finite set of integers 77 which satisfies min{< G 77} > n. 4. For any bounded sequence of real numbers (aB)"_1; the series SjLj aixi con- verges, when 2°!, xi converges. The proof will be omitted. A basis (xn)^=x in X is said to be unconditional if for every X ^ x = ~2*LXa¡x,, the sequence (anxn)™=, verifies one of the equivalent assertions of Lemma 1.1. If 0 < inf„ ||x„|| < supj|jcj| < + oo, we say that the basis (xn)^_x is bounded. The following corollary is also known.
Corollary 1.2. Let (x¡)^Lx be an unconditional bounded basis of X. Then
x\\\ = sup 2 a,btxt <+oo. (1.1) IM< i í=i We have also
Lemma 1.3. The space X with the p-norm is a p-Banach space.
Proof. Let (xk)k"=x be a Cauchy sequence in (A', ||| |||) and let e > 0. Then there exists the sequence of integers (nk)kx>=xso that |||jcn — xn'||| < e for n > nx and Ujjfifc_ x"*+i||| < e/2* for every k G N. Since (X, || ||) is a /7-Banach space and ||jc|| < 111*111,there exists x = Sn°_,(x','+' - x"") + x"' G X and |||jc < 2í for n > nx. □
Corollary 1.4. There exists a constant 0 < M < +oo such that
2 a>bixi< M 2 «,•*,■sup|è,|' (1.2) i=i í=i where (x,-)," , is an unconditional bounded basis of X and x = 2," x a¡x¡ G X.
Proof. It follows by Lemma 1.3 and by the open mapping theorem. □ Two bases (xn)™=, and (y„)^ x of X are equivalent and we write (x„) — (yn), if for every sequence of scalars (an)™=x, S^L, anxn converges if and only if S^ anyn converges. A basis (xn)™=x of X is called symmetric if every permutation (x^n))^_, of (xn)™=, is a basis of X equivalent to (xn)™_,. Let w = (w,-)," , G c0\ /,, where 1 = w, > w2 > • • • > w„ > • • • > 0. For a fixed/7 with 0 < p < 1, let
) = \a = (a,.)", G c0: \\a\\p,w= sup 2 |äWof ' w¡ < +0° I *■ i =i where 77is an arbitrary permutation of the integers. Then X = (d(w,p), || || ) is a/7-Banach space and the canonical basis (xn)"_,, is a symmetric basis of X (see [4]). If p > 1 we can define analogously d(w,p), which is a Banach space under the norm
(00 \ i//> 2 I«„(,/ ■w, , a - (a,)"., e ). 1 = 1 /
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A sequence (y„)^ x in a /7-Banach space with a basis (x„)"_, is called a block basic sequence of (x„)^°_,, if there is an increasing sequence of integers (p„)^x such that.y„ = 2f_+;„+ i <*iX,with (aX-i scalars. It is known that (>»„)"_, is a basis of Sp{.y„: n G N} (see II.5.6 of [10]). If E and F are two Banach spaces and 0 < r < + oo, we say that the linear bounded operator T: E —»F is r-absolutely summing if, for any finite set of elements (jc,)"_ , of 7s, there is c > 0 such that (¿ll/Xlf) Denote the set of all r-absolutely summing operators from TJ to F by Pr(E, F). If T G Pr(E, F) and r > 1 then wr(T) = inf{c > 0: c verifies (1.3)} is a norm on Pr(E, F), which is a Banach space with respect to this norm. We recall here two more theorems which we need: If A is an infinite matrix of real numbers (ay-)"_, and 0 < p < 1, then we denote by i/P IM || ooj, = SUp'(2 AÍJ ajkXk <1\ 7 where \\x\\x = supteN|^|. Theorem 1.5 (see Theorem 2 of [2]). Let 0 mf{\\d\\p/{X.p)-\\B\\xy.A = (diagrf) ° B) < K\\A\\X,P (1.4) where 2KAI"'}) diag í/ ù //ie matrix which has on the diagonal the numbers dn, and K is a positive constant depending only on p. Theorem 1.6 (see Theorem 94 of [9]). Let (X, p) be a measure space and H a Hilbert space. Then any linear bounded operator from LX(X, p) on H is p-absolutely summing for all 0 < p < oo. (In fact Maurey proved a stronger version of Theorem 1.6.) 2. The unicity of the unconditional bases of lp, 0 Theorem 2.1. Any two unconditional bounded bases of lp, 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 434 NICOLAE POPA (assumed normalized) basis and let^ = 2°1, aixi an element of lp, 0 lp be the operator defined by the infinite matrix A = (a¡by)°°_x. By (1.2) it follows that A is a continuous linear operator, and moreover !//> !//> ooj, = sup 2 \atxt A/ 2 a,*, (2.1) W< i Theorem 1.5 shows us that for every e > 0 there are K = K(p) > 0, the diagonal matrix D = (didy)?J=x, where d = (4)», G lp/{1_p) and the matrix C = (cy)°°_, such that 4 - D ■C, CO / CO \ ||C||oo,i - sup 2 2 V/,k < +00, (2.2) I\I<1 7=1Vi=l / !4./(i-,>iq«,i<*lMIU,+ e- Theorem 2.b.7 of [8] says that each linear bounded operator T: c0 —»/, is 2-abso- lutely summing, consequently tr2(C) < KG\\C\\^X (2.3) where C is the operator defined by the matrix C and KG is a universal constant. Note now that by Holder's inequality we have |/7||= suP (ÍI4AI") '<|A- ai-/') (2.4) iiftii, (oo \l/2 / oo \l/2 = (hy(2.2)) = lf1\\DC(ei)\\2/p\l \,= i / <||ß||(f ||Ce,.||j < (by (2.3) and (2.4)) 1/2 < KG(K\\A\\X,P + e) < (by (2.1)) »/> < KKGM 2 a¡X¡ + eKG. i = i Since e is arbitrarily small it follows that, for every y = SJLi aixi G 7,, 1/2 i/> (¿N < MKKn 2 a,-*,l (2.5) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LORENTZ SEQUENCE SPACES 435 Consequently the operator U: lp —*l2 defined by U(x¡) = e¡, i = 1,2,. is continuous and verifies IMI,,2= SUP 2 -M : a = 2 «/*,■>\\a\\p < 1 [ < KKGM, (2.6) i= i i=i Denoting by Sq the unit ball of lq, 0 < q < oo, and by T^S^) the closure for the topology of /, of the convex and balanced hull of Sp, it is easy to see that S, £ r(1)(Sp. Then, by (2.6), it follows that £/(/, n sx) c u(f^(sp) n lp) Q (KKGM)-lTw(S2) c (KGKM)-XS2, and this relation implies that U can be extended to an operator V: lx -* l2 such that we have || V\\ < KGKM. (2.7) But Theorem 2.b.6 of [8] says that each linear bounded operator V: lx —»l2 is a 1-absolutely summing operator and ttx(V) < KG\\ V\\ < KGKM. Applying Theorem 1.6 it follows that V G Pp(lx, l^, hence there is a positive constant Mx depending only on p such that ■n(V) < KGKMMX = A\. (2.8) (2.8) implies that (oo \l//> / oo \i/p 2kr) =(2iin«,*,)if) P\x/P < *i sup 2 |a, 2 V» (2.9) |X,|<1\,= 1 7=1 On the other hand, denoting by C the adjoint of the operator C, we have ,p\ i/> sup 2 io,r 2 V. = (by (2.2)) |\|\»//> = SUP 2 \di 2 V,7 < (by Holder's inequality) IM < sup 2 2 Vy.•HIp/O-p) IM License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 436 NICOLAE POPA 3. Symmetric basic sequences in d(w,p), 0 Lemma 3.1. Let (xn)^°_, be the canonical basis in d(w,p), 0 Pn><>'j For the sequence of scalars (A,-),™x we have 2 V, 2 a, 2 «,*,+ 2 «,*, 7=1 7=1 =/>„ + ! / \''=0 + ' oo I Pt+i 2 M 2 <*txi 2 \ 2 a¡X¡ 7=1 \'=0+l 7=1 V'=/>,+ ! Pm+l 2 \\ 2 a,xt - 2 IV 2 «i^i 7=1 \i-0+l 7-1 > (since {/* - ^ + 1, . . . ,Pak + l - Pnk) n {rj - p»j + l> ■■ ■ -/Vi ~ p",) = 0and {rk + 1, ...,7\ + i} D {/) + 1.*Vm} - 0 for k*j) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LORENTZ SEQUENCE SPACES 437 > 2 fcfl 2 («,r--Q.,_0+,.- 2 (e/^ri^r 7=1 \'=o+l / 7=1 > 2 |a/( 2+'(«,r--,.-J -\ei->iv- ir']-8up|^ 7=1 \' = r,+ l 7 7 > 2 |A/[l - ep2-p(i+X)- t¡n^(lf - I)"1] 7=1 ep(2p+x - 1) > 1 - 2 iV- 22'(2'' - 1) 7-1 On the other hand we have 2 V* < 2 IM>Ji = 2 \h\p-a 7=1 7=1 7=1 We shall use forward a special block basic sequence. Let (•*„)"_ 1 be a symmetric basis in the /7-Banach space X. If 0 =?*=a = 2~_i a,,*,, G A' and if (/»,-),*! is an increasing sequence of integers, let >>^a)= 2^ +1 a¡_p x¡, n G N. Then (y„a))n°„x is a bounded block basic sequence of (xn)"_,, and we shall call it a block of type I 0/ (*X=,- Theorem 3.2. Every bounded block basic sequence of (xn)n°^x in d(w,p), 0 ■ ■ ■ > ap > 0 for every n G N. If supn(pa+l — pn) < + 00, then it is clear that (y„)n ~ (*„)„, hence (.y„)j¡°_! is equivalent to a block basic sequence of type I of (xn)x-v Assume now that sxxr>n(pn+x - pn) = 00. Let b¡ = sup„|aPn + ,| for 1*G N. It is easy to prove that lim, b¡ = 0. Case I. Assume that for every e > 0 there exists m such that \\\ZP"JP+m a¡x¡\\ < e for every n so that/7„+1 — />„ > m. Since supn(//n+1 — /?„) = +00, we may assume that pn + 2- pn+x > Pn+X- Pn for every n G N. Define now z„ = 2fi+1'";'" «/+,„-*,•> « G N. Then ||zn||PjM,= H.yJL^H,= 1 for n G N. By hypothesis and using the fact that there is a subsequence (nk)^x such that the sequences (aj+p )"_, converge simultaneously for i < m, we can find a Cauchy subsequence of (z„)*_,. Hence we may assume that limn zn = z = 2°L] c,*, G d(w,p). It is clear that z =£ 0. Since (y„)%L! is a bounded block basic sequence of (.*„)£_, it is well known that K = supJI-PJI < +00, where 7>n(2°°_ia¡y¡) = 2"_, aiyi (see III.2.11 of [10]). Conse- quently we can find a subsequence (z )* such that 2" < 1/2 JC. Define now p^+i Ui = Zd Ck-pnXki i = L 2, . . k=P« + l License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 438 NICOLAE POPA Then (w,-)," , is a block basic sequence of type I of (xn)na_x and 2°l1||>'n,. — u¡\\pw < 2/liH^ - z\\PiW< 1/2K, hence by the Krein-Milman-Rutman Theorem (see The- orem III.2.13 of [10]) it follows that (y„)^x ~ («,),_,. Case II. There exists an e > 0 such that for every m G N there exists n(m) such that pn+x — p„ > m and ||2f^¡ + m a¡x¡\\PtW> e. Then there exists an increasing sequence of integers (i,-),™i such that Pn,*\ t>n,+i-Pn,>i and 2 > e for some i G N. 7 =/>* + ' P.w Since lim, b¡ = 0 we may assume that (a)JL \ is a decreasing sequence. Let Pm+l z, = 2 ajXj for i G N. 7=/%+l Then e < \\z¡\\p„ < H^H^ = 1 for i G N, also (z,)°l, is a bounded block basic sequence of (*„)"_! and the coefficients of z, converge to zero. By Lemma 3.1 it follows that there exists a subsequence (r,),"] of (z,)°l, which is equivalent to the canonical basis (>w < oo for every sequence (è,-),",) then l|2J_! ¿¿yJI^H. < °o implies that '2'*Lx\di\p < oo for every sequence of scalars ( Proposition 3.3. A bounded set A c d(w,p), 0 0 there is n G N such that oo sup 2 |«U./-W< Proof. If d(w,p) d; /,, let a = (ai)xmXG d(w,p) \ /„ A = {x¡: i G N} and / G d(w,p)'. We shall show that lim /(*,.) = 0. (3.3) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LORENTZ SEQUENCE SPACES 439 Indeed if (3.3) is not true, there exist the sequence (/*)"_ i of integers and a > 0 such that \f(x¡ )\ > a for every k G N. We consider b = (b¡)fix, where _ j aj sign fixh), i = ij,j G N, 0, i ¥* ij,j G N. It is clear that b G d(w,p) \ /,. But oo = a. 2°l,|a,| < |2°l,/(x,)è,| < oo, which is contradictory. Then (3.3) is true and A is weakly relatively compact. On the other hand (3.2) is not verified for A and p = 1, hence A is not a relatively compact subset of d(w, 1). Since the canonical mapping i: d(w,p)—> d(w, 1), 0 0 such that, for every a G d(w, p), we have i=i2 H < *|M|#. (3.4) It is clear that it suffices to prove (3.4) only for positive decreasing sequences (a,),°l, G d(w,p). If (3.4) is not true, then for every n G N there is the positive decreasing sequence a{n) = (a/*0),", G d(w, p), such that 2"||a(n)|^ < 2 a\n\ (3.5) i = i Denote by £<">= l-lé%jL*¿». Then ||2-., &<">|L„< 2r_1||Z><%„ < 2"_11/2"P < oo, consequently it follows that b = 2"_, b(n) G /,. But (3.5) implies that 1 < 2°°_i bfn) for every n G N, hence ||2~_1¿>(,,)||1= 2~_, 2°°_i bfn) = + oo, which is a contradiction. Hence / is continuous. On the other hand the canonical mapping J: I ^>d(w,p) is continuous and consequently the extensions /: lp —» d(w, p) and J: d(w, p) -» /, are continuous. Since clearly lp = /, for 0 < p < 1, and J ° I = id,,it follows that d(w,p) = lx. □ We shall give an example of space d(w,p), 0 1.) Example 3.5. Let wn = (1 + |log «|)_1 for n > 1 and 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 440 NICOLAE POPA the locally convex Orlicz sequence space lM = {a = (a,)*,: \\a\\M = 2°1, M(\a¡\) < + a}. We shall show that lM = d(w,p). (3.6) It is clear that lM = lM , and that d(w, 1) = lM¡ implies that d(w,p) = lM, where f f6(0,l], 1 + |log t"\ ' Mq(t) = 0 Theorem 4.e.2 of [8] says that d(w, 1) = lM if and only if there exists y > 0 such inat 2 l/W'\ywn)<+oo (3.7) n = l where W(x) = (I + log x)-1 for x > 1. In our case W~x(ywn) = ex/^y~i)nx/y, hence, for every 0 < y < 1, (3.7) and also (3.6) are true. But Theorem 3.3 of [5] shows us that d(w,p) = l^, where M is the largest Orlicz convex function on [0, 1] such that M(x) < M(x), x G [0, 1]. Since K(p)t < tp/(l + |log i|) < tp for / G (0, 1], where K(p) > 0 depends only on p, it follows that M is equivalent to the function N(t) = t, hence dJ^Tp)= lû = /,- (3.8) By Proposition 3.4 it follows that d(w, p) c /,. □ Remark 3.6.1. Since M(t) is not equivalent to N(t) = tp (i.e. there are not the relations 0 < inf,e(0 X]M(t)/N(t) < sup/e(0 x]M(t)/N(t) < + oo) the previous space d(w,p) is a/7-Banach space, other than lp, whose dual is lx. 2. This space is moreover an example of a /7-Banach space X with a unique unconditional basis, other than lp, for which X has a unique unconditional basis. Indeed limx_>0M(x)/x = +oo, hence by Theorem 7.6 of [5], d(w,p) = lM has a unique unconditional basis. □ There are spaces d(w,p), 0 Proposition 3.7. Let (w,.)(" , G c0\ lxbe a decreasing sequence of positive numbers such that there exist an increasing sequence of integers (nf)JL x, the scalars b > 0 and 0 < y < 77/(1 - p) (where 0 "7+1 _ "7 < "7+2 ~ M/+i> / e N- (3-9) nj+x-nj License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LORENTZ SEQUENCE SPACES 441 Proof. Let if i < nx, " \j-ß/p(nJ+x - nj)-l/p Un, < / < «,+ 1, where 0<ß "7+i w \\a\\P,w= 2 wt + 2 2 ,_i j=x , = „.+ ! jß(nJ+x - nj) "l OO 1 < 2 y>i+2 -^z < + ». 1=1 7=1 / On the other hand "j OO B7+l HI.= 2i+2 2 , = 1 y.i , = „, + 1 jP/P(nj+l-jß/p(„ _ „.)„ v/p = «,+ 2 1 , = > /"'(n,+I - -,)(,/'> CO > (by (3.10)) > nx + bx~x/p 2 , ,£/>. ,-Y 1, satisfies the conditions of Proposition 3.7. In Theorem 4 of [1] it is proved that every two symmetric bounded bases of a subspace of d(w,p),p > 1, are equivalent. Similarly we can state the following still open problem. Problem 1. Let X c d(w,p), 0 „)"_ i and (z„)~_, equivalent? We are unable to give an answer to Problem 1, however the following theorem is true: Theorem 3.8. Let d(w,p), 0 0 such that a < \x'n (>»_)| for every i G N. Then for every scalar (a¡)?=x, there are e, = ± 1, / = 1,2, ... ,n and License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 442 NICOLAE POPA 0 < M < oo, such that a 2 Kl < x,ni 2 wJ a¡e¡y» i=i2 IM <^2i=i where ||x||~h, is the norm of the element x G d(w,p). This inequality shows us that d(w,p) ä! /,, which is a contradiction. (3.13) implies by Proposition 3.1 of [5] that there exists a subsequence (yn)°Lx of (y„)™„x which is equivalent to a block basic sequence of (.x„)^L i- The basis (>»„)"_x is a symmetric basis, then (y„)fL \ ~ (y„)™=i and consequently we may assume that^ = 2?™*' +x b¡x¡, m G N. Moreover by Lemma 3.1 it follows that lim supjaj * 0. (If not then (yn)n ~ (e„)n.) Since (x„)£L, is symmetric, we may also assume that there is e > 0 such that for every m G N there ispm + 1 < km < pm+x so that ¡¿/¿J > e. If 2"_, a^y„ converges in d(w, p), then for every n G N, 1 1 < — M**, Theorem 3.9. Let X c d(w,p), 0 ", = y«.+i■=«,■ ..a,*,, n = 1, 2, ... , and (zn)"_, is equiva- lent to a block basic sequence vm = 2^;rm + i b„yn, m = 1,2, ... , of (yn)„°. More- over we may assume that lim sup,,.^ bn =£ 0 (otherwise Lemma 3.1 implies that (z„)„ — (e„)„ which contradicts our assumption). Reasoning as in Theorem 3.8 we obtain that (z„)"-i domainates (y„)n°^x. By interchanging the roles of (>»„)"-1 an<^ (zn)^°_, we deduce the conclusion. □ Let us mention the following problem. Problem 2. Is there a subspace X c d(w,p), 0 4. Complemented subspaces and sublattices of d(w,p), 0 < p < I. We prove now the version for 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LORENTZ SEQUENCE SPACES 443 Lemma 4.1. For every bounded block basic sequence (yX~\ °f (xn)n = i there is a block basic sequence of (yX-1 which is equivalent to the canonical basis of I. Proof. Let yn = 2f=¿ +1 a¡x¡, n = 1, 2, ... . Let us remark that, since íujJI.t'JUh. > 0, 2°li a¡xi does not converge in d(w,p). On the other hand, if sup„||2"=) a¡x¡\\pw < oo, then 2°_! a¡x¡ converges in d(w, p). Hence suPk sup 2 ä = 00. •Pn+i Also let Pn+\ l/P 2 y> 2 y, « G N. '=Pn+ 1 i=p„+ 1 Then the block basic sequence (zX= i or (.On3-1 satisfies the conditions of Lemma 3.1 and, consequently, there is a subsequence (zn)°l, of (z„)"=1, which is equivalent toOX-i- □ Using Lemma 4.1 we can prove Theorem 4.2. Let X c d(w,p), 0 Proof. By Proposition III.2.15 of [10] it follows that X contains a bounded basic sequence (yX-i which is equivalent to a block basic sequence (zj*., of (xX-v Then Lemma 4.1 gives us a subspace of Sp{z„: n G N) which is linearly homeo- morphic to lp. Consequently X contains a subspace Y « lp. □ Remark 4.3. In Corollary 17 of [3] it is shown that every (closed) subspace of infinite dimension X of d(w,p), 1 < p < oo, contains a subspace Y complemented in d(w,p) and linearly homeomorphic to lp. This assertion is not true in the case 0 /2y2+xe¡, n = 1,2,. . . , does not contain any infinite dimensional subspace which is complemented in lp. Consequently, let A' be a subspace of d(w, p) linearly homeomorphic to lp and let (z,),°l, be a bounded basis of X. We consider the block basic sequence u„ = n~x/pHf2^n-2\)/2+\ zr n = •> 2,- Then Y = Sp{«„: n G N} « lp does not contain any complemented infinite dimensional subspace of d(w, p). □ Remark 4.3 shows us that there are subspaces linearly homeomorphic to lp (in fact isometric to lp) which are not complemented in d(w,p) for 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 444 N1COLAE POPA Proof. By Example 3.5 and Remark 3.6 it follows that d(w, p)lM sé lp, where t M(t) = 0 if t = 0. Moreover, the canonical basis of d(w, p) is the unique unconditional bounded basis of d(w, p). But a theorem of Kalton (see Theorem 7.2 of [5]) shows us that if lF is a nonlocally convex Orlicz sequence space, then any two unconditional bounded bases of lF are equivalent if and only if any complemented subspace with an unconditional basis is isomorphic to lF. We conclude applying this result. □ Example 4.4 motivates the following question: Problem 3. Let d(w,p), 0 ||for every a, b G X. (4.1) We call such a space X a p-Banach lattice. (Analogously it is defined a Banach lattice.) It is clear that, extending the order relation to X (whenever the last space exists), A' is a sublattice of X. Let A"be a vector lattice and Y c X a sublattice of X (i.e. x G Y implies that \x\ G Y) which has the property that x G X, y G Y and |jc| < I.yI imply that x G Y. Then we call Y to be an order ideal of X. If A c X is a subset of X, then we denote by IX(A) = {x G X: 3a G A and 0 < X G R, such that |jc| < X\a\}, the order ideal generated by A. It is clear now that d(w,p), 0 Lemma 4.5. Let X be a p-Banach lattice which is an order ideal of u>,(vX-1 G co a sequence of positive real numbers, 0 < yn = 2,<=0 a¡x¡, and 0 < zn = 2ye+ BjX„ n G N, two bounded basic sequences of X (where xn = (dn¡)fLx for n G N) such that °n n »rm = 0 for every n, m G N. If there exists a positive and continuous projection P from X onto Sn{vnyn + zn: n G N}, then the sequence (zn)n dominates (v„yn)n. Proof. Let P(y¡) = 2°°_, c/)(W + zj) and 7>(z,)= 2°1, dfi\^r + z¡), i = 1,2,..., where cj¡) > 0 and ¿W > 0. Since the basis (xX-1 is clearly an uncondi- tional basis of X, there is a positive and continuous projection Q such that Q(xn) = xn for n G a„ where / G N, and Q(xn) = 0 otherwise. Then QP(z¡) = 2J1, dJ-'^Vjyj,i G N, and g/1 may be considered as an operator from Sp{z,: i G N} to Sp{j>,,: n G N) defined by the infinite matrix with positive entries ( License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LORENTZ SEQUENCE SPACES 445 D is obviously positive and 0 < D(x) < QP(x) for every 0 < x G Sp{_y,: i G N}. Since X is a/7-Banach lattice we have \\Dx\\ < Hß/'xll < Af||x|| for every 0 < x G Sp{.y,: i G N}, consequently 7) is a continuous operator. Assume now that 2„°_1 a„z„ converges in Sp{z,: i G N). Then Z)(2"_, a„zn) = 2?-i «„^V^ converges in fjr(^{^: Í G N}). Since 0 < c„")(V„ + z„) < P(yn) for every « G N, then |c„n)| < M, for n G N. On the other hand vncnn) + d^n) = 1, n = 1, 2, ... , hence lim„ i7„(n)= 1 and, consequently, 2°f_, anvnyn converges. □ We can now prove Theorem 4.6. Let X be an order ideal ofu, which is a p-Banach lattice and assume that its canonical basis (*,,)"_ i 's symmetric. If (yX-1 's a positive block of type I of (xX=i then (yn)n ~ (xn)n '/ and only if E =Sp{y„: n G N} is a positive comple- mented sublattice of X (i.e. there exists a positive and continuous projection P from X onto E). Proof. Let 2„°_, a„xn G X, a, ¥= 0 and yn = 2fb¿+i <%-£*/ for every n G N, where a¡ > 0. We may assume that all a¡ < 1. Since (xX-1 is a symmetric basis of X, there exists M > 1 such that J_ 2 KXP.+\ 2 M» < m 2 *«j^+p.+i M n=l j=i for every 2„ Ä„x„ G X. Suppose now that (y„)„ ~ (x„)„- Let 7C > 0 such that II2"-i*wJI 2 «rôPm +7 if/>m + ' and (ym - zm)/\\ym - 'mí ¿y», + *- o iiym = z„ Let vn = ||.y„ - z„|| for n G N. Then i>m,*-., zm are positive, ym = » License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 446 NICOLAE POPA 2 bmzn 2 2 bmZn n = \ , = 1 m = n, OO "i+l-l Pm+¡-Pm 2u 2-¡ "m 2-i XP„+j ,= 1 m = n¡ 7=1 oo ";+l"i*i- 1 < (since (tc,)°°_, is a symmetric basis) < M 2 2 bmXpm+ l , = 1 m - n, < A/2 2 */v'i Proposition 4.7. Let X be a separable p-Banach lattice with the property that every order interval [0, x], where X 3 x > 0, is compact. Then there is in X a normalized sequence (xX=\ of positive pairwise disjoint (i.e. inf(x,, x) = jc, A x¡ = 0, i ¥=j) elements such that (xX-\ & a basis simultaneously of X and of X. Particularly (xX=\ /J an unconditional basis of X. Moreover X is an order ideal of X. Proof. We prove first the second assertion. Let y G X and z G X such that 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LORENTZ SEQUENCE SPACES 447 every sublattice of d(w,p) has a basis of positive pairwise disjoint elements. We can now state the analogue of Corollary 14 of [31. Theorem 4.8. Let E be a positive complemented sublattice of d(w, p), 0 0 for n G N such that Sp{z„: n G N} is a positive complemented sublattice of d(w,p). (yX-i being a symmetric basis it follows that (y„)„ —(zn)„. Then we can apply Theorem 3.2, consequently if E sé lp, there exists a positive block of type I ("/.)*"»i OI (xn)«°=i sucn that (y„)„ — ("„)„• Using again the proof of Proposition l.a.9 of [8], we may assume moreover that Sp{w„: n G N) is a positive comple- mented sublattice of d(w,p). By Theorem 4.6, we obtain now that («„)«-1> a'so and (yX- v is equivalent to (xX-. • D In connection with Theorem 4.8 we can state a weaker version of Problem 3. Problem 3a. Let d(w,p), 0 E is a contraction, i.e. ||P|| < 1. 5. Positive and contractive complemented sublattices of d(w, l/k), k G N. In this section we shall characterise, under certain conditions, the positive and contractive complemented sublattices of d(w, l/k), k G N. We prove first an inequality: Lemma 5.1. Let k G N, (w,),°°., G c0\ /, such that wx > w2 > ■ ■ ■ > 0 and a, = • • • = a, > a/+1 > a/+2 > • • • > am > 0 for some I < m. Then denoting by sn = E;=1tv,fy c(i) = (¿)(«,+ ,)1A(«/-1A - «iV.'^hw,*-1 and by Í 1 for n > I + I, 10 forn K = 2 *,*(«, - «, + ,) + snka„+ d,(n)C(l) < (I, 2 "''*«,- -w, = Bn for every n < m. (5.1) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 448 NICOLAE POPA Proof. We use the induction for n. If n < / it is nothing to prove. Now let n = I/ + i I.1 ThenTU a!+x = a,,* + (£, - S*)«,+I + (i )«■/*(«;--a _ a.+-.A)vW/*-. = «a*+ «/..((^"/v^-2 + • • • +<.) + (lyxw-^-^sr1 < (since o/+, < a,,/ = 1, . . . , /) +((ikw"^'"?^*-VV /c/ •• +«/.,<.)/ /+i \* (2 «,IA--,| (i; «,-,)' Thus (5.1) is proved in this case. Assume now that (5.1) is true for n — 1 > I + 1. We have A -4,-1 +(í* - i*-lK < (since a, > a,+, for /' < n) *-i <^-.+(DC?:«'•w. «y*-^nn n + • • • +«1.^n n < (by induction hypothesis) < s «,"*■», +(¿)( % ,■"-,)">• ,i/* = (¿ «T •-,)"= *,. D Lemma 5.2. If/r/i /Ae notations of Lemma 5.1, if (ßi)T-\ are positive numbers such that 2 A- < ** f°r every n < m, (5.2) 1-1 then 2«,A + 3/(")c(/)<(2«/A->v,) /or n < m. (5.3) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LORENTZ SEQUENCE SPACES 449 Proof. If n < m, we have 2 «,A+ d,(n)C(l)= «„(2 a) + («»-. - «j("¿' a) + • - - + (a, - «j)A + 9/(")C(/) < (by (5.2)) < <(by(5.1))<(2«,,A-^) • D Using Lemma 5.2 we can prove now Proposition 5.3. If p = 1/A:, 1 < k G N, a«<7 Tf ¿s a positive and contractive complemented sublattice of d(w, p), then there is a sequence of finite pairwise disjoint subsets (sX=i of N such that un = (2""_,w,)"*(2,e^ X¡) for n G N, constitute a normalized basis of E, dn being the cardinal number of a„. Proof. By Proposition 4.7, E has a normalised basis (uX-\ OI positive pairwise disjoint elements of E. Let un — 2,eo otinx¿, where (oX-\ is a sequence of pairwise disjoint subsets of N and a,„ > 0 for / G a„, n G N. If dn = + oo, then we write a„ = {ty: /' G N}, and, since (uX-\ is normalised, it follows that (ain)°Lx G d(w,p). Consequently there exists a permutation of integers ir„ such that 7/n(a„) = on, (ct^ (i)X= i is a decreasing sequence, and trn(i) = /' for every / G o„. If an < oo, such a permutation mn exists obviously. Since the norm || || is invariant under permutations of integers, we can find an isometry T: d(w,p)^d(w,p) such that T(uJ= tm, where tm = 2^., a„m(ij)m for every m G N. Then Q = TPT~X: d(w,p) —»Sp{im: m G N) is a positive and contractive projection whenever P is a positive and contractive projection onto E. Conse- quently, we may assume that the coefficients (a,m),eo of um are decreasingly ordered for every m G N. Since P is a positive projection, we have />(*,.) = ßinun where ßm > 0 for every / G an, n G N, (5.4) and moreover (CO \* 2 ( for every element (y,)fl, G d(w,p). Consequently, taking y¡ = y¡ = • • ■ = y, = 1 and y, = 0 fory =£ /'„... , im, where m < än, we obtain m 2 /V<*¿ forA« License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 450 NICOLAE POPA If an = oo, then, since lim, ain = 0, there is an integer / G N such that a¡¡n > o, „ and a, „ = a, „ for/' < / — 1. By (5.3) we have 2 a,nA„ + C(l) < for every m > I + 1. 7=1 (!<»,)' Now, by passing to the limit over m —»oo in the previous inequality, we obtain, in view of (5.5). OO / 00 \* i = 2 v8*»+c<'> < I 2 «#k'"A =1- Since C(/) > 0, this is a contradiction, hence ön < oo for every n G N. (5.8) Then, reasoning as above, we get that ain = ajn for every i,j G a, n G N, conse- quently a,„ = (2°"_! w,)-* for every r G an, n G N. □ We study now the existence of positive and contractive projections onto a one dimensional sublattice of d(w,p),p = l/k. Proposition 5.4.(a) Let p = l/k, 1 < k G N, and u = 2,eo a,*,, where \\u\\pw = 1 a«r/ a, > 0 /or every i G a. 77iert //tere existe a positive and contractive projection P: d(w, p) -»Sp{«} if and only if the following conditions are satisfied: a = n, a, = • • ■ = a, = (2"=1 w¡)~k, and there exist the positive numbers /?, > • • ■ > 0 such that r« / m \*,k = sk for m a/zi7 2 A = tf. (5.10) i=i Then P(2r_! «,*,) = (2,go aßjufor every 2°°-, «,*, G ¿O*,/»). (b) // w = s~k(Z"_x x,) ant/ /„ = (l/n)sk < tm for every I < m < n, then the operator defined by It, a'*H(.t."M*' ¿y a positive and contractive projection onto Sp{u}. (c) If wx < tn for every n > 1 and u is as at the point (a),jhen there is a positive and contractive projection P: d(w, p) —»Sp{ u) if and only if a = 1. Proof, (a) If P~Ld(w,p) -> Sp{u} is a positive and contractive projection then, by Proposition 5.3, b~= n < oo and a, = • • • = <*,•_= s~k. Denote now by (/?;)/= i the coefficients of (P(x¡))"_, decreasingly ordered. Then 2 ßj- 2 A)« =H2 *») 2 x¡ = sk, 7=1 7=1 Ve« ' je H License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LORENTZsequence spaces 451 where H G a with 77 = m < n. If m = n, then 2 A= p[ 2 xA = 2 *4 -«*. 7 = 1 / p,w 7=1 Consequently all the conditions are satisfied and obviously P(2°l] a,x,.) = (2,6o a,-A)« for every 2°°=i «,*, S ¿(w,/?). Conversely, by (5.3), 2?_, a, A < (2?-i aV* • w,.)* for every a, > a2 > • • ■ > 0, and, consequently, |2"_, a,A| < supw(2"_! \avii)\l/kwiyk for all scalars (<*,.)?_,.Thus the operator defined by P(Zf=x a,x,) = (21(Ea a,A)M is well defined, ||P|| < 1 and P > 0. Since P(u) = s~k(Z"=x ß,)u = m (by (5.10)), it follows moreover that P is a projection. (b) If tn < im for every m < n, then, putting ßx = • • • = ß„ = tn, the relations (5.9) and (5.10) are verified, consequently P(2°l, a,.*,.) = (l//i)(2"_, a,)(2"_, x,) is a positive and contractive projection. (c) If there exists P: d(w,p) —»Sp{w} a positive and contractive projection and if ä > 1, then by (a) it follows that (l/n)sk = (l//t)2"_, A < ß\ < w\ for « > 1, which is a contradiction. □ We can now state the main result of this section: Theorem 5.5. If p = l/k, 1 < k G N, and wf < (l/rt)(2"=1 wf for every n > 1, í/ien the positive and contractive complemented sublattices E of d(w, p) coincide with the (closed) order ideals of d(w,p). In particular any positive and contractive complemented sublattice of d(w, p) cannot be linearly homeomorphic to lp. Proof. If £ is a closed order ideal of d(w,p), then E =Sp{x,: /' G A c N}, hence there is a positive and contractive projection P: d(w, p) —*E. Conversely, if £ is a positive and contractive complemented sublattice of d(w, p), by Proposition 5.3, it follows that E =Sp{wn: n G N), where un = 2,eo a,x. > 0 for n G N and (aX= i is a sequence of finite pairwise disjoint subsets of N. By Proposition 5.4(c) it follows that dn = 1 for every n G N. Consequently E is a closed ideal of d(w, p). The second assertion is an obvious corollary of the first. □ The conditions of Theorem 5.5 are verified for example by the spaces d(w,p), p = l/k, 1 < k G N, with w, = l/¡°, 0 < a < \. Indeed, 2 ", > f"+1 —a =7-^— [0 +»)'-"- 1] >«,/* for every n> 1. ,=i 7, x 1 — a d(w, l/k), where 1 < k G N and w„ = l/na for 1 > a > 1 - 1/rc and « G N, are examples for spaces d(w, p) for which there is a one dimensional positive and contractive complemented sublattice which is not an order ideal of d(w, p). More precisely, for a fixed m G N, there are n > m and a positive and contrac- tive projection onto Sp{w}, where u = (2"_, w,)_1(2"=1 x,). Indeed, using the notations of Lemma 5.4(b) we have 1 / r" dx\k „*(i-«> 1 r_ < —I I —- I =-;-<-;- for every n G N. «Wo x") (\-a)k-n (I - a)k ■nx~k+ak License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 452 nicholae popa Consequently, for 0 < e < | min, m with tn < e, hence tn < tj for every 1 < j < n. We conclude applying Proposition 5.4(b). □ Theorem 5.5 suggests the following version of Problem 3. Problem 3b. Let d(w, p), 0 < p < 1. Is there a positive and contractive comple- mented sublattice of d(w, p) linearly homeomorphic to lp1 6. The representation of d(w, l/k) for 1 < k G N. In this section we give a representation of the dual of d(w, l/k), 1 < k G N, and also a representation of the Mackey completion of this space. These representations seem to be useful in the study of the d(w,p)'s structure. Remark first that the dual of d(w,p), 0 d(w,p)' = |Í (o,),°l,: 2 |ai| \K(o\ < 0° for every permutation 7T I i= i of the integers and for every a + (a,)~=, G d(w, p) \. (6.1) Then we have Proposition 6.1. Let d(w,p), wherep = l/k, k > 1. Then d(w,P)'n c0= c0n j (6,)r_,:sup I 2 ¿,*)/ ( 2 *,] < ooJ = E, IHi;,„= sup(2¿,*)/(2 *,)> (6.2) w/iere 6 = (6,-)," j G c0 and b* = (p,*),0^ is a decreasingly rearrangement of b. Proof. Let b G d(w,p)' n c0, i„ = 2"_, o* and j„ = 2"_, wt. Suppose first that (6.3) n S Since lim„ sn = + oo, it follows that lim„ tn = + oo. Let «0 = 0 and sn = 0. If m G N is fixed, then (6.3) implies that t„ - tm sup —-~-n m = + oo. (6-4) Choosing nx arbitrarily we can find n2 G N, n2 > nx such that s„ — s„ > s„ — s„ (since lim s„ = oo), "2 "i "i "o V n ' t„2-,n¡>22k-x(sn2-sn)k (by (6.4)). By induction we get an increasing sequence of integers («,-)," x such that 1+1 1 1 I-i (6.->) for> =1,2,_ ^+, - 'n, > (J + 1)2*"'(V, ~ S)" (6.6) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LORENTZ SEQUENCE SPACES 453 Now let «,_, < i < rtj and y, = j 2k(sn•» — s 1- ) k. Then y = (y¡)fLx is a decreasing sequence (by (6.5)) and moreover 2 ^y*• w,= 2 2 = 2 ¡/J2 < oo, 1 = 1 7=1 f-l$_| + l i (\ - V.) 7 that is_y G d(w,p). On the other hand 2 ytb?=2 2 i-i > (by (6.6)) > 2 2 -=+oo, = 1 , = «,_, + ! y (j - j ) 7=1 i = ny_, + l 7 i=i 1r-|7 which contradicts the fact that b G í7(w, 77)'. Consequently (6.3) is not true and then Xk = sup — < + 00. (6.7) " sk Hence d(w,p)' n c0 G E. Now let o G E. If the decreasing sequence^ = (y¡)fL\ e d(w,p), then n-l 2 M* - 2 t,(y,- yi+i)+ v« 1=1 1-1 < (by (6.7)) < Xk 2 sk(y, -yi+x) + sfrn 1 = 1 <(by(5.1)) Hence \\b\\'pw = ||o*||^)K, < sup„ tn/sk and b G d(w,p)' n c0. On the other hand, for every e > 0, there is n G N with t„/sk > Xk — e. Let if 1 < n, .V, 0 if i > n. Then IMI^ = 1 and \\b\\'p,w= ||6%,„ > 2?_,M = /„/*„* > Xk - e. e being ar- bitrarily small, it follows that ||o||p)M,= sup„ tn/sk, consequently the equalities (6.2) are satisfied. □ Corollary 6.2. Letp = l/k, k > I. (a) Ifd(w,p) +./, then d(w,p)'= (¿>,) £c0:sup(2¿,*)/(2 »,) < 00 = {(¿>,)r_i:suP(2 v)/(i *.■) (b) If d(w,p) c /„ then d(w,p)' = lx. Proof. The case (b) is clear by Proposition 3.4. (a) Let b G d(w,p)'. If o G c0, then there are e > 0 and («,•)_,", such that \b \ > e for every j G N. Reasoning as in the proof of (3.3) we get a contradiction. Hence License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 454 NICOLAE POPA d(w, p)' c c0, consequently, by (6.2), it follows the first equality. By the last part of the proof of Proposition 6.1 it follows the second equality. □ We state now the main result of this section: Theorem 6.3. Let p = l/k < 1 and d(w,p) c£ /,. If there is a positive decreasing sequence (vX= i e co sucn mat n i n \ * 2 v,■~ 2 w¡) M every n G N (6.8) ,=i \,=i / (i.e. there are constants A, B > 0 such that ^(2"_i o,) < (2"_, w,f < 5(2"_, «,.) for every n G N), then d(w7ïj^d(v, 1). (6.9) Proof. Theorem 11 of [4] says that d(v, 1)' = {(è,),",: sup„(27_! of)/(2"_, o,) < oo}. Then by the hypothesis and by Corollary 6.2(a) it follows that d(w,p)' = d(v,l)'. (6.10) It is easy to see that the canonical basis (xX-i or d(w' P) is a symmetric basis of d(w,p) too. Then, denoting by d(w,p)x the Köthe dual of d(w, p), that is the space {(0,)°!,: 2°°_,|a,o,| < +00 for every a = (ai)°°=x £\d(w,p)), it follows that d(w,p)x = d(w,p)x = d(w,p)' = 'c<|| = 0). (2) Every order interval [0, x] c E is a(E, £")-compact. (3) E is an order ideal in E". Finally, a theorem of Ando (see Proposition IV. 11.1 of [11]) says that a Banach lattice E has an order continuous norm if and only if every closed order ideal of E is the range of a positive projection from E. Consequently d(w, p) is a complemented subspace, with a symmetric basis (y„)„, of the space d(w,p)" = d(w,p)xx = d(v, 1). Since d(w,p) sé /„ by the Corollary 14 of [3] it follows that d(w, p) tu d(v, 1). □ Remark 6.4. (1) d(w, l/k) with k > 1 and wn = l/n", where (k - l)/k < a < 1, are examples of the spaces d(w, l/k) which verify the conditions of Theorem 6.3. Indeed, if a = 1, we take in Proposition 3.7, y = 0, b = 1 and n, = j for every j G N. Hence d(w, l/k) +. /,. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LORENTZ SEQUENCE SPACES 455 If (k - l)/k < a < 1, let 0 < y < l/(k - 1), rtj = jx/a for every j G N, and b = supj[(j + l)x/a -j]/a]/jy < + oo. Then, by Proposition 3.7, it follows that d(w, l/k) +_/,. Suppose now again that a = 1 and put 1 if i <[ek~x] + 1, (log 0 ' if; >[**-»]+ i where [e* '] is the entire part of ek '.Then " f"(iogx)k~l , n .k ( ¿ \* . ¿j u, ~ I J- dx ~ (log n) — ¿i wi\ ror every n. 1=1 J3 x \,= i / Moreover limn v„ = 0 and vn > vn+x for every n G N. If (k - l)/k < a < 1, then we put vn = l/nß, where ß = k(a - (k - l)/k) < 1 for every n G N. Clearly limn u„ = 0 and (vX-1 is a decreasing sequence. Moreover " r" dx I " \k .2 °<~/ ^~«1_/3~«M1_a)~ 2 w, for every/j G N. i=l ix \ ; =i / (2) We do not know if the condition (6.8) is superfluous. Let us mention the following problem: Problem 4. Let p = q/k, 1 < q < k. Is there a positive decreasing sequence (©,)£, G c0 \ /, such that d(w, p) sa d(v, q)1 More generally: Problem 4a. Let 0 References 1. Z. Altshuler, P. G. Cassaza and B. L. Lin, On symmetric basic sequences in Lorentz sequence spaces, Israel J. Math. 15 (1973), 140-155. 2. G. Bennett, An extension of the Riesz-Thorin theorem, Lecture Notes in Math., vol. 604, Springer-Verlag, Berlin and New York, 1977. 3. P. G. Cassaza and B. L. Lin, On symmetric basic sequences in Lorentz sequence spaces. II, Israel J. Math. 17(1974), 191-218. 4. D. J. H. Garling, On symmetric sequence spaces, Proc. London Math. Soc. 16 (1966), 85-106. 5. N. J. Kalton, Orlicz sequence spaces without local convexity, Proc. Cambridge Philos. Soc. 81 (1977), 253-277. 6. G. Köthe, Topological vector spaces. I, Springer-Verlag, Berlin and New York, 1969. 7. J. Lindenstrauss and A. Peltzyñski, Absolutely summing operators in £,p-spaces and their applica- tions, Studia Math. 29 (1968), 275-326. 8. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I. Sequence spaces, Springer-Verlag, Berlin and New York, 1977. 9. B. Maurey, Théorèmes de factorisation pour ¡es operateurs linéaires à valeurs dans les espaces Lp, Astérisque 11 (1974), 1-163. 10. S. Rolewicz, Metric linear spaces, PWN, Warszawa, 1972. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 456 NICOLAE POPA 11. H. H. Schaefer, Banach lattices and positive operators, Springer-Verlag, Berlin and New York, 1974. 12. W. J. Stiles On properties of subspaces of lp, 0 Current address: Mathematisches Institut, Universität des Saarlandes, Saarbrücken 6600, West Germany License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use