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L I B R A R. 33 Michigan State University This is to certify that the thesis entitled Kb'THE FAl-lILIES II\‘ VECTOR LATTICES presented by Charles Good Denlinger has been accepted towards fulfillment of the requirements for Ph.D. degree in Mathematics flame/m fl Major professor Date February 22, 1971 07639 ABSTRACT KoTHE FAMILIES IN VECTOR LATTICES BY Charles Good Denlinger In 1934 Kothe and Toeplitz introduced the notion of the a—dual Ax of a vector space A of real sequences: Ax = {{y }: le y I < w, {x } c A}, n n n n and defined a "perfect" sequence space A to be one such that A = AXX. In many important cases, the a-dual and the Banach dual of a sequence space coincide. In any case, the ideas suggested by the notion of a—duality have stimulated extensive investigations into the topologi— cal and order properties of real sequence spaces. This theory, with its various generalizations, has become known as the theory of Kothe spaces. In the present thesis we proceed from the original notion of Kothe sequence spaces, along lines of generalization different from those heretofore undertaken. We consider families [Xi’ I] of elements xi in a vector lattice E, indexed by an arbitrary set I. We let wI(E) denote the collection of all such families, and consider certain vector sublattices of wI(E), analogous to the familiar subspaces ¢, c c, and Qp (1 §_p :_W) of the space w of all real sequences. The 09 order properties of these vector sublattices are studied. Charles Good Denlinger The notion of an "order summable” family [Xi’ I] is of fundamental importance in the development of this thesis, and appears not to have been exploited before, in the manner in which it is used here. Consider the collection V(I) of all finite subsets of I, partially ordered by inclusion, as a directed set. If we let {TJ} denote the net of sums T = E. J ilein (J E V(I)), then a family [Xi’ I] is said to be order summable if the net {IJ} is order bounded in E; equiva— lently, {TJ} is order convergent in the Dedekind completion E of B. The collection of all such [Xi’ I] generalizes the sequence space 21, and is denoted 9%(E). This concept of summability leads to a generalization of Kothe's a—dual. We embed E in its universal completion E# and choose a weak order unit 1 in B#. By the work of Vulikh and others we know that there is a unique multiplication operation in E#, relative to 1. We use this multiplication to define the "X-dual" of AI(E) wI(E) as the collection A§(E) of all [yi, I] in wI(E) such [Xiyi’ I] is an order summable family of elements of E for every [Xi’ I] in AI(E). In addition, we use this multiplication operation in E# to define the vector lattice Q§(E, l) (1 < p < c°) as the collection of all [Xi’ I] in wI(E) such that [XE, I] e fl%(fi). Analogues of the Holder and Minkowski inequalities are established, and attention is directed to conditions which will guarantee ¢I(E) .5. QELE, l) C_'-_ 12%(3, l) <_=_mI(E). if 1 :p < q < °°. and WEE: HJX = 9%“, 1), if %-+ %-= 1. Corresponding to each y c X§(E), the mapping y" A 5': from AI(E) into E defined by y (x) = 2. x.y. is shown to be a 161 l l Charles Good Denlinger positive, order-continuous linear mapping. Throughout the thesis considerable attention is given to the case in which E is a normed vector lattice. We define Q-norms on £%(E) and p-norms on 2§(E, l), which behave like the usual norms on 21 and 2p, respectively. Conditions under which these norms are essen— tially unique are studied. These considerations involve the problem of extending a monotone norm from E to its Dedekind completion B. Semi—continuity and continuity of the norm on E are relevant. Another dual, the y-dual of a subset AI(B)€E wI(E) is defined and discussed briefly in the final section of the thesis. It is similar to the X-dual, but is independent of the choice of unit 1 in E#. An interesting result of a different nature is obtained at the end of Chapter I. Using an approach to Banach limits for bounded sequences of real numbers, developed in a paper by Simons, we are able to define, and prove an existence theorem for, Banach order limits of order bounded sequences in vector lattices. Several criteria for almost order convergence are developed. VKOTHB FAMILIES IN VECTOR LATTICBS By Charles Good Denlinger A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY ‘ Department of Mathematics 1971 To Gloria and Joey ii ACKNOWLEDGMENTS I wish to thank Professor John J. Masterson for introducing me to the theory of vector lattices, for stimulating me to pursue research in this area, and for his patient guidance along the way. I also wish to thank Professor George W. Crofts for several enlightening conversations. In fact, the original question which led me to define and consider "order summable families" was raised by Professor Masterson in a seminar on nuclear spaces, led by Professor Crofts. I also wish to express my indebtedness to those many people - family, friends, colleagues, and teachers — whose individual and collective influences have led me to undertake an academic career, with its accompanying responsibilities. iii TABLE OF CONTENTS page INTRODUCTION 1 CHAPTER I. GENERAL FAMILIES IN A VECTOR LATTICE 11 Section 1. The fundamental spaces wI(E), mI(E), and ¢I(E). 11 2. Cartesian products and direct sums. 18 3. Order properties in subspaces of wI(E). 20 4. Dedekind completion and universal completion. 28 5. Banach o-limits in Archimedean vector lattices. 31 II. SUMMABLE FAMILIES IN A VECTOR LATTICE 40 6. Some types of summable families. 40 7. The space Q%(E). an 8. 9%(E) as a normed space. 49 III. A KOTHE-TYPE DUALITY THEORY 64 9. Multiplication in Archimedean vector lattices. 64 10. The spaces 92(E, I), 1 §_p §_w. 7O 11. p—norms on the spaces Q§(E, l). 79 12. Kothe X-dual spaces. 85 13. Linear mappings from AI(E) into E. 91 1M. Convergent families in a vector lattice 9M 15. Kathe y-dual spaces 99 iv APPENDIX Representation theory and universal completion 103 BIBLIOGRAPHY 106 INTRODUCTION A vector lattice is a vector space E over the real scalar field R, equipped with a partial order §_(which is reflexive, anti- symmetric, and transitive) such that (l) x :_y implies x + z :_y + z for all z in E, (2) x :_0 implies ax :_O for all real a :_O, and (3) E is a lattice under 5, Thus every two-element set {x,y} in E has a supremum, denoted sup{x,y} or x‘v y, and an infimum, denoted inf{x,y} or x A.y. Since its inception around the year 1930, the theory of vector lattices has greatly enriched the field of functional analysis. A brief account of its origins and early contributors may be found in the forward of Vulikh [18] and in the introduction to Zaanen [21]. For expositions of some of the major results of the theory of vector lattices, the reader is referred to Vulikh [18], Luxemburg and Zaanen [6], and Nakano ([7] and [8]). Vector lattices have been called by various other names; for example, "linear lattices" or "semi—ordered linear spaces" in the works of Nakano, "Riesz spaces" in the works of Bourbaki, Vulikh, Luxemburg and Zaanen, and "K—lineals" in the works of various Russians. The basic definitions and fundamental properties of vector lattices may be found scattered throughout the literature; for example, see Namioka [9], Peressini [10] or Vulikh [18]. As a convenience to the reader a few of these basic facts will be given here. The set E+ = {x e E : x :_O} is called the positive cone of E, and has + + the properties E+ + E+§ E , AB 9- E+ for all A e R+, and E+(\ (—E+) = {0}. If x e B we let x+ = x v O, x_ = (—x) v O and le = x+\/ x-; then x = x+ — x-, lxl = x+ + x— and x+rx x_ = 0. If |x|/\ lyl = O we say that x and y are disjoint and write x.l.y. Vector lattices are necessarily distributive lattices. If x, y, and z are arbitrary elements of a vector lattice E, the following relations hold: (4) —(xvy) = (-x) A.(-y), -(XAy) = (—x) V (-y); (5) (xvy)+ = x+xv y+, (XAy)+ = X+/\ y+; (6) (xvy)_ = X“\J y“, (XAY)~ = x_/\ y—; (7) if aeR+, then a(xvy) = ax v my and a(XAy) = ax A ay; (8) (xvy) + z = (x+z) V (y+z), (XAy) + z = (x+z) A (y+z); (xvy) + (XAy), Ix-yl = (xvy) - (XAy); (9) x + y (10) x V y (x-y)+ + y, x A y = x - (X-y)+; (11) (x+y)+ 1..“ + y+, (x+y)- _<_ x" + y'; (12) |X+Y| i. IXI + lYI: ||X| -|Y|| : IX-YI; (13) |(xvz) - (yvz)| §_lx—yl, |(XAz) - (yAz)I j_|X-y|; (14) if x, y, z 3_O, then (x+y) A z :_(XAZ) + (yAz); (15) x _g y if and only if x+ _<_ y+ and x_ _>_y'; (16) [x] §_y if and only if —y §_x j_y; (17) xlyif and only iflxlv lyI = |x| + lyl; (18) if x = y - z for y, z 3_O, thenx+ §_y and x— :_z; if in addition y 1.2, thenx+ = y and x_ z. Proofs for these well—known relations may be found in the literature cited above. Given vector lattices E and P, a linear isomorphism w: E + P such that x :_0 if and only if w(x) 3_O is called a vector lattice isomorphism; we then write w: E = F. If w: E + P is a vector lattice isomorphism and A.§;E, then sup A exists in E if and only if sup w(A) exists in F; in that case, w(sup A) = sup ¢(A). The analogous state- ment for infima also holds. Unless otherwise specified, the term "isomorphism" will, in this thesis, mean "vector lattice isomorphism". A linear subspace M of a vector lattics E is said to be a vector sublattice of E if, for each x, y in M, the supremum of x and y in E is also in M. It then follows from (4) above that the infimum of x and y in E is also in M. A (lattice) id§§l_of a vector lattice E is a linear subspace I S;E such that y e I whenever Iyl §_Ix| for some x e I. An ideal is necessarily a vector sublattice. A vector sublattice M of E is said to be order closed if it contains sup A, for every subset A of M having a supremum in E. An order closed ideal is sometimes called a band. A vector lattice E is said to be Archimedean if every x e E+ satisfies the equation inf{%x: n e N} = O; equivalently, if x, y e E+ such that Xx :_y for all A > 0, then x = 0. It is well-known (see Vulikh [18], p. 75) that any n-dimensional Archimedean vector lattice is isomorphic to Rn. Every vector sublattice of an Archimedean vector lattice is Archimedean. Any vector lattice of real—valued functions, with the usual pointwise-defined linear operations and order, is Archimedean. A vector lattice E is Dedekind complete (resp. Dedekind o-complete) if every subset (resp. countable subset) of E which is bounded above in E has a supremum in E. Either of these conditions is sufficient to imply that E is Archimedean. Every ideal in a Dedekind complete (resp. o-complete) vector lattice is also Dedekind complete (resp. o—complete). A Dedekind complete vector lattice in which every set of pairwise dis— joint elements is bounded is said to be universally complete. Any order—closed vector sublattice of a universally complete vector lattice is also universally complete. A vector lattice E is order separable if every subset A of E+ having a supremum in E contains a countable subset with the same supremum. Any ideal in an order separable vector lattice is again order separable. An element 1 in a vector lattice E is called a weak ordgr_unit if x A 1 > 0 whenever x > O in E, and is called a strong 9£de§_unit if for every x e E+ there exists a scalar A > 0 such that XX :_1. Given vector lattices E and P, E is said to be order-dense in P if every f :_0 in F+ satisfies the condition f = sup{e e E: 0 :_e §_f}; E is said to be 322§i7239237§32§2.if for each f > O in P there exists an e e E such that O < e :_f. It is well—known that in an Archi— medean space F,order-denseness and quasi—order-denseness are equiva— lent. We list here a few of the common vector lattices. The space w of all real sequences, with its usual (coordinate-wise defined) linear operations and order relation, is an example of a universally complete vector lattice. Real Euclidean n-space Rn is isomorphic to an order-closed ideal of w, and hence Rn is also universally com- plete. The familiar sequence spaces 1p (1 §_p §_m), cO (the space of all zero—convergent sequences), and o (the space of finite sequences) and ¢ are Dedekind complete are ideals in w; hence 2p (l §_p §_w), cO vector lattices. The space Am of bounded sequences is often denoted m. The vector sublattice c, consisting of all convergent real sequences, is not an ideal in w, and is not Dedekind complete. The vector lattice P(X) of all real-valued functions on an abstract set X, with linear operations and order defined pointwise, is universally complete. A universally complete vector lattice of key interest in current research is the space‘WKX, O, u) of equivalence classes of almost— everywhere finite—real—valued, u-measurable functions on the measure space (X, 9, u), again with linear operations and order defined point- wise. The spaces LP(X, Q, n) (l §_p §_m) are ideals in°WKX, 9, u), and hence are Dedekind complete vector lattices. Finally, the space C(X) of continuous real-valued functions defined on a compact Hausdorff space is a vector lattice which need not be Dedekind complete. If {xa} is a monotone increasing (resp. decreasing) net in a vector lattice E, with supremum (resp. infimum) x in B, then we 0 write x + x (resp. x + x ). A net {x } in E is said to order a O a O a -——-—- converge to X0 in E if it is order bounded and if there exists a net {ya} in E+ such that lxa — xl §_ya + 0. If {Xe} order converges ° .3 to x0, we write Xa x0, or x = o-lim x . O a a If {Xe} order converges then it has a unique o-limit, and every subnet of {Xa} o-converges to the same o-limit. Order convergence preserves sums, scalar products, suprema and infima; more precisely, . o o o o If XOL_—9X and yer—5y, then Xoc + ya—sx + y, lxa—a Ax (l e R), xav yon; xv y, and XOLA ya—ng y. Given a bounded net {Xe} in a Dedekind complete space, we define lim x = inf sup x and lim x — 0‘ 0‘ 0‘ B_>_a B T on sup inf x8; then x —2>x.if and only if lim x = x = lim x . 0L B_>_0L 0L 0!. 0!. T 0!. It is easily seen that if I is an ideal in a vector lattice E and {Xe} is an order bounded net in I, then xaig»x in E if and only if 0 . xa._a.x in I. A sequence {xn} in a vector lattice E is said to order *—converge to an element x e E (denoted xn_£>x) if every subsequence of {xn} has a subsequence that order—converges to x. The properties of order convergence described in the preceding two paragraphs hold for order *—convergence as well. If we consider the vector lattice7n of equivalence classes of finite (a.e.) Lebesgue—measurable functionson [0,1], with the usual order, then order convergence is equivalent to convergence almost everywhere and order *-convergence is equivalent to convergence in measure. A net {Xe} in a vector lattice E is said to be y-uniformly convergent to an element x E E (for a given y c E) if for every positive real number 6, there exists a such that |xa - xl §_5y for 0 all a.: a 0° A net {Xe} in E is said to be uniformly convergent if it is y-uniformly convergent for some y E E; in symbols, xagx. In an Archimedean vector lattice, uniform convergence implies order convergence. Many of the vector lattices studied in functional analysis are also normed spaces. If H.“ is a norm on a vector lattice E such that [xl §_|yl impliesllx“ §_“y“, then H.\ is called a monotone norm, and (E, i, H.“) is said to be a normed vector lattice. A Banach lattice is a norm—complete normed vector lattice; if E is a Banach O + 0 0 O lattice, E 18 necessarily norm-closed. An M-space 18 a Banach max{\|x\\ . ll Y“ } s an lattice in which x, y 3_O implies “X\J y“ L—space is a Banach lattice in which x, y :_0 implies “x + y“ = \lx" + fly". In his famous paper [3] Kakutani proved that every L—space is isomorphic and isometric to L1(X, D, u) for some locally compact Hausdorff space X and some positive Radon measure u defined on X. In another famous paper [4] Kakutani proved that every M-space is isomorphic and isometric to a vector sublattice of C(X) for some compact Hausdorff space X. Vulikh ([18], theorem VII.7.1) has shown that every L—space is Dedekind complete and has the property that xa t O Implies “X,“ + O. In the present thesis it is shown that the theory of (real) K6the sequence spaces (see Kothe [5], § 30) can be generalized in a meaningful way to spaces of families of elements in a vector lattice. Given a vector lattice E and a nonempty set I, the notation [Xi’ I] will denote a family of elements xi of E, indexed by the set I. This is a generalization of the notion of a sequence [Xn’ N] in E. Accordingly, the question arises as to whether it is possible and fruitful to define spaces of families in E analogous to the familiar sequence spaces w, ¢, CO, C, and AP (1 :_p :_W), using only the theory of vector lattices. It is this question which gave rise to the author's investigations, culminating in this thesis. The basic spaces wI(E), ¢I(E) and mI(E)_generalizing w, o and m, respectively, are introduced in the beginning of Chapter I. A large portion of Chapter I is devoted to an investigation of the order properties of these, as well as of more_general, vector sub- lattices XI(E) of wI(E). Order convergence and order *-convergence in AI(E) are related to order convergence in E. Possession by E of properties such as the Archimedean property, Dedekind completeness, universal completeness and order separability is related to the possession by AI(E) of the same properties. Dedekind and universal completions are discussed in this context. Chapter I ends with an application, somewhat unrelated to the remaining material in the thesis. A paper [16] of S. Simons is extended to yield a definition of, and an existence theorem for, "Banach o-limits" in vector lattices. Several criteria for "almost o-convergence" are developed. Chapter II introduces and develops a theory of order summable families in a vector lattice E. Of primary interest are families [Xi’ I] for which there exists an element u e E such that for every finite subset J€; I, Z Ixil :_u. The collection of all such families iEJ is a generalization of £1, and will be denoted 21I(E). For each [Xi’ I] in 111(E) there exists a "sum". leil in E, the Dedekind completion of E. This sum is seen to E: an order continuous function of [Xi’ I]. Several other types of summable families are considered. In Chapter III representation theory, universal completion, and multiplication in Archimedean vector lattices are used to generalize the theory of 1p spaces and Kothe dual sequence spaces to the context of Archimedean vector lattices. For 1 :_p < w, the space,QpI(E,l) is defined as {[xi, I] e wI(E): Z Ixilp j_u, for some u e E}, where the icI multiplication is performed in E , the universal completion of E, relative to a fixed weak order unit I in E#. Analogues of the Holder and Minkowski inequalities are established, and some attention is directed to conditions which will guarantee ¢>I(E)g £PI(E,I) g. ,QqI(E,l) EmI(E) if1_<'_pwI(F) whose restrictions to mI(E) and ¢I(E), respectively, are isomorphisms onto mI(F) and ¢I(F). (1.3).Propos1tion. If E is a vector lattice, and I and J are 13 two sets of the same cardinality, then there exists an isomorphism w: wI(E) + wJ(E) whose restrictions to mI(E) and ¢I(E), respectively, are isomorphisms onto mJ(E) and ¢J(E). The converse of (1.2) is false. As a counterexample, consider the spaces wN(R) and wN(R2). The map [(an’ Xn2)’ N] + [X(n,i)’ NX{1, 2} ] is clearly an isomorphism from wN(R?) onto wa{1,2}(R), and by (1.3) the latter space is isomorphic to wN(R).. Thus wN(R) = wN(R2). However, R # R . Suppose E is a vector lattice on which there is defined a normll.". We call a family [Xi’ I] in wI(E) norm—bounded if there is a positive real number M such thatllxi“ :_M for all i e I. In general, there is no relationship between the order-boundedness and norm—boundedness of a family [Xi’ I], as the following examples will demonstrate. Consider the vector lattice o of finite real sequences, with its usual order and "sup" norm. For each n e N we use the customary notation en to denote the "nth basis unit vector" en = (O, O, ..., 1, 0,...) whose kth term is 1 if k = n, 0 if k ¢ n. In wN(¢) the family [en, N] is norm-bounded but not order-bounded. Let BVO[O,1] denote the vector space of all real—valued functions f of bounded variation on the unit interval [O,1]such that f(0) = O. The set {f e BVO[O,1]: f(t) :_O V<3 : t.: 1} is a positive cone, inducing a partial order under which BVO[O,1] is a vector lattice. We take the total variation as norm; i.e.l|f“ = Té(f). Then the family [fn’N] defined by the following formula, l4 .1 .1 Sln'E' lf‘B-itil, f(t): n o ifO:t<%l-, cannot be norm—bounded. Howeverlfn(t) §_2 for every n E N and O §_t j_1; thus [fn’ N] is order bounded. Thus order—bounded sets need not be norm-bounded. If E is a vector lattice on which there is defined a norm H.“, and I is a nonempty set, we let NI(E) denote the collection of all norm-bounded families of elements of E; N (E) = {[x., I] e w (E): 3M5 R+ allx.“ < M \di 5 I}. I l I l _- Now NI(E) is a vector subspace of wI(E). Moreover, if we define "[Xi’ I]“m = :2? “Xi” for each [xi, I] in NI(E), then NI(E) is easily seen to be a normed space. With the ordering induced from wI(E), NI(E) is a partially ordered vector space, but need not be a vector lattice. To see that NI(E) need not be a vector lattice, consider again the vector space E = BVO[O,1]. This time let the positive cone be if E BVO[O,1]: f is non-decreasing} and let the norm be "f” = sup {|f(t)|: O §.t §_1}. We first show that for this ordering, f+ is the positive variation of the function f: n f+(t) : P; (f) = sup {iZ1[f(xi) — f(xi_1)]+: n E N and O = x1 < x2 < ... < xn = 1 }. Now 0 3 s _< t _g 1 implies PEG?) - 1936) = P: (f) _>_ f(t) - f(s) which implies Pg(f) — f(t) :_P:(f) — f(s). Thus P$(f) — f(t) is a non- decreasing function of t for O :.t §_1; i.e. Pé)(f) :_f. Also, 15 () o () P (f) :_0 since PO (f) is non-decreasing. Now suppose g Z_f and g :_0; that is, g and g—f are both non—decreasing on [0,1]. Then for any 0 §_8 < t j_1 and any partition 3 = t0 < t1 < ... < tn : t of the interval [s,t], we have for each 1 §_i §_n, so that g(t.) — g(t. ) > f(t.) - f(t. ). We also have 1 l-1 —- l l— 1 g(ti) - g(ti_ ) 3_O. Therefore 1 g(ti) - gdi_1) :_[f(ti) - f(ti-1)]+’ and hence g(t) — g(s) = P:(g) 3_P:(f) = Pg(f) - P:(f). We thus have g(t) — Pg(f) 3_g(s) - P:(f). From this we see that g — Pé)(f) is an increasing function on [0,1]; i.e. g :_Pé)(f). Therefore, f+(t) = PEG). By this result, we conclude further that If] is the total variation function of f: lf|(t) = Té(f), for O §_t :_1. Now consider the family [fn’ N] E wN(E) (E = BVO[O,1]), given by .... 2. if .1. fn(t) = ‘t n IA t: 0 if 0 §.t < —:3|I—=H Then [fn’ N] E NI(E) since anu §_1, but [fn’ N]l = [Ifh|, N] E NI(E) since “lfnlfl + w as n + w. Therefore, NI(BVO[O,1]) is not a vector lattice. (1.4) Proposition. If (E,j) is a vector lattice on which there is defined a norm H.“, and if I is an arbitrary set, then (a) (E,§J“JD is norm—complete if and only if (NI(E),§Juflm) is norm-complete; (b) “.H is monotone on E if and only if (NI(E),§_, “.Hm) is a normed vector lattice; 16 (C) (E, §_, I) is an M—space if and only if (NI(E), :, “.lm) is an M—space. Proof. (a) Suppose (E, n.u) is norm—complete, and let {[xgn), I]} be a Cauchy sequence in NI(E). Then for each i in I, (n) 0 o 0 {xi } is a Cauchy sequence In E; hence it norm—converges to some element xi in E. Consider the family [Xi’ I]. Let 6 > 0. Since 3 M > 0 such that n, m-: M implies “[xém) — xén),I]um < 6, we also have “[xén) , I]“m < ”[x§M), 1]"m + 6 whenever n 3_M. For each i in I, we then have (n) (M) , "Xi N 5."Xi H + 63 thus letting n + m we have _ . (n) (M) 11ng - 1,... “X, n 5. “X, n + 6. Therefore [Xi’ I] E NI(E). It remains to show that {[xEn), I]} “'“m — converges to [xi, I]. Let 6 > O and choose M as above. Then m :_M implies “x. — X(m)“= lim‘\x€n) - X(m)“ < 6 for all i in I. Hence, m > M l l n+w l l - implies “[xgm), I] - [Xi’ fl“m < 6, which establishes the desired convergence. To prove the converse, suppose that (NI(E), H.Hm) is norm— complete, and let {xn} be a Cauchy sequence in E. The sequence of constant families [x§n), I] given by xgn) = xn is then a Cauchy sequence in NI(E), and thus must converge to an element [Xi’ I] in NI(E). Thus for every 6 > 0 there exists M > 0 such that n :_M . . (n) _ (n) lmplles “[xi , I] - [Xi’ I]”m §_6; hence, “Xn — Xi“ - “Xi - xi“§_6 (i E I). Therefore, {xn} is norm-convergent in E, proving that E is norm—complete. 17 (b) Suppose that}\.“ is monotone on B. Let [xi, I] E NI(E); then there is some u in R such that “Xi“ §_u for all i E I. Hence, “Ixil“ = "Xi“ §_u for all i E I, so that I[xi, I]| = [Ixil, I] E NI(E). Therefore NI(E) is a vector lattice. We shall next show that “'“m is monotone on NI(E). If [[xi, I]| §_I[yi, I]I in NI(E), i.e., [Ixi|, I] §_[Iyil, I], then Ixil §_lyi| for all i E I; thusllxi“ §_“yi“ for all i E I, which implies “[Xi’ I]“m §_“[yi, 1]“m' The converse is easily seen by the fact that NI(E) contains con- stant families. (0) If (E, §J\LU) is an M-space, then for [Xi’ I], [yi, I] in NI(E)+, “[Xi’ I] V [yi, IJHm = "[Xi\J yi, I]um = sup (“xi v yiH: i E I} = sup {“xiu V “yiH: i E I} ,g-(igg “XiH)\/(§:gllyin) = “[Xi, IJNm V 'lEyi, 1]“m, which, together with (a), impliis that (NI(E), j_, “'“m) is an M-space. The converse is easily seen, since NI(E) contains con— stant families. (1.5) Observations. If (E, i, H.H) is a normed vector lattice, then mI(E) and NI(E) are normed vector lattices and are ideals in wI(E). The proof of (c) above shows that if E is a normed vector lattice, then “'Hm preserves suprema on mI(E)+ if and only if “.N preserves suprema on E+. If (E, jJ “.H) is a normed vector lattice in which every norm- bounded set is order-bounded, then the norm u." is said to be mono; tone bounded. Note that in such a case, mI(E) = NI(E). 18 Section 2. Cartesian products and direct sums. It is well-known that the Cartesian product gEa of an arbitrary collection {Ea: a E I} of vector spaces, i.e., the set of all maps f: T + L&Ea such that for all a E F, f(a) E Ea’ is a vector space under the operations (f + g) (on) f(a) + g(oc), (Rf) (a) K(f(0L)). The subspace consisting of all f such that f(a) = O for all but finitely many on is called the direct sum of the spaces Ed and is denoted % Eoc' If each of the spaces Ed is a vector lattice, then gEa is also a vector lattice, under the ordering: f :_g if and only if f(a) §_g(a) for all a E I. The positive cone in HE is thus H(E+). Moreover, GE is an ideal in a a a a a a HE , with positive cone $(E +). a a a a If Ba = E for all a E F, then the space NEG may be denoted EB or Er. (2.1) Proposition. Under the ordering described above, ~ _ I ~ wI(E) - E (- E ) and ¢I(E) — f§EE° iEI (2.2) Proposition. If {Eaz a E F} is a collection of vector lattices and I is an arbitrary set, then wI(dgFEa) 2 agTwI (E ). a Moreover, if Ea = E for all a in F, then both of the above spaces are (E). isomorphic to wIXI Proof. Let [fi’ I] E wI(ag Eq)° Then for each 1 E I, fi 13 a I mapping from F toléEa such that fi(a) E Ea for all a E I. Let 19 w([fi, I]) be the map F: F + LéwI(Ea) such that P(a) = [fi(a), I] for all a E P. That is, ¢, I], and hence, w: wI( g Ea) + g wI(Ea)' It Is a routine matter to verlfy that w is a (vector lattice) isomorphism. (E) In case Ea : E for all a E P, we use the map H: wI( 3 Eu) + waT given by the formula H([fi, I]) (i, a) = fi(a). Again, it is routine to verify that N is an isomorphism. (2.3) Remarks. If {Ba: a E P} is a collection of vector lattices and I is an arbitrary set, then m ( H E ) = H m (E ). I def a aEI I 9 The restriction of the map w given in (2.2) establishes the desired isomorphism. The second assertion of (2.2) does not extend to the spaces mI(.); (E) is not necessarily isomorphic to H m (E). in eneral g ’ mFXI GET I For example, mN( H R) ¢ mNXN(R)- nEN This may be seen by noting that mNXN(R) has a strong order unit (any positive constant map on NXN), while mN( H R) does not have a strong nEN order unit (since HR does not). 20 Section 3. Order properties in subspaces of wI(E). In this section we investigate relationships between various order properties which may be possessed by the vector lattice E, and those possessed by wI(E) and its vector sublattices. (3.1) Proposition. If E is an Archimedean vector lattice, then any vector sublattice of wI(E) is Archimedean. 2322:, Since a vector sublattice of an Archimedean vector lattice is Archimedean, it suffices to consider wI(E). Suppose E is Archi- medean. If [Xi’ I], [yi, I] E wI(E)+ with l[xi, I] §_[yi, I] for all A :_0, then Axi §_yi for all A :_O and i E I; hence, xi = O for all i E I, so that [Xi’ I] = 0. Therefore, wI(E) is Archimedean. (3.2) Definition. A vector sublattice wI(E) will be said to have the regular supremum property_if, for arbitrary UWE.AI(E), an element [zi, I] in AI(E) is the supremum of U only if, for every i E I, O zi0 = sup {xi0: [Xi’ I] E U}. Note that the phrase "only if" may be replaced by "if and only if", since the addition of "if" merely adds a condition which is trivially satisfied by every vector sublattice of wI(E). (3.3)Prcposition. If E is a vector lattice, every ideal XI(E) in wI(E) has the regular supremum property. Proof. First we show that wI(E) itself has the regular supremum property. Let UE wI(E) and [Zi’ I] = sup U. Let iO E I. Then zi0 is an upper bound in E for {Xi : [Xi’ I] E U}. Suppose y is another 0 upper bound for this set. Then the family [yi, I] in mI(E) given by 2l y if i = i0, zi if i # i0, is an upper bound in mI(E) for U. Thus [yi, I] Z-[Zi’ I]; in part- lcular, y = yi0 Z'Zio. Therefore zi0 = sup {xiO: [Xi’ I] E U}. Now let AI(E) by any ideal in wI(E), and let U E AI(E) with [Zi’ I] = sup U in AI(E). Then [Zi’ I] = sup U in mI(E), since XI(E) is an ideal. Thus zi = sup {xi : [Xi’ I] E U}, as shown above. 0 0 Therefore, AI(E) has the regular supremum property. (3.4) Proposition. If E is a vector lattice, every vector sub- lattice AI(E) of mI(E) containing ¢I(E) has the regular supremum pro— perty. Proof. Suppose ¢I(E)‘E XI(E), and let U be a subset of AI(E) having supremum [Zi’ I] in AI(E). Let i E I. Then zi is an upper O O bound for {xi : [Xi’ I] E U}. Suppose y is another upper bound; then 0 %(y + z. ) is still another upper bound, since y + z. > x. + x. 10 lo _ l0 1o for all [Xi’ I] E U. Define the family [yi, I] by y - 210 if i = i0, 0 If l # l0. Then [yi, I] €:¢I(E)€E lI(E), so that the family [(1/ayi + Zi)’ I] = % [yi, I] + [Zi’ I] belongs to AI(E) and is an upper bound for U. Thus [(%y. + z.), I] > [z., I]; in particular, %y + %z. > z. , which I l —- I I0 - lO implies y Z-Zi . Therefore, zi = sup {Xi : [Xi’ I] E U}. 0 O 0 To find examples of vector sublattices AI(E) which do not have the regular supremum property, one must thus go beyond the familiar sequence spaces. Consider E = R, and I = [0,1], the closed unit 22 interval. As a vector sublattice of wI(R), the space C[0,1] does not have the regular supremum property. Indeed, the family U = {fhz n E N} 1 given by fn(x) = xp (0 j_x §_1), has the constant function 1 as its supremum in C[O,1]; but sup {fn(0): n E N} = O. In view of the fact that wI(E) has the regular supremum property, we see that a vector sublattice XI(E) has the regular supremum property if and only if any subset U having a supremum in AI(E) has the same supremum in wI(E). (3.5) Proposition. Let E be a vector lattice, let AI(E) be an . . (a) . ideal in wI(E), and let {[xi , I]&€T be a bounded net in XI(E). ) Then [xi(a , I].Jg.[xi, I] in AI(E) if and only if xi(a)_3,xi (in E) for all i E I. Proof. First note that each net {xi(a)} is order—bounded in E. Suppose [xi(a), I]-99[xi, I] in E. Then there is a net {[yi(a), I]} in AI(E) such that [[xi(a), I] — [Xi’ I]l §_[yi(a), I] + 0. (a) _ (a) That is, [lxi Xil’ I] _f_[yi , I] + 0. Thus for each i E I, (on) §_yi + 0, using the regular supremum property. Hence, lxi(a) " Xil xi(a)-—2»xi in E for each i E I. (a) 0 Now suppose that for each i E I, xi .——9xi. Then for each i, (a) — x.| §_yi(a) + 0. Then . a . . there eXIsts a net {yi( )} in E With Ixi 1 him, I] + o in (.163), and |[xi(°‘), I] _ 5.1, I]; 18,”). I]. Therefore, [xi(a), I]—9>[xi, I] in wI(E), as well as in the ideal AI(E). (3.6) Proposition. Let E be a vector lattice, let AN(E) be an (k) n ideal in wN(E), and let {[x , N]} be a bounded sequence in AN(E). ,- 23 (k) , . . . (k) l . Then [xn , N] —£L)[Xh’ N] in AN(E) if and only if xn ._;9 xn in E for all n E N. Proof. Suppose [x;k), N] —I9 [Xn’ N] in AN(E). Let n E N, and . k conSIder any subsequence {x (i) w (k) E n }i=1 of {xn }. By —convergence, k(i) m k'(i) m . {[xn , N]}i=1 has a subsequence {[xn , N]}i=1 order converging k(i)}0° i: . k'(i) W to [Xn’ N] in AN(E). Then {xn }i-1 1 has the subsequence {xn (k) k n order converging in E to xn, by (3.5). Therefore x ———+ xn in E. (k) n Now suppose that for every n E N, x ——:a»xn. Let {[X:(l), NJ}:=1 be an arbitrary subsequence of {[xék), N]}. By *_convergence there 00 oo k1(i) O eXists a subsequence {k1 (1)}i:l of {k(i)}i=1 such that x1 -——e x1. Continuing inductively, for every n = 2,3,... there exists a sub- w k (i) (i)}. such that x n ——29 x . For each i 1 n n sequence {kn (1)}i=1 Of {kn-1 = ( kn(i) n E N we thus have a sequence {ynl)}:.:1 in E such that Ixn - x I n j_y;l) + 0. Let n E N. Then for j :.n, kj(j) is a term of the sequence {kn(i)}; i.e., kj(j) = kn(i) for some i 3_j. Then for j 3_n, k (j) k (i) - x j - x = x n — x < y(1) < yin) + 0. That is, for each n n n n - n —- i k-(i) k-(i) n E N, xnl ——3§ xn. Therefore, by (3.5) [xnl , N]-Ji> [Xn’ N] in AN(E). But this means that every subsequence of {[xék), N]} has in turn a subsequence order converging in XN(E) to [Xn’ N]. There- fore, [x(k), N] —I» [x ,N] in AN(E). n n It may be of some interest to note that the proofs given for the "only if” parts of (3.5) and (3.6) remain valid under the less 24 restrictive hypothesis that_AI(E) (resp. AN(E)) is a vector sub- lattice of wI(E) with the regular supremum property. (3.7) Definition. Given a vector lattice B, we define 3&(E) = {[Xi’ I] E wI(E): xi = O for all but countably many i E I}. We note that 8&(E) is an ideal in wI(E). (3.8) Proposition. Let E be a nontrivial vector lattice. (a) If AI(E) is an ideal in $i(E) containing a i (E) for some 1.0 i0 E I, then E is order separable if and only if XI(E) is order separ- able. (b) wI(E) is order separable if and only if E is order separable and I is a countable set. Proof, (a) Since ideals in order separable spaces are order separable, it suffices to prove that 6&(E) is order separable when— ever E is. Thus suppose that E is order separable, and let A be a subset of 6&(E)+ having supremum [Zi’ I] in 6&(E). Then zi = 0 for all but countably many i, say i = i1,i2,.... Since for each j, zj = sup {sz [Xi’ I] E A}, we know that for each [Xi’ I] in A, xi = 0 except for i = i1, i2,.... Thus by order separability of E, for (n,j) each n = 1,2,..., there is a countable set {[xi , I]: j = 1,2,...} in A such that (n ') . zi = sup {xi ’3 : j = 1,2,...}. n n (h.i). - . . _ . Then zi §_sup {xi . h — 1,2,..., 3 - 1,2,...} j_sup {xi . [Xi’ I] n n n E A} = zi , since $i(E) has the regular supremum property. Therefore, n [21’ I] = sup {[x(h’j) i ,I]:h=1,2,...;j=1,2,...}, 25 and we see that A has a countable subset with the same supremum. (b) In view of (a) it suffices to prove that if I is uncountable, then wI(E) is not order separable. Suppose I is uncountable. Pick any e > 0 in E, and let [e, I] denote the constant family [Xi’ I] where xi = e for all i E I. For each iO E I define [e(io), I] to be the family [yi, I], where yi = 0 if i # i0, and yi = e if i = i Let 0. A = {[e(i0), I]: iO E I}. It is clear that sup A = [e, I] and that A has no countable subset whose supremum is [e, I]. Therefore wI(E) is not order separable. (3.9) Proposition. Let E be a vector lattice. Then E has a weak order unit if and only if wI(E) has a weak order unit. E has a strong order unit if and only if mI(E) has a strong order unit. Proor. (a) If 1 is a weak order unit in E, a weak order unit [ei, I] in wI(E) may be obtained by defining ei = 1 for all i E I. Conversely, if [Zi’ I] is a weak order unit in wI(E) we may pick i E I O and let e = zio; then e is easily seen to be a weak order unit in E. (b) Suppose 1 is a strong order unit in E, and again define ei = 1 for all i E I. Then [ei, I] E mI(E). If [yi, I] E mI(E) then there exists y E E+ such that lyil §_y for all i E I. There exists A > 0 in R such that Ay §_1; hence, lA[yi, I]| j_1. Therefore, [ei, I] is a strong order unit in mI(E). 0n the other hand, if [Zi’ I] is a strong order unit in mI(E), pick iO E I and let e = zi . Let x E EI. Defining xi = x for all i E I, we have [Xi’ I] E mI(E). Then there exists A > 0 in R such that O < A[x., I] < [z., I]. In particular, Ax = Ax. < z. = e. - l - 1 l0 —- lO 26 Therefore, e is a strong order unit in E. (3.10) Proposition. If E is a vector lattice, and AI(E) is any ideal in wI(E) containing a (E) for some i E I, then I l 9 O O (a) E is Dedekind complete if and only if AI(E) is Dedekind com- plete; (b) E is Dedekind o—complete if and only if AI(E) is Dedekind o—complete. Proof, We prove (a) only; the proof of (b) is analogous. Since Dedekind completeness is inherited by ideals, it is sufficient to prove that wI(E) is Dedekind complete whenever E is. Suppose E is Dedekind complete, and let A = {[xga), I]: a E P} be a subset of wI(E) having an upper bound in wI(E). Then for each i E I, { . a E I} is bounded above in E; hence it has a supremum A”). i z in E. By the regular supremum property, [zi, I] = sup A in wI(E). Therefore, wI(E) is Dedekind complete. The condition that AI(E) be an ideal in wI(E) was included in (3.10) to facilitate the ”only if" part of the proof. Consider the example E = R, wN(E) = s, AN(E) = c. Then c is a vector sublattice of wN(E) which contains aN i (E) for every i E N, but E is Dedekind ’ 0 complete while 0 is not Dedekind complete. Notice that c even has 0 the regular supremum property by (3.4). (E) for some i E I was The condition that AI(E) contain a 0 1,10 included in (3.10) to facilitate the ”if" part of the proof. If we leave out this condition, we can easily find a counterexample to (a). Take E = c, I = {1}, and AI(c) = 0 Then AI(E) is Dedekind complete, 00 27 while E is not. (3.11) Proposition. If E is a vector lattice, and AI(E) is any order closed ideal of wI(E) containing GI l (E) for some i E I, then 0 O E is universally complete if and only if AI(E) is universally complete. 0 Proof. Since universal completeness is inherited by order closed ideals, it suffices to prove that wI(E) is universally complete when— ever E is. Suppose E is universally complete. Then wI(E) is Dedekind com— (a) plete by (3.10). Let {[xi , I]: o E F} be a set of pairwise disjoint positive elements of wI(E). Then for every i E I, {x§a): a E I} is a set of pairwise disjoint elements of EI, so this set must have an upper bound ui in E: ui :_x§a) (for all a E P). But then [ui, I] :_[x§a), I] for all a E I. Therefore wI(E) is universally complete. (3.12) Proposition. Suppose E is a vector sublattice of a vector lattice P. Let A (E) denote one of the spaces w (E), o (E), a . (E) I I I 1,10 or mI(E), and let AI(F) denote the corresponding space for F. Then, considering AI(E) as a vector sublattice of AI(F), (a) if E is an ideal in F, then AI(E) is an ideal in AI(F); (b) if E is order dense in F, then AI(E) is order dense in AI(F); (c) if E is quasi order dense in F, then AI(E) is quasi order dense in AI(F); (d) if E is order closed in F, then AI(E) is order closed in AI(F). Proof. (a) Let [Xi’ I] E AI(E) and [yi, I] E wI(F) such that 28 I[yi, I]I §_|[xi,I]l; i.e., lyil §_Ixil for all i E I. If E is an ideal in F, then yi E E for all i E I, so that [yi, I] E AI(E). (b) Suppose E is order dense in F, and let [fi’ I] :_0 in AI(F). Then for each i E I, fi = sup {e: 0 §_e E'fi’ e E E}. Since AI(E) has the regular supremum property, we thus have [fi’ I] = sup {[ei, I]: 0 :_[ei, I] i-[fi’ I], [ei, I] E AI(E)}. But this means that AI(E) is order dense in AI(F). (c) The proof of (c) is trivial. (d) Suppose E is order closed in F. Let A §EwI(E) have supremum [fi’ I] in wI(F). By the regular supremum property in wI(F), fi = sup {eiz [ei’ I] E A} for every i in I. But then fi E E for each i. Therefore [fi’ I] E wI(E). Therefore wI(E) is order closed in wI(F). Section 4. Dedekind completion and universal completion. The relationship between the Dedekind completion of E and that of wI(E) is simple and natural. In this section we establish this relationship and its ramifications for certain subspaces of wI(E). Similarly we examine the universal completions of E and wI(E). These results will find application in the next chapter. Propositions (4.4) and (4.7) are the only original results in this section. Definition (4.1) and proposition (4.2) may be found in Luxemburg and Zaanen [6, section 32]; proposition (4.3) may be found in Nakano [8, §30], Peressini [10, pp. 151—154] or Vulikh [18, pp. 108—113]. (4.1) Definition. Given a vector lattice E, a Dedekind completion 29 of E is a vector lattice E such that (a) E is Dedekind complete, (b) there eXists a one-one linear w: E + E such that w(x) i w(y) whenever x §_y, (c) for every n > 0 in E there exist x, y in E such that 0 < 0(x) _<_? :My)» (u92) Proposition. Let E, E and w be as defined in (4.1). (a) The map 0 preserves arbitrary suprema and infima. II X> II (b) Every 2 in E satisfies sup {w(x): x E E, w(x) 5_2} inf {W(y): y E E, W(y) 3_2}. (c) Condition (c) of (4.1) may be replaced by the pair of con— A A ditions: (i) for each 0 < x E E there exists x E E such that 0 < x < 2, and (ii) the ideal generated in E by E is E. (d) If 1 is a weak (resp.strong) order unit in E, then w(1) A is a weak (resp. strong) order unit in E. (4.3) Proposition. A vector lattice E has a Dedekind completion if and only if E is Archimedean. Moreover, any two Dedekind completions of E are isomorphic. Henceforth we shall regard an Archimedean vector lattice E as already a subspace of its Dedekind completion E, with the map 0 being merely the inclusion mapping. (4.4) Proposition. If E is an Archimedean vector lattice, then (a) Q?) = wI(E); (p) Q?) = ¢I(E); /\ A (c) mI(E) = mI(E). 30 A Proor. Since E ,C-_E, we may regard wI(E), ¢I(E) and mI(E) as vector sublattices of wI(E), ¢I(E) and mI(E), respectively. (a) By (3.10) wI(E) is Dedekind complete. Now let 0 < [2i, I] E wI(E). Then for every i E I such that xi # 0, there exist Xi’ yi A in E such that 0 < xi i-Xi §_yi. Defining xi = yi = 0 for those i such that 21 = 0, we have 0 < [Xi’ I] i [$21, I] i [371’ I]. Therefore, wI(E) 2 wI(E) by the uniqueness result of (4.3). (b) The proof of (a) carries over to this case. (c) Only a slight modification needs to be made to the proof of (a) to cover this case. If [xi, I] E mI(E), then there exists 9 E E such that 21 < y for every i E I; but then by (4.1) there exists y E E such that 9 < y. Then take yi = y for all i E I, and proceed as in the proof of (a). The notion of universal completion may be found in Nakano [8, §34] or Vulikh [18, pp. 142-144], although Vulikh prefers the term "maxi— mal extension". Both (4.5) and (4.6) are drawn from these sources. (4.5) Definition. A universal completion of a vector lattice E is a vector lattice E# such that # (a) E is universally complete, # (b) E is isomorphic to an order—dense vector sublattice of E . (4.6) Proposition. A vector lattice E has a universal completion if and only if E is Archimedean. Moreover, any two universal com- pletions of E are isomorphic. In View of proposition (4.6), we shall hereafter, for the sake 31 of convenience, regard E as a vector sublattice of its universal completion E#. It is not hard to see that E is then the ideal gener— ated by E in E#. (4.7) Proposition. If E is an Archimedean vector lattice, then ) = wI(E)#. wI(E# Proof. By (3.11) wI(E#) is a universally complete vector lattice containing wI(E), and by (3.12) wI(E) is order dense in wI(E#). Apply the uniqueness part of (4.6). Section 5. Banach o—limits in Archimedean vector lattices. In a recent paper by S. Simons [16] we find a novel approach to the theory of ordinary Banach limits for bounded real sequences. This approach may readily be adapted to the context of vector lattices, leading to the establishment of ”Banach o-limits" for the space of order bounded sequences in a vector lattice E, i.e., mN(E). The results (5.4) - (5.12) of this section are thus generalizations of results already known for ordinary Banach limits, as presented in Simons [16] or Goffman and Pedrick [2]. Throughout this section, let E represent an arbitrary EEEEE? medean vector lattice. The elements [Xn’ N] of mN(E) are merely sequences in E, and thus for convenience will often be denoted {xn}. (5.1) Definition. (a) o: m (E) + m (E), called the ”shift L N N operator? is defined by C(Xn} = {Xn+1}' (b) L: mN(E) + E is defined by L{xn} = lfim xn (= ipf REE Xk’ in E). 32 (c) S: mN(E) + E is defined by S{xn} sop x (in E). n (d) C(E) = {{x } E mN(E): II; X = lim x , in E}. n 1'1 n —n-n Note that C(E) is the set of all sequences in E which order converge in E; C(E) is a vector sublattice of mN(E). (5.2) Definition. A Banach o—limit on mN(E) is a linear map g: mN(E) + E such that socigisa i.e., g o o (x) §_g(x) §_S(x) for all x E mN(E). We let ®I(E) denote the collection of all Banach o—limits on mN(E). A mapping T: E + F from E into a vector lattice F is said to be sublinear if (a) T(Ax) = AT(x) for all x E E, A E RI, and (b) T(x + y) :_T(x) + T(y) for all x,y E E. As an example, observe that the standard argument used for real sequences applies here to show that the map L: mN(E) + E is sublinear. It is easily seen that a sublinear map T: E + F will have the following additional properties: (C) T(O) = 0; (d) -T(x) j_T(-x) for all x E E; (e) if f: E + F is linear and f(x) §_T(x) for all x E E, then —T(—x) :_f(x) :_T(x) for all x E E. It has been noted (see e.g. [10], p. 78) that the proof of the classical Hahn—Banach theorem remains valid if R is replaced by an arbitrary Dedekind complete vector lattice. 33 (5.3) Proposition. (Hahn-Banach). Let E be a vector space, F be a Dedekind complete vector lattice, and p: E + F be sublinear. If f: E + F is a linear map defined on a linear subspace E of E, 1 1 with f(x) §_p(x) for all x E E1, then there exists a linear extension f'of f to all of E, with f(x) :_p(x) for all x E E. (5.4) Proposition. If T: E + F is a sublinear mapping from E into a Dedekind complete vector lattice F, then for every x E E, T(x) = sup {g(x): g E L(E, F), g :_T}, where L(E, F) denotes the space of all linear maps from E to F. Proof, Let x E E. We need only show that T(x) §_sup {g(x): g E L(E, F), g §_T}. Let [x] denote the linear span of x, and define f: [x] + F by f(ax) = aT(x). Then f is linear and dominated on [x] by T, since (i) a :_0 implies f(ax) = aT(x) = T(ax), and (ii) a < 0 implies f(ax) = aT(x) = (-a)[—T(x)] §_(-a)T(—x) = T(ax). Thus by the Hahn—Banach theorem, there exists g E L(E, F) with g :_T and g(x) = T(x). Therefore T(x) j_sup {g(x): g E L(E, F), g :_T}. (5.5) Proposition. If g E @dKE), then (a). g 0 O = g, (b) giL, (c) {xn} e C(E) implies g(x) equals the order limit of {xn}, (d) g 3_O. 3322:: (a) Let x E mN(E). Then -(g o o)(x) = (g o o)(-x) g(—x) = -g(x) by (5.2) and the linearity of g and 0. Thus (g o o)(x) |/\ | v g(x). Combining with (5.2), we have g o o = g. 34 (b) Let x E mN(E). For each n E N, g(x) = (g o o) (x) = = (g 0 on) (x) :_sup {xkz k 3_n} by (5.2). Thus g(x) §_iof REE xk = L(x). (c) Let x = {x } E C(E). Then lim x = lim x in E, so -L(-x) = L(x). n -—n—I’l n n But since L is a sublinear map satisfying (b), -L(-x) j g(x) §_L(x). Therefore, g(x) = L(x), which equals the order limit of x. (d) x > 0 implies -x < 0; whence, g(—x) < L(-x) = inf sup x < 0, so that g(x) = -g(—x) :_0. Observe that if‘g E L(E, F) satisfies (a) through (d) of (5.5), then g E 0 in E, then L o (o — I) (e,0,e,0,...) = L(—e,e,-e,e,...) = e > 00 (5.8) Proposition. (Existence of Banach o—limits). The map _-—.—1 n Q({xn}) - l%m E. k=1 xk is a sublinear map 0: mN(E) + E which generates Banach o—limits. Proof. Sublinearity of Q follows from sublinearity of lim. Let x = {xn} E mN(E). Then Q(x) §_l m fi-E:=1(S(x)) = S(x), and n .O 8‘ >< I E, Q o (O - I) (X) . 1 n 1.1.”) H Zk=1< n+1 I X1) 36 < m U'H % [S(x) - x1] = 0. Therefore by (5.7), Q generates Banach o—limits. (5.9) Proposition. If we define the map W: mN(E) + E by W(x) = sup (g(x): E 66316)} and let T denote a sublinear mapping from mN(E) into E, then (a) W is sublinear, generates Banach o-limits, and dominates all Banach o—limits; (b) T generates Banach o-limits if and only if T :_W; (o) T dominates all Banach o—limits if and only if W :_T; (d) a sequence x = {Xn} is mN(E) is almost o—convergent if and only if -W(—x) = W(x). £33555. That W is sublinear is clear. Since each g E EIIE) satisfies the conditions of (5.7), so must W. Thus W generates Banach o-limits. That W dominates all Banach o—limits is inherent in its definition. Then (b) follows directly from (a) and (5.4); (c) follows from (a) and the definition of W. By our remarks following (5.2), we know that for all g E 6I(E), —W(-x) :_g(x) :_W(x). Thus —W(—x) = W(x) implies that x almost o-converges to W(x). Conversely, suppose x almost o-converges to 2 E E. Then -x almost o—converges to -2, so that W(-x) = g(-x) for arbitrary g E @fl1E). But then the linearity of g implies —W(—x) = —g(—x) = g(x) = W(x). Therefore, (d) holds. (5.10) Proposition. If we let SN(E) denote the collection of all x = {xn} E mN(E) such that the sequence {sn} of partial sums 37 8n = 2::1 xk is order bounded, then for every x E mN(E) W(x) = inf {S(x + z): z E SN(E)}. Proor. Let V(x) = inf {S(x + z): z E SN(E)}. By (5.9) it suffices to prove that V is a sublinear mapping which both generates and dominates all Banach o-limits. Let x, y E mN(E). For every zl, z E SN(E) we have 2 + z E SN(E) 2 1 2 and S(x + y + z + 22) _<_S(x + Z1) + S(y + 22). Thus 1 inf {S(x + y + z): z E SN(E)} §_S(x + Z1) + S(y + 22); then taking infima, first over 2 in SN(E), then over 2 in SN(E), 1 2 we obtain V(x + y) i. V(x) + V(y). Therefore, V is sublinear. Now V :_S, since 0 E SN(E). Let x E mN(E). Then ox - x E SN(E) and every 2 E SN(E) satisfies V(z) j_0; hence, V o (o — I) (x) :_0. Therefore, (5.7) implies that V generates Banach o-limits. Let g E @i(E) and x E mN(E). Each 2 E SN(E) may be written 2 = oy — y, where y = {0, 81’ s n 2, ..., Sn’ ...} E mN(E) with 3n = X Zi° i=1 Thus S(x + z) = S(x + oy — y) 3_g(x + 0y - y) = g(x) + (goo)y - g(y)= g(x). Taking infimum over all z in SN(E), we obtain V(x) 3_g(x). Therefore, V dominates all Banach o—limits. (5.11) Proposition. For every x = {xn} E mN(E), W(x) = inf {lim jem n E N}. l k k-E' x . . : k E N; n1, n2, ..., k [Equivalently, W(x) = inf {%’8(Z§:10ni(x))i k EN ; n1, n .., nk E N}.] 29 38 k x i=1 n.+j 1 Proof. Let p(x) = inf {lim 2-2 E N}. __ W k 3 By (5.9) it suffices to prove that p is a sublinear mapping which : k E N; n1,n ,n 2’°°° k both generates and dominates all Banach o-limits. (a) We first show that p is sublinear. That p(Ax) = Ap(x) for all A :0 is clear from the definition of p. To establish subadditivity, let x = {xn} and y = {yn} belong to mN(E). Let k,l E N and n1,n2,..., nk, m1,m2,...,ml E N. By definition of p, p(X + Y) < lim 1""- ],< l (x + y ) —t->co kl i=1 j=1 n.+m.+t n.+m.+t ’ l J l J and using the subadditivity of lim, p(x + y) < lim-—I ZI [I . X + lim-—I k I y —'t+m kl 1:1 3:1 ni+mj+t t+m kl i=1 j=1 ni+mj+t l 11 —.——1§ 1k 1 5'l Ij:1(%IE ki=1xn +m +t) I k 2i 1(li3 InZ yn +m +t)' 1:1 Since this inequality holds for arbitrary k E N and n1+mj, n2+mj,..., nk+mj E N, upon taking the infimum over these variables, we obtain the inequality k 1 p(x + y) :I It, p(x) + p I, _ (lim 3 -1 t+m $.21 ) l j=1 yn.+m.+t ’ l J and of course %-Z]=1 p(x) = p(x). Then taking the infimum over l E N and ni+m1, ni+m2,..., ni+ml E N, we obtain p(x + y) §_p(x) + p(y). 2, ..., nk E N. Then by (5.5) g(x) =-]%-g(z};___1 oni(x))_<_%8(z};:1 onion). Hence, taking (b) Let g E 81(E), and let k, n1, n the infimum over k, n ., nk, we obtain g(x) §_p(x). Therefore, 1’ p dominates all Banach o-limits. (c) To show that p generates Banach o-limits we use (5.7). 39 Clearly p :_S. Let x E mN(E); then there is some u E E such that Ian §_u for all n E N. For arbitrary k, n1, n2, ..., nk E N we have p(ox — x) < lim-I-zk (x . - x .). In particular we may take j+m k i=1 ni+1+j ni+j ’ ni = i for i = 1, ..., k, and obtain the inequality -1—-1 k _-——— 1 p(ox ' X) —<- 3%.)"; k 2II=1(Xi+1+j I Xi+j) - 37.1.3 k (Xk+1+j X1+j)' Thus p(ox - x) §_%u for all k E N. By the Archimedean property in E, we thus have p(ox - x) :_0. Therefore by (5.7), p generates Banach o—limits. (5.12) Proposition. A sequence {xn} E mN(E) almost o—converges to 2 E E if and only if 2 = lim 1-(x + x + ... + x ) p+w n n+1 n+p holds uniformly in n. This proposition may be proved by a straightforward alteration of the proof for ordinary Banach limits, presented by Goffman and Pedrick [2]. The alteration is required in order to remove the depend— ence of their proof on the linearity of the order. Proposition (5.11) was proved here by a similar alteration of proofs given in Goffman and Pedrick. CHAPTER II SUMMABLE FAMILIES IN A VECTOR LATTICE Section 6. Some types of summable families. We continue to let E denote an arbitrary vector lattice and I denote an arbitrary nonempty set. The theory of summable families will require the use of convergent nets in wI(E). The basic notions of order convergence were presented in the introduction. Let V(I) denote the collection of all finite subsets of I, and ifJ QJ. Givena partially order V(I) by inclusion: J 2 1 2 15-“I family x = [Xi’ I] in wI(E), we shall be concerned with the nets {oJ(x)} and {TJ(x)} defined over V(I) by oJ(x) = iIJXi’ and TJ(X) = iéJlxi We say that [Xi’ I] is order summable (to an element x E E) if oJ(x) 3x0; in symbols, We say that [Xi‘ I] is absolutely order summable, if {IJ(x)} order converges to some element of E. Thus x is absolutely order summable if and only lxl is order summable. If OJ(X)-—E)XO, then we say that [Xi’ I] is uniformly summable to x if {TJ(x)} converges uniformly 03 to an element of E, we say that [Xi’ I] is absolutely uniformly sum- mable. 4O 41 (6.1) Definition. Given a vector lattice E and an arbitrary nonempty set I, we let R;(E) denote the collection of all [Xi’ I] in wI(E) which are order summable, and we let Q¥(E) denote the collection of all [Xi’ I] in wI(E) which are uniformly summable. Finally, we let 1 21(E) = {[xi, I] e wI(E): tau e E a ingxil :;u‘dJ e V(I)}. (6.2) Proposition. 2;(E) and 2¥(E) are vector subspaces of 01(3); 2%(E) is an ideal in wI(E). Proof. It is clear that each of these spaces is closed under . . . o scalar multiplication. Let x = [Xi’ I], y = [yi, I] E 21(E). Note that for each J E V(I) o (x+y) = (x. + y.) = x. + y. = ’ J iIJ l l igJ l igJ l oJ(x) + oJ(y). Since the nets {oJ(x)} and {oJ(y)} are order conver— gent, say to x and y0 respectively, they are order bounded, and there 0 exist nets {uJ} and {VJ} in E such that §_v + 0. IOJ(X) — xOI §_u J + 0 and IoJ(y) — y J 0' 0| §_uJ + VJ I 0. Then |0J(x+y) — (x0+y0)| 1|0J(x) - x0] + |0J(y) - y Thus x + y E Q§(E). Therefore 2; is closed under addition. The proof that £¥(E) is closed under addition is similar. Therefore Q§(E) and Q§(E) are vector subspaces of wI(E). The proof that ($(E) is an ideal is easy, and will be omitted. In contrast to the space 2§(E), the collection of all absolutely order summable families [xi, I] in wI(E) does not even form a vector subspace of wI(E). As a counterexample, consider E = c, the space of convergent sequence. For each n E N we continue to let en = (0,0, ...,1,0,...) whose kth term is 1 if k = n, and 0 if k # n. We let 42 l = (1,1,...,1,...) each of whose terms is 1. The families x = [en, N] lel N n n . k n |(-1) enl = I. On the other hand, given k E N, anllen + (-1) enl and y = [(—1)nen, N] are each absolutely order summable, with an :EnEN = (0,2,...,0,2,0,0,...); thus TJ(X + y) is not order convergent in 0. Hence x + y is not absolutely order summable. Therefore, the collec- tion of all absolutely order summable families [Xn’ N] E wN(c) does not form a vector subspace of wN(c). (6.3) Proposition. If E is an Archimedean vector lattice, then XII1(E) 9'.- RCID(E) 9'. 2%(1‘3). Proof: In an Archimedean vector lattice, uniform convergence implies order convergence. Thus,Q§(E)<; fl§(E). Let [Xi’ I] E Q;(E). Since order convergent nets are bounded, there exists u E E such that IZIEJ x.| j_u for every J E V(I). 1 As shown in the appendix, an Archimedean vector vector lattice E is isomorphic to a vector sublattice of Cw(Q), for some extremal compactum Q. For convenience we identify E with its image in Cm(Q). Then for every J E V(I) and every t E 0, I2. lEJ xi(t)| : IZiEJ Xil(t) j_u(t). Remember that each xi(t) is a real number. Consider an arbitrary J E V(I). For each t E Q, let Jt = {iEJ: xi(t) 3_0}. Then Zi€J xi+(t) = ZIEJ xi(t) :_u(t). Therefore, for every t J E V(I), zi€J in :_u. Similarly we can show that XIEJ xi :.u. Hence for every J E V(I), x < 2n. + ziEJ i '— IIeJIXII: IieJ Xi Therefore, [Xi’ I] E 2%(E). We conclude that Q?(E)<§,£%(E). 43 (6.4) Proposition. (a) If E is a finite dimensional Archimedean vector lattice, then Q$(E) = 2¥(3)' (b) If E is a Dedekind complete vector lattice, then Z§(E) = 9%(E). Proor. (a) It is well known that any finite dimensional Archi— medean vector lattice E, say of dimension n, is isomorphic to Rn. In Rn, order convergence and uniform convergence are equivalent. Thus ,Q‘I’ n , 0 nm - 0 00 < w. (ii') for every n, 2m=1lxnm| 9 that is, the rows are all A -sequences, and from some row on, all 1 the rows are zero. It is thus apparent that TI<¢> = ¢N<21). (7.1) Proposition. 00 (a) (NW {(Xnm):3u e R 3 sz m=1lxnml fin}; (b) ,Qfi(m) contains a proper ideal isomorphic to mN(£1). Proof. (a) A matrix (xnm) belongs to [§(m) if and only if (i) for every m there exists um E R such that Ix I §_um for every for every n. nm n E N, (ii) for every n there exists yn E R such that 2;=1|Xnm yn, and (iii) there exists u E R such that yn :_u 0 0 Observe that (i) follows from (ii) and (iii). Then (a) is apparent. (b) It is clear that mN(£1) is isomorphic to the space M = {(xnm): (l) Vn’zm=1lxnm| < m, (li)‘v’rr13um E R 5 lxnml : um‘Vn, and (iii) 2;:1 um < W.}, with the pointwise-defined linear operations and order. If (xnm) E M, then for every n, zm=1lxnml i-Zmzl um < w; 1 . . . 1 c. hence (Xnm) E QN(m). Therefore, mN(£l) is isomorphic to M _,QN(m). From its definition it is clear that M is an ideal in,Q§(m). Now let I denote the identity matrix I = (Inm) with Inm = 0 if n ¢ m, and 1 if n = m. Then by (a) I €,Q§(HU. 0n the other hand I E M. Hence, M is a proper ideal in,flfi(m). 46 k :1 (7.2) Proposition. 1,},(11) = {(xnm): 311 € RSVkA 9 {n=1 m=1|xnml < u}. Proof. By the remarks at the beginning of this section, a matrix (Xnm) belongs to 2fi(21) if and only if (i) for every m there exists R 00 : ' ° ° um E such that anllx um, (ii) for every n there EXlsts yn E R nml such that 2;:1Ixnml = yn, and (iii) I:=1 yn < w . 1 . _ m If (xnm) E QN(£1), we define yn as above and let u - Zn=1 yn. k l k w k Then for every k,l , {n=1 Zm=1lxnml j-Zn=1 Zm=1lxnml = In=1 yn :_u. 0n the other hand suppose (xnm) is a matrix and u E R such that k l . .. for every k, l, En=1 zm=1lxnm| §_u. Then (i) and (ii) clearly hold for (x ). We also have (iii) since Em y = lim 2k (Em IX I) = nm ’ n=1 n k+m n=1 m=1 nm . . k l 1 flog II: Zn=1 Zm=1lxnm| §_u. Therefore (Xnm) E QN(£1). As a consequence of (7.2) we see thatI7fi(21) is the space of all (Xnm) such that the double series 2 is absolutely convergent. x n,m nm We turn now to 1%(E) for a general Archimedean vector lattice E. A Recall that we regard E as embedded in its Dedekind completion E. . . . 1 . ConSIder an arbitrary [Xi’ I] in 21(E). Since [Xi’ I] may also be regarded as an element of.Q%(E), we know by (6.4) that there exist elements R, 9 in E such that A x x. and y = E : ZiEI i iEIIXi|° (7.3) Proposition (order continuity of the sum). Let E be an Archimedean vector lattice, and suppose {[xga), I]}oEI is a net in 47 9%(3) SUCh that [X§a), I] -29 [yi’ I] E 21(E). Then in E, .(a) _ (a) - _ . where x — EiEI xi and y - ZiEI yi. Proof. (a) First suppose [x§a), I] :_O for all aEI, and (on) . (a) [x§a), I] I [yi, I]. Then clearly a 1 and sop x = (CI) . su su . x. = su su . x. = su o-lim . x. OLp JP IleJ l Jp OLp I JP ( a I g(a) + . sup 2 yi = 9. Therefore, y. J iEJ (b) Now suppose [x§a), I] I 0. Pick an arbitrary do E I. Then [Xia0) - xga), IJQZQO = [Xgao), I] - [X§a)s I] I [ng0), I] . 1 ( ) ( ) ( ) ( ) 1“ (I(E)+' Thus by (a) Iiel Xiao ‘ IieI Xia ‘ IieI(XiaO ' Xia ) I ZiEI XIGO) (for a 3_do)- That is, 2(a0) — 2(a) I 6(a) (93.90), S,(on) . (on) , and hence by cancellation, - t 0 (a :_a0). Therefore, x waux' Since this is true for every a0 E I, we conclude that 2(a) + 0. (c) Finally suppose [x§a), I] —2+-[yi, I]. Then there exists (on) a net {[ui , I]} in 9%(E) such that [Ix§a) — in, I] :_[u§a), I] I 0. (a) By (b) Zi€I ui + 0. Then x - yI - IZiEI xi — ziEI in :_ .( - ZieIlXia) - in -Ziel uia) + 0. Therefore, x a)._£; ya (7.4) Proposition. If (Ea: GET} is a collection of vector lattices and I is an arbitrary nonempty set, then N 1 Ed) — HaEF2I(Ea)' 48 The proof of (7.4) amounts to showing that mapping w defined in the proof of (2.), when restricted to II y(l) :_1. Thus n :_max {n1, n n __p; 1'1 . . p 2, ., np}implies zk=1 y hence, yn :_p. This result contradicts the eventually constant nature ofIyn}. Therefore, [{xék)}, k E N] E Q§(E). Section 8. ,Q%(E) as a normed space. Throughout this section we shall assume that (E, j, H.”) is a normed vector lattice as defined in the introduction. We shall examine several natural norms induced on I%(E) by the norm I.“ in E. . . . 1 (8.1) Definition. An Q—norm on 91(3),-relative to “p“, is any monotone norm “.Ni on Q$(E) such that (a) [Xi’ I] e ¢I(E) implies “[X1’ 1]“2 = II IXiIN3 iEI (b) if XiEJIXiI §_y E E for all J E V(I), then ”[Xi’ 1]”2 §_uy“. 50 For each [xi, I] E Q%(E) we define “[Xi’ fl“1 sup {“ziEJIXiI“: J E V(I)}, “[Xi’ 1]“1' ZieJ Existence of the real numbers “[Xi’ IJ“1 and “[Xi’ 1]“1, follows inf {uuu: Ixil :_u E E ‘dJ E V(I)}. from the monotonicity of “.h on E, the order boundedness of the set {ziEJIxilz J E V(I)}, and the completeness of R. (8.2) Proposition. If (E, :JlMD is a normed vector lattice, then 1 (a) “.“1 and N.H1, are £-norms on QI(E); (b) any R-norm “‘“2 on 2%(E) satisfies the inequality I\.\\1é\|.1|£é u .Ill. Proof. (a) We first establish the triangle inequality for M1 and u.u1,. Let [xi, I]. [yi. I] e 2§. Then ntxi,IJ + [yi.I]H1 : Sgpiizingxi + Vii“ f--S‘13p\\ziEJ|xi|Jr ZiElei l“— < SUP (“XiEJm I“ + Mm“) triangle inequality. Let u,v E E such that for every J E V(I), 5-i[xi’ 1]"1 + “[yi, I]“1. Thus H.“1 satisfies the ZiEJIXiI §_u and ZiEleil §_v. Then ZiEJlxi + yil :_u + v and hence, by monotonicity of “.H on E, “[xi, I] + [yi, Ijulv :.uu + V“.i \iuH + “v”. Taking infima in this inequality, first over all such u 11“,. and then over all such v, we obtain “[Xi’ I] + [yi, 1]“1, §_' +l‘[yi, 1]“1,. Therefore, “."1 and‘l.“1, satisfy the triangle inequality. It is then quite clear that “'“1 and H.“1, are norms 1 onRI(E). That the norms “.“1 and “'“1' are monotone is also clear. Now 51 suppose [Xi’ I] E ¢I(E). Let JO = {i E I: xi # 0}. Then by monoton- icity of H. “[Xi, 11131 =uzi€JO|xi|u = “[xi, 11||1,. That u .u, and “."1, satisfy (b) of (8.1) is obvious. Thus n.“ and H.” are 1 1' l-norms on£%(E)- (b) Suppose that “.H is an l—norm on E, and let [Xi’ I] E £%(E). £ For each J E V(I) define [Xi(J)’ I] E ¢I(E) by Ixil if i E J, xi(J) = 0 if i ¢ J. Then [lxi|, I] : 33p [Xi(J)’ I]. .As 2%(3) is a normed vector lattice, "[Xi’IJHQ : “[lxila IJU£ =i‘sngxi(J), IJNz 3--S}]le[xi(J)’ IJHQ : sgp“ZiEJlin“= ”EXi, I]"1' That “[Xi’ 1]“2 §_“[xi, I]“1, is clear from definition (8.1). Therefore, (b) is true. (8.3) Proposition. The norm N.” is additive on E+ if and only if u.“1 is additive on I%(E)+. Proof. Suppose H.“ is additive on E+, and let [Xi’ I], [yi, I] E U) Q%(E)+. Then “Exi, I] + [yi, IJH1 = sgpl|zi€J(xi+yi)“ = gp“ZiEJ xi + ZiEJ yiN : 33p (“EieJ Xi“ + “EIEJ Vi“) : SgpliZiEJ Xi“ + SEP “EiEJ yi“s since ”33p" here represents the o—limit of a net of real members, and o-limits preserve sums. Thus H.H1 is additive on Q%(E)+. . . . + . For the converse, suppose n." is additive on,Q%(E) . Given 1 x, y E E+, pick arbitrary iO E I and define [Xi’ I] and [yi, I] by x if i = i0, y if i = i0, Xi : yi : 0 if 1 ¢ 10, 0 if i # i0. Thenl\x + y” : “[Xi’ I] + [yi’ 1]“1 : ”[Xi’ 1]”1 + “[Yi, IJNl : 52 “x“ + “y“, and hence, “.u is additive on E+. (8.u) Example of a normed vector lattice (E, :3 h.“) for which the norms “.H and H.“ are not equivalent. Let E = m, with the 1 1' usual order but with the norm x ___. “ix }n = supl—2J+ limlx I. n n n n n Recall from section 7 the representation of the elements of.2§(m) as matrices. For each k E N let Sk denote the matrix whose i, j entry is 0 if i # j or if i = j < k, and is 1 if i = j :_k: (00 ) Sk : o o . 1 o o - kth row 00 010 00 001 \00 00° /. Then {Sk} is a sequence in 9§(m), the nth column of which is a sequence {s;k)} E m; S = [{sik)}, nEN]. By definitionl‘Sk“1 = k 1 1 . . sgpi‘ZnEJlsé )I“ = k’+ O = k3 while "Skul' = inf{luH: VJ E V(N), EnEJlsék)l :_u} : %-+ 1. Thus “Sknl + 0, while “8k“1' + 1. Therefore, the two norms are not equivalent. (8.5) Proposition. Let H. denote an Q-norm on £%(E). (a) If {[xgn), I]}::1 “.“2-converges to an element [yi, I] of ( 9 %(E), then for each i E I,{xin)} H.H—converges in E to yi. Hg (b) (9%(E), EJHiN) contains a norm—closed subspace, namely aI i (E), which is isometric to E. [Thus (9%(E), H.H2) is norm ’ 0 complete only if E is “.“—complete.] 53 Proof. (a) Since H.H1 é H.“£, it suffices to prove the result for “.“1. If “[xgn), I] - [yi, IJ“1 + 0, then sgp\‘zi€J|x§n) - yiln + O, (n) hence‘di E I‘\x Vi“ = “Ix§n) - yiln + O. i (b) Let iO E I. We already know that the map y + [yi, I], where yi = y and yi = O for i # i , is an isomorphism of E onto a (E). O 0 By definition ”[yi, I]“£ = y . Thus this map is also an isometry. That GI i (E) is norm closed follows from (a). 9 O In many important cases there is only one suitable fi—norm on,Q%(E). It is thus of interest to consider conditions under which this will happen. Two concepts which prove to be important in this investigation .1\ to the Dedekind completion E. These concepts are presented more fully are semi—continuity of H.“ on E and uniqueness of extension of in the thesis of R. Reichard [12], and I shall present here only those details which seem relevant or enlightening in the present context. (8.6) Definition. The norm'i.H in a normed vector lattice (E, jJ [.I) is said to be (a) semi-continuous if 0 §_xT + x implies sup\\xTH = ”x“; (b) continuous if xT + 0 implies ipfl‘xT” = O; (c) sequentially continuous if xn + 0 implies igf uan = 0. Notice that a continuous norm is also semi-continuous, for if “ .“ is continuous and xT + x, then x - xT + O and ”x — XI“ + 0; hence, O §.Hx“ - sup’le“ = inf {“xfi-“Xgfl §_ inf Hx - XI“ = O, and therefore,||xH = sqp|\x4{. 5” Also note that in a normed vector lattice with continuous norm, order convergence implies norm convergence. For if Xa + x, then there ' ' — < . - < eXists a net {ya} With Ix Xal __ya + 0 But thenl‘x Xa“ __“ya“ + 0. Therefore, we have in particular, X : ZiEI xi implies x = ”.H-lgm EiEJ xi. As noted in the introduction, every L—space has a continuous norm. The spaces Lp and RP are additional examples of normed vector lattices with continuous norm. In the space m with the usual ordering, the usual normilx“ = suplxnl is semi—continuous but not continuous. Semi-continuity may n .. _ (I) _ be seen as follows. if sT — {xn } i {yn} y, then SgpiisT" ' (T) _ (T) _ - . sup (sgplxn I) - 33p sup lxn I - sgplynl -lly“. That “.N 18 not continuous may be seen by considering sk = (0,0,...,1,1,...,1,...), all of whose terms are 1 except the first k, which are 0. Clearly sk + 0, but [\skH = 1 \Ik. Consider again the space m with its usual ordering, but this time with the norm “xu* = sgpixnl + IEE’xn . If for each kEN we let sk denote the sequence (1,1,...,1,0,0,...) whose nth component gkn is 1 if n j_k and is 0 if n > k, then 3k + 1 = (1,1,...,1,1,...). But Hsk“* = 1 for every k, while H1“* = 2. Thus we have an example of a norm H.H* which is not even semi—continuous. (8.7) Proposition. Suppose (E, :3 fl. ) is a Dedekind complete normed vector lattice with semi—continuous\[.", and “.“2 is an fi—norm 1 (a) . . 1 . on,(I(E). If {[xi , I]}aET is a net in QI(E) which “."g—converges 1 . . (a) (a) to [yi, I] E QI(E), and if we define (for all a E F) x = ziEIIXi I 55 and y = Eiellyil’ then X(a)JL$»y. That is, [x Ia). IIIUI [yi . I] ....... Z | XI“)I-—I I, iEI Proof. Since H.II1< II.IIz it suffices to prove the result for II.“1. Suppose “[xga), I] — [yi , 1]"1-+>0; that is supIIEieJm (a )- II _> (O!) ((1) . . yi I O. For every a E F, Ixi I < Ixi — in + Iin (i E I), hence for each J E V(I), sup ziEJm XI )l < .3. (2.5.Il ‘“> - y, | + IieJIin) Interchanging ngu)I and Iin, we obtain a similar Jinequality, and :sup 2.... —y. I + I...I. combining these two inequalities we obtain sup EiEJMi o0| - sup Zieleil < sup XiEJIfi (a) - yila that is, Ix(a)— y I < sup EiEJlxi (a ) — in, from which it follows that IX“) - yII : Isup XiEJIX XW) - y III N (OI) x(OI) , ow for each a E T, EieJle - yi I + sup ZiEJm — in, hence, by . . . (a) I II X(a) N semi—continuity of H. II, supIIEieJm — y. lII= sup EiEJm — yi I Therefore, “x[en,N]. On the other hand, for each k E N, 2n EN V(k ) = (1,1,...,1,0,...) whose first k terms are 1, while (E nEN en = (1 ,1,...,1,...), all of whose terms are 1. ThusIIZ — EnEN e n“ = H(0,0,...,1,1,...)“ l = k:I-+ 1 f 0. We therefore have the desired counterexample: [Vék)’ nEN] II,[en, N], while ZneNIvik)I JN?’ ZnENlen nENv (8.8) Proposition. If (B , 5) MD is a normed vector lattice with semircontinuous “.I, then, relative to ".n, (a) U.H1 is semi—continuous on I%(E); (b) H.Il is the only semi—continuous R—norm on I$(E); (c) x = ZiEIIXiI implies “[xi, 1]“, =IIxH for every l—norm on 9%(E). Proof. (a) Suppose O < [X(T), I] f [yi, I] in 21(E). By (3.3) for each i E I, O :_x§T> t yi; hence, for all J E V(I), O < Z iEJX x(T) + EiEJ yi. Then by semi—continuity of “.I I, supIIziEJ X(T)“= ( ) ( ) NXieJ yi". We therefore have sup II[xiT , 1]“1 = sup supIIziEJxl T I ( ) = 53p SLIP IIZiEJ XiT II : 33p IIEiEJ inI = ”Eyi’ IJIIl' Therefore’ I."1 is semi—continuous. 57 (b) Let H.H2 be a semi—continuous z—norm. Define the net {[xi(J), I]} as in (8.2). Since [Ixi(J)I, I] I [Iin, I] = JEV(I) IEX., IJI, we have by semi—continuit and definition of i—norm that 1 Y uni, nu, = IIIIxil. nu, = suplItlxiI ; III, ”31) 2...»: II= Itx .11“ Therefore,II.“£ = IH.H1. (c) ZiEJm + x over the directed set V(I). Hence, by semi— continuity of H.II, 33p NEiEJIXiIH = “x"; i.e., “[xi, 1]“1 = “x“. On the other hand, ZiEJlxil _<_ x VJ E V(I)im91ies “Exi, I)”, : IIxII. Thus we have ".“1 :_“.H1, which in combination with (b) of (8.2) yields the desired equality HI = H. “1- (8.9) Corollary. If (E, jJ H.“) is a Dedekind complete vector lattice with semi-continuous N.”, then there is one and only one Q-norm on 9%(E), relative to H.“; namely H.“1. The proof follows directly from (6.4) and (8.8). (8.10). Example: We shall exhibit here a normed vector lattice (E, j, “.H) with semi-continuous ”.N, such that not all l-norms on 9%(E) are semi-continuous. By Virtue of (b) in (8.8) we need only produce an example in which “'Hl # “.“1,. Note that in our example of (8.4) we did not have a semi—continuousII.“. Let E denote the space of eventually constant sequences of real numbers, with the usual ordering and with the norm IIXH = Sgp 5(n) Ixnl, where S(n) = 1 if n is even, and 2 if n is odd. That I.“ is a norm 58 is easy to see; we show only the triangle inequality: “x + y“ = 83p 5|Xn+yn| isgp Mn) (lxnl + lynl) 5.8:}? 501) lxn| + 8:19 Mn) lynl = “x“ + Hy“. Semi-continuity of u.” is also easy to see. Suppose O :_x(T) + y in B. Let x(T) = {X£T)} and y = {yn}; then for each n, ('t) _ (I) _ 0 j'xn + yn by (3.3). Thus sup|\x — sup sgp 6(n) xn — s$>Mn)s$>éf)=s$>Mn)yn=“yW (T)“ For each n e N, let en denote the sequence en = (0,0,...,1,0,...), having nth term 1 and all others zero. Consider [e2n, N] e 9%(3). We have ”[e2n, NJ”1 = 53p “EneJ e2nH = 1, Since each e2n has all its odd terms equal to 0. But if zieJ e2n :_u e E for every J a V(I), then u must have its even terms equal to 1, from some point on; hence flu” :_2. Thus “[e2n’ N]“1, :_2, and therefore u.“1 # Il-l\1.~ (8.11) Monotone extensions 2:.i;fl.£2.§3 In this number we present results without proof. For details and references, consult R. Reichard [12] or v. A. Solov'ev [17]. Suppose (E, :J “.H) is a normed vector lattice. By a (monotone) extension R."* of u.“ to the Dedekind completion 8 of B we mean a norm on 8 which agrees with u.” on E and is monotone on 8. Then A (E, j) H.u*) is clearly a normed vector lattice. Given an arbitrary 2 e 8 we define \lauu = inf {flyH: |2| :_y e E}, and p(x) = sup {Hzn: O :_z :_|2|, z e E}. Clearly every monotone extension “.H” of H.” satisfies p(.) ill.”N ill-ll» u 59 The following facts will be useful in the sequel. (a) Proposition (Vulikh). fl.“u is a monotone extension of “.\ to B. (b) Proposition (Nishura — Lozanovski). If (E, :9 l|.H) is a Banach lattice, then (8, f) ”'“u) is also a Banach lattice. (c) $333. the map 0: 8 + R+ is not generally a norm. (d) Proposition. If “.\ is semi—continuous on B, then p’is a semi—continuous norm on B, which we denote “'“p' In fact, “““p is the unique semi—continuous monotone extension of R.“ to B. (e) Proposition (Solov'ev - Reichard). If H.“ is continuous on B, then H.“u is continuous on 8 and is the unique monotone extension of H.H to B. (f) Proposition (Solov'ev). If H.” is sequentially continuous on E, thenll.“ is continuous if and only if E is of countable type (i.e., every bounded subset of pairwise disjoint elements of E is countable). (g) Proposition (Vulikh [18], Chapt. VII, E16). If (E, i, \\ .H) is Dedekind o—complete, with sequentially continuous “.n, then 1.“ is continuous. (i) (ii) (E, j) is Dedekind complete. (8.12) Proposition. If (E, <,(|.H) is a normed vector lattice, and [xi, I] egos), with 2 =2 Ixil in E, then iEI (a) “[Xi’ IJHl, = ”2””; A (b) if H.” is semi—continuous, then “[Xi’ 1]”1 = Hxnp. Proof: (a) “[xi, IJHl, = inf {Hunz Z Ixil j_u VJ e V(I)} = ieJ 60 inf {lluflz ziEIlXil : )2 :u g E}:“§“u. (b) In the semi—continuous case we have, using (8.11) (d), 11x1. :111 = sgpllzieJlXiln = sgp HzieJIinHp = “53p zieJIxiH\p =1\anp. (8.13) Corollary. If H.” is a continuous norm on B, then there exists one and only one fi—norm on.(%(E), relative toll.”. Proof. In the case of a continuous 1.“, We have H.uu = H." on 0 fi by (8.11) (e). Apply (8.12) and (8.2). It is clear, in view of (8.12), that if the norm]\.” is semi— continuous, then any property on (E, jJ fl.H) which implies p(.) = “.Hu on 8, will imply that ($(E) has a unique 2—norm. Such properties were investigated by R. Reichard, in [12]. In particular, the "projection property” in E is one such property. We next consider the question of continuity of the norm “'Hl on 2%(3) for a given continuous norm]].“ on E. It is clear that continuity of n.“ alone does not imply continuity of H.u1. For in View of (8.11), continuity of ].u1 entails that.Q:(E) be of countable type, which cannot be true if I is an uncountable set. l.”) is a normed vector lattice (8.14) Proposition. If (E, jJ with continuous]].H, then “.“1 is sequentially continuous on 0%(E). Proof. Suppose {[xén), I]} is a sequence in.g%(E), with 9(n) [x§n), I] + O. For each n e N there exists a B such that (n) _ y — 53p ZieJ xi (J a V(I)), and by (8.13) 19% = 11.90. 111.. 61 Note that for each n, the net {iiiJ Xin)}JgV(I) decreases in B to 0, since . (n) _ . (n) (n) 13f E x — 13f (Xiel x. - ZieJ x. “ (see (6.5)) iéJ i i i _ (n) (n) ' 9 53? 215.1 Xi = 0. Thus by continuity of H.“ for each n we have inf]]z x(n)1 = 0. U, J i¢J i Let 6 > 0. Then there exists J a V(I) such that “E. xii)“ . O iiJO i 6 ’ . (n) . (n) . . < 2. For each i s I, xi 1 0, hence, ZieJO xi + 0. By continuity of “.“, l2. Xgn)“ v 0. Thus there exists n e N such that for all lEJO i O x?“ < mlm n 2‘ nO’ “ZiEJO X(n) + E x<11)“ For every n.: n : “XiEJO i iafJO i .9“ . 0’ we thus have “901)“u x91 2] 13f “[xén), 1]“1 = 13f “9“)”u = o. < %-+ “ZiéJo xii)“ < 6. Therefore, Z'eJO “ZiéJO Given a norm-complete normed vector lattice (E, :3 “4D, one may ask whether fl%(E) is norm—complete for some l—norm “'“l' The following two results are partial answers to this question. Recall that the norm\\.“ on the vector lattice E is said to be monotone bounded if everyH.H—bounded subset of E is order bounded. We say that a norm “.1 on a vector lattice E is monotone complete if, for each sequence {xn} in B such that O i-Xn 1 and “xn“ :_M < m for all n e N, there exists an element x in B such that xn 1 x. (8.15) Proposition. If (E, jJ ““) is a Banach lattice with a monotone bounded norm [.“, then 1(B) is “.“ -complete. I 1 62 Proof. Let {[xgn), I]} be a “.“1—Cauchy sequence inIQ:(E). Then for each i e I, {X§n)} is a N.“-Cauchy sequence in E. Thus, by completeness of E, for each i e I there exists xi 5 B such that (n) “xi - xi“ + 0. Now consider the family [Xi’ I]. Let 6 > 0. Then there exists M e N such that n :_M implies m;§n)_ ng), fllll < 6, by the Cauchy xen> - .901“ < .. property. That is, n :_M implies s3pl,ZiEJ| 1 Observe that for each J a V(I), “ZieJlxi _ XEM)[H :}\ZiEJ|l%m xgn) _ xéM)‘“ = l%ml]ZiEJlx§n) — XEM)|\ jfi. Since \.I is monotone bounded, XEM) :J c V(I)} is norm bounded, this and Since the set{ZiEJlxi — set must also be order bounded. Further, (M) (M) 215.11in :ZieJlxi “ Xi I I ZiaJlxi l’ and [x§M), I] e “%(E). Therefore, [Xi’ I] e Y%(E). Finally, “[xgn), I] — [Xi’ I]“1 + 0, since for each J a V(I), n :_M implies “[Xi - Xgn)’ 1]“1 f.“[Xi ‘ XEM). I]“1 + “[XEM) — xgn), I]“1, and it has already been shown that each of the two terms on the right side of this last inequality are less than 6. Therefore,“[xi, I] — [x§n), IJ“1 .l + 0. Therefore, (2%(E),] l) is norm complete. (8.16) Proposition. If (E, 53“.“) is a normed vector lattice with semi—continuous, monotone complete norm “.“, then every l—norm “'“l on,Q§(E), relative to “.H, is monotone complete. Proof. It is clearly sufficient to prove that “.“1 is monotone (k), N] 1 and for all k a N, n complete. Suppose that O :_[x 63 “[xék), N]“ i M < w. Then for each n there exists an element xn of (k) B such that xn 1 xn by our hypothesis on E. Then [x;k), N] 1 [xn, N] in wN(E). It remains to show that [Xn’N] belongs to “g(E). _ r = r For each r e N, let Tr — Zn" 1lxn l ( Zn- 1 xn). Then 0 :_TP 1, and for every r,\\rr“ =\“Z;1(o—lim xn k)=)“ “0— lim Zn—1xf1k)“ = l|sup 2r x(k)“ = sup“zr x(k)“, sincel\. “ is semi— ~continuous. n=1xn n=—1Xn (k) nll But spp]]2:zl x = “[xék)’ N]“1 §_M, for every r. Therefore, by the monotone completeness of H.“ on B, there exists an element x a B such that Tr 1 x. Therefore, [Xn’ N] €.€§(E). CHAPTER III A KéTHE-TYPE DUALITY THEORY Section 9. Multiplication in Archimedean vector lattices. In Chapter III we consider only Archimedean vector lattices. It will be seen that Archimedean vector lattices all have a general multiplicative structure (however weak) which is sufficient to enable us to formulate a general theory of what we may reasonably call Kothe—family spaces, analogous to real Kothe sequences spaces. (9.1) Definition. Let E be a Dedekind o—complete vector lattice with a weak order unit 1. We say that E admits a_multiplication operation relative :2_l_if, corresponding to certain pairs x, y of elements of B, there exists a "product" xy, which obeys the conditions: (M1) for every x e E, x] exists and equals x; (M2) if xy exists, then yx exists and equals xy; (M3) if xy, (xy)z and yz all exists, then x(yz) exists and equals (xy)z; (Mu) if xy and xz exists, then x(y + 2) exists and equals XY + yZ§ (M5) if xy exists and a e R, then (ax)y exists and equals a(xy); (M6) if x, y :_0 and xy exists, then xy :_0; (M7) if xy exists, [x'l §_lx| and Iy'l §_|yl, then x'y' exists; (M8) x.L y if and only if xy exists and equals 0. 64 65 (9.2) Observations. For any multiplication operation relative to a weak order unit 1 on a Dedekind o-complete vector lattice E, the following additional properties hold: (M9) if O :_x :_y, z :_0 and yz exists, then 0 :_xz §_yz; (M10) if xy exists, then (xy)+ = X+y+ + x'y‘. (xy)_ = x+y_ + x-y+’ IXYI = |X| lyl; (M11) if 1 is a strong order unit, then B is closed under this multiplication. In 19H0 B. Z. Vulikh ([19] and [20]) demonstrated a method for constructing, within an arbitrary Dedekind complete vector lattice with weak order unit 1, a product xy satisfying (M1) through (M8). Vulikh's method has been made accessible to the English—reading mathematician in the papers ([13] and [14]) of Rice. We give here a brief outline of this approach. Let E be a Dedekind complete vector lattice with weak order unit 1. Define the collection of "unitary” elements of E by u(E, 1) = {e e E: e A (1-e) = 0}. Every element x e E+ is the supremum of all the scalar multiples of unitary elements that lie below it. The multiplication is then defined as follows: (a) for e, e' e u(E, 1), ee' = e A e'; (b) for x, y :_0, xy = sup {a8ee‘: 0 :_ae :_x, 0 :_Be' §_y, e,e' e u(E, 1), a,B e R+} if this supremum exists (otherwise xy is not defined); 66 (c) for x, y e E, xy = x+y+ — x+y_ — x_y+ + x—y_ if all the products on the right side exist (otherwise xy is not defined). (9.3) Proposition. (Uniqueness of the product; Rice [13], theorem 5.1). In a Dedekind complete vector lattice E with weak order unit 1, if x * y denotes another multiplication on E satis— fying (M1) — (M8), then x * y exists if and only if xy exists; more— over x * y = xy. (9.9) Proposition. (Rice [13], lemma 5.2 and theorem 5.3): Let 1, 1' be two units in a Dedekind complete vector lattice E. Denote the product relative to 1by xy and the product relative to 1' by x * y. Then u(E, 1') = {1'ez e e u(E, 1)}, and for every x, y e E, if xy and x * y both exist, then X *y : (xy) *1. The development of the representation theory, outlined in the appendix, provided another proof that every Dedekind complete vector lattice admits a multiplication operation, and made possible further elucidation of the nature and properties of such multiplications. Let Q denote an arbitrary quasi—extremal compactum, as defined in the appendix. It can be shown that the usual (pointwise) multiplication of functions is an operation on Cw(Q) satisfying (M1) through (M8), relative to the usual 1—function: 1(t) = 1 for all t s Q. Moreover, Cw(Q) is closed under this operation; Cm(Q) is a commutative ring with identity. The same remarks hold for C(Q). 67 Using the representation theory, and this multiplication in Cm(Q), we have immediately the following proposition: (9.5) Proposition. Every Dedekind c—complete vector lattice E with a weak order unit 1 admits a multiplication operation relative to 1. If, in addition, 1 is a strong order unit, then B is closed under this multiplication. (9.6) Multiplication in_Archimedean vector lattices. If E is an arbitrary Archimedean vector lattice, we may embed E in its universal completion E#, and thus by picking an arbitrary weak order unit 1 in E# we obtain, as a consequence of (9.5), a ”product” defined on certain pairs of elements of E, which has the properties (M2) through (M6) and (M8). Of course, if E already contains a weak order unit 1, then 1 is also a weak order unit for E#; if we take the multiplication relative to this 1, we will then have property (M1) in E. From (M7) we see that if E is closed under the multiplication induced from E#, then so is 6. Proposition (9.5) showed how a change of unit elements affects the definition of the product of two elements. The following propo— sition will show that, given two different universal completions of E (necessarily isomorphic), the unit elements may be chosen in such a way as to induce the same multiplication in E. (9.7) Proposition. Suppose Bf and E: are universally complete # vector lattices, and suppose w: E1 + E: is an isomorphism. Let 11 denote a weak order unit in Bf; then 12 = w(11) is a weak order unit 68 # # in E2. Denote the products in E1 and E2, relative to 11 and 12 respectively, by xy and x*y, respectively. Then W(xy) = W(x) "‘ MY). 3322:. In View of (Mid it suffices to prove this identity for # x, y :_0. Note that u(E:, 12) = {f e E2: f A (12-f) = 0} = {¢(e): e a Bf, e A (1l—e) = O} = w[u(Ef, 11)]. Now xy = sup {a8(e A e'): O §_ae :_x, O j_Be' §_y; e,e' e u(Ef,11); a,B e R+}. Since w is an isomorphism, we thus have 111(xy) = sup {a8(¢(e)/\ w(e')): O _<_ae :x, 0 :Be'1y, etc.} SUP {aBNKe} "" W(e')):0 E. 0L(11J(e)) : 111(X), 0 :B(¢(e'))_<_1J)(y), etc.} = sup {a8(f:':f1): o :u‘f i 111W), 0 _<_pf' :My); f,f' e: u(E:.12); a,B e R+}. = ¢(x) =‘= My)- By theorem 3.1 of Rice [13], and observing that E E E, we have the following proposition. (9.8) Proposition. If E is an Archimedean vector lattice with weak order unit 1, then for every x e E+ and every positive ingeger n, there is a unique positive nth root of x (denoted xi) in E. Later in this chapter we shall wish to raise elements of our vector lattice to arbitrary positive real powers. If E is an Archi- medean vector lattice, x e E+ and p is a positive real number, the symbol xp is ambiguous, in the sense that it depends upon the choice of multiplications taken in E#. Nevertheless, corresponding to each representation of E# as a space Cw(Q) and each choice of weak order unit 1 in E, the symbol xp is well-defined by the representation theory. 69 (9.9) Proposition. If E is an Archimedean vector lattice closed under the multiplication induced on E by multiplication relative to some unit 1 in E#, then for every x g E+ and every positive real number p, xp 6 E. ngxzfi. Let n denote the least positive integer greater than or equal to p. Representing x as an element of Cw(Q), with 1 represented by the function 1(t) = t, for all t e Q, we see that xp — an Xn—l’ /\ since x(t) < 1 im lies xp(t) < xn_1(t), while x(t) > 1 im lies xp(t) P _, __ P :_xn(t). But E is closed under the multiplication, and E is an ideal in E#. Therefore, xn'V xn_1 e E; hence, xp 5 E. The following proposition and corollary may be found in Rice [13] as theorem 10.3 and corollary 10.3.1. (9.10) Proposition. (order continuity of the product) Let E be a universally complete vector lattice, and let {Xa}’ {ya} be two nets in E indexed by the same directed set. If xa—Jg x and yu—Ssy in E, then xaya—ngy. (9.11) Corollary. Suppose {Xa}’ {ya} are two nets in a 1 Dedekind complete vector lattice E, indexed by the same directed set. If xa—3>x, ya.fgy, xaya g E for all a, and there exists 2 g E such that lxaya| :_z for all a, then the product xy exists in E and x —2> aye xy. Many of the spaces encountered in applications of vector lattice theory have a natural multiplication defined on them already, without 70 reference to their universal completion. Many of these spaces are not Dedekind complete, or do not contain a unit element. For example, the space C[0,1] and the sequence space 0 are vector lattices which are not Dedekind complete, while the space ¢ of finite sequences does not have a weak order unit. In each of these spaces the pointwise (or coordinatewise) product provides a natural multiplication operation. In cases such as these it is natural to ask whether the given multiplication coincides with that obtained by embedding the vector lattice in its universal completion and choosing some appropriate weak order unit. Suppose E is a vector lattice of real—valued functions f: X 9 R defined on some abstract set X, with the vector lattice structure induced from wx(R), and with the pointwise multiplication operation. Assume further that for every 5 s X, there exists f e E such that f(E) # 0. Then E# = wX(R). Let 1 denote the constant function, 1(5) = 1 for all E e X. (We do not assume 1 a E.) Then the point— wise multiplication operation on E# satisfies (M1) through (M8), and hence by (9.3) is the pply_such operation that can be defined # on E with1 as unit. Therefore in this case, the multiplication on E does coincide with multiplication induced from E#, relative to this unit 1. Section 10. The spaces Q§(E, 1), 1 :_p §_w. We continue to assume that E is an Archimedean vector lattice, # embedded in its universal completion E . The vector lattice E# has a weak order unit 1, which of course is not unique. 71 (10.1) Definition. Given an Archimedean vector lattice E, and an arbitrary weak order unit 1 in E#, for each 1 < p < m, we define P - . P gi(E, 1) - {[xi, I] e wI(E). 3 u e B such that V J e V(I), EieJlxil .1 u}. Here the powers lxilp are taken in E#, relative to 1. Thus neither lxilp nor 2 p can be expected to belong to E. But if ieJlxil [Xi, I] e 2§(E, 1), then, since E is an ideal in E#, all the powers Ixilp and all the sums E P belong to E. Thus ieJlxil p _ . P 1 A Q I(E, 1) — {[xi, I] e wI(E). [Xi’ I] e QI(E)}. For the sake of uniformity of notation, we define 9%(E, 1) = Q%(E), and 21(3, 1) = 11(E) = mI(E). .r (10.2) Example. Let E = m; then E# = 3. Consider the family [en, N] e wN(E), where en = (0,0,...,1,0,...), as defined in the examples following (1.1). If we take 1 = (1,1,...,1,...) then [en,N] e,pfi(E, 1) for every 1 :_p :_m. 0n the other hand, suppose we take 1' = (1,1/2,1/3,...). Then 1' is a weak order unit in E#. For each n e N, fien e u(E, 1‘). Relative to 1', (fien)p = fien, and lenlp = P 1 _ -1 . p . n (hen)p - np en. Thus relative to 1', E lenl t E if p > 1. neN Therefore, [en, N] é Qfi(E, P) if 1 < p < w. This example shows that g§(E, 1) does indeed depend upon the choice of unit taken in E#. This example exhibits another phenomenon which will be of some significance later. E is closed under multiplication relative to 1, but not under multiplication relative to 1'. Let us denote the product of two elements x, y s E#, relative to 1', by x * y. Then 1 * 1 = sup {aBee': O §_ae :_1, 0 :_Be' :_1, e, e' e u(E, 1')} :_sup {n2(%en) * (%e ): n = 1, 2, ...} = sup {nenz n = 1, 2, ...}. I'l 72 Hence, 1 u 1 ¢ m = E. Thus E (= E) is not closed under multiplication relative to 1'. The following proposition shows that definition (10.1) is neither vacuous nor trivial. (10.3) Proposition. If I is an infinite set, and 1 :_p < q :_w, then (a) [g(E, 1) — ¢I(E) is nonempty; (b) Q%(E, 1) —,Q§(E, 1) is nonempty; (c) if E is closed under multiplication, then ¢I(E) E £§(E’ 1); (d) if 1 a fi, then1§(3,1)ep°l°(s); (e) if 1 e E and E is closed under multiplication, then (1)103); 95(3, 1) 912%(E, 1) 22:03). (f) if ¢I(E)EQ§(E, 1) for some 2 :p < 0°, then E is closed under multiplication. 3322:. Since I is an infinite set we may assume, without loss of generality, that N §=I. (a) The result is clear if p = 00; hence, we assume p < m. Pick an arbitrary u > O in E. Since E is order—dense in E#, we may 1/p p pick an x e B such that 0 < x < u and consequently 0 < x < u. Define [Xi’ I] e wI(E) by 11/(2p) (% x if i e N, l 0 if i d N. Then [Xi’ I] e Y§(E, 1) since for every J 6 V(I), ZieJlxilp = — -2 ZneJnNn 2 xp i-(XneNn ) u e E. Clearly, [xi I] i ¢I(E). (b) First suppose q < m. Pick an arbitrary u e E and x e E with 0 < xq < u as in (a) above. Define [Xi’ I] e wI(E) by 73 1 1/p (I) x if i e N, 0 if i ¢ N. _ w 1 For every J 5 V(I), EieJlxilq = ZnsJ N n q/pxq i <2n=1 .97Pj u e E, n P _ eJlxil - since q > p. Thus [Xi’ I] e 2%(E, 1). HOWever, ii 2 [n-lxpl = E n—lB xp which by the Archimedean property in neJnN neJnN ’ E, cannot be bounded by an element of E. Thus [Xi’ I] ¢ Q§(E, 1). Now let q = m. Pick an arbitrary u > 0 in E and define u if i s N, o ifi¢N. m P - P Then [yi, I] e 2I(E), but [yi, I] é (1(E, 1) Since the sums Zielei1 , being integral multiples of up, cannot be bounded in E, again by the Archimedean property. (c) Suppose E is closed under multiplication. If [Xi’ I] e ¢I(E), let JO = {i e I: xi # 01; then 2. ieJIXilp i-EieJ Ixilp, which belongs O to E by (9.9). Thus [Xi’ I] E 22(E, 1). (d) Let 1 e E and [Xi’ I] e QECE, 1); say EieJ 1x]p < u e E, l ._ for all J a V(I). By representing xi and u as functions in Cm(Q), we see that lxilp :_u Vi implies 1Xil :_u V 1 e E 1i 5 I. Hence, [Xi’ I] 61116). (e) Suppose 1 e E and E is closed under multiplication. Let [Xi’ I] a Q¥(E, 1). In View of (d) we assume q < w. Then there exist u, v e E such that for every J e V(I), X lxilp :_u, and for every isJ i e I, Ixi] :_v. By (9.9) vq_p a E. We thus have, for any J a V(I), 2 1Xin =2 Ixi1q_p 1inp di_pIi€J1Xi1p :vq'p u 6 fi. ieJ ieJ 71+ Therefore, [Xi’ I] e 9%(E, 1). (f) Suppose ¢I(E) E Q§(E) for some 2 :_p < w. Since xy = %;Ex+y)2 — (x—y)21 we need only show that E is closed under squaring. Let x e E, x # 0; then there exists x s E such that le < x. Pick an arbitrary i0 8 I and define x if i = i0, 0 if i ¢ i0. Then by hypothesis, [Xi’ I] E,Q§(E, 1), so there exists u e E such that lxilp :_u for all i e I. From the representation theory we see that x2 < xp\J x. Thus “2 __|“[2 < x2 :_ xp v x :_ u V x e E. DC /\ Thus 22 e E, since E is an ideal in E#. (lO.M) Corollary. If 1 is a strong order unit for E or E, and 1 :p < q :°°. then ¢I(E)<_:_QII’(E, 1);??(3, 1) 912?}3). Proof. If 1 is a strong order unit, then by (M7), E is closed under multiplication. (10.5) Examples. It is not hard to find spaces E which satisfy the condition <+> ¢I. Consider R, Rn, the space RI of all real—valued functions on an abstract set I, the space C(X) of all bounded, continuous, real—valued functions on a topological space X, the space m of all bounded sequences, the space c of all convergent sequences, and the space of eventually constant sequences. All these spaces satisfy condition (e) of prop— osition (10.3); thus they satisfy (1). 75 On the other hand, many common spaces do not satisfy condition (T). with termwise multiplication, and take 1 to be # Consider E = £1, the usual constant sequence 1 = {1,1,1,...} in E (= m). For each n e N define the sequence S = {x 1 with n nk if k = n, X l = k nk 0 if k ¥ n. If q > 1, then [Sn’ N] e Qfi(E, 1), since the series 2::1 (iqq converges l k there is no 3 a £1 such that s 3 sn Vn. Thus [8,1, N] ¢Q;(E). We 1 w and hence, [82, N] a PN(E)~ But the series Zk=1 diverges; hence, therefore have in this case, 93$, 1) $1,305, 1), for all q > 1. Now consider E = L1[O,1], with the linear operations and order defined pointwise, and with the usual constant function 1. Recall that L1[O,1] :2 Lp[0,1] ; Lq[0,1] ; Lm[0,1], for 1 i p < q < .._ Thus there exists a function f e E+ such that f e Lp[0,1] — Lq[0,1]. Pick any i0 5 I and define [fi’ I] e wI(E) by setting f if i = i0, 0 if i # i0. Since fp e E but fq is not bounded above by any element of E, [fi’ I] a 9§(E, 1) and [fi’ I] e ¢I(E), but f ¢ Q%(E, 1). For this example, therefore, ¢I(E) $ 1%(E, 1), and 2§¢1§ :2 <51in +zi€JIyi1 >. (10.8) Proposition. If [x§a), I]—2>[yi, I] in f§(E, 1), then (a) p p . A EieIlXi I —£§ziellyil in E' Proof. Since [ x(c) P, I] and [ly. p, I] belong to 1(E), we ——-—- 1 1 I may apply (7.3) to yield the desired conclusion. (10.9) Proposition. (a) If E is Dedekind complete, so is Q§(E, 1). (b) Q§(E, 1) is order—dense in,9§(E, 1); YE?;;:\13 is the ideal generated by,p§(E, 1) in [g(E, 1). 2322:. (a) As shown in (3.10), if E is Dedekind complete, then mI(E) is Dedekind complete. By (10.7) £§(E, 1) is an ideal in wI(E); hence, it must be Dedekind complete. (b) The proof of (3.12) (b) remains valid, with AI(E) = Q§(E, 1) and F = E, showing that R§(E, 1) is order dense in fl§(E, 1). The ideal generated by Q§(E, 1) majorizes 25(E, 1) in the sense of defini— tion (4.1) (c); hence, by (4.1), it must equal]T§(E:\1). There are spaces which satisfy the condition (t) of (10.5), but which fail to satisfy the criteria of (10.3) (e). We shall see as a result of (10.12) that among such spaces are the sequence space o, the space of all real-valued step functions on R, and the space C (R) of real-valued continuous functions on R with compact support. We now consider a property which will guarantee condition (t) in a vector lattice. 78 (10.10) Definition. A subset {ea: c e I} of a vector lattice E is said to be a generalized strong order unit for E if (a) ea A e = 0 whenever a ¢ 8, and B (b) iven an x e E+, there exists T E T and 0 < A < 1 8 Y such that sup {ea: c e T} e E and Ax :_sup {ea: c e T}. (10.11) Proposition. Let E be an Archimedean vector lattice # with a generalized strong order unit {ea: c s I}. Then in E the element 1 = sup {ea: c e F} is a weak order unit, and for any T §.P, #,1). sup {ea: a e T} e u(E Proof. First note that 1 e E# since every set of pairwise # disjoint elements of E has a supremum. # # Now let x > 0 in E . There exists an element x e E such that 0 < x < x#, and there exists a subset T E P such that 0 < XXV: # sup {ea: c s T} :_1. Hence, 1 A x :_1 A x :_1 A Ax = Ax > 0. Thus # 1 is a weak order unit in E . e ) Let a e P. Then 1 — ea = (288? B — ea = iB¥a eB = sup {e8: 8 ¢ c}. Thus ed A (1 — ea) = ed A (sup {e8: 8 # dB = sup {(ea A e8): # 8 ¥ a} = 0; thus, ed a u(E , 1). Now u(E#, 1) is a complete Boolean algebra (see Vulikh [18], 1‘, 1). theorem IV.2.1). Hence, for every T E T, sup {ea: c e T} e u(E (10.12) Proposition. If E is an Archimedean vector lattice with generalized strong order unit {ea: c e F}, and if 1 = sup {eaz a e F} . # in E , then (a) E is closed under multiplication relative to 1; (b) ¢I(E)911)(B’ 1)§9%(E, 1)§2:(E), for any 1 :p < q i°°~ 79 Proof. (a) Let x,y e E. Then x _<__A1f1 and y _f_A2f2 for some f1 = sup {ea: c 6 T1} 8 E, f2 and o < A2 < 1. Then xy :_(A1f1)(A2f2) = (A1A2)(f1f2) = A1A2(f1/\ f2) = sup {ea: c a T2} 8 E, 0 < A1 < 1, e E, so that xy 6 E. (b) That ¢I(E) ;;Q§(E, 1) follows from (10.3)(c). Let [xi, I] e [g(E, 1). Then there exists u e E such that E. p :_u for llein every J a V(I). There exists f = sup {ea: c e T} e E and O < A < 1 such that u :_Af. Then for all i e I, Ixilp j_Af. Note that fl/p = f # since f e u(E , 1). Thus for all i E 1; lxil §_A1/pf e E. Therefore, [Xi’ I] e Q:(E). Therefore, 9§(E, 1) E Q:(E)- Again let [Xi’ I] e Q§(E, 1). Then there exist u,v e E such that for all J e V(I), Z lxilp :_u and for all i e J, 1X11 < v. ieJ ._ Now v :_A1 for some A > 0. Thus for all J c V(I)’ ZieJlxilq = Zi€J1Xi1q—p1xilp f. Vq-p ZiEJIXilp §_ (Aq-p1)u = Aq_pu e E. Therefore, [xi, I] e 9%(E, 1). We have proved 2§(E, 1)<; Q%(E, 1). Section 11. p—norms on the spaces Q§(E, 1). Throughout this section we assume that (E, EJH.H) is a normed vector lattice. It is natural to question whether it is then possible to define a norm on 05(E, 1) which behaves like the usual norm “x“ = (2::1ixan51/P for x = {xn} in the familiar sequence space 2p. In section 9 we saw that there is possibly more than one R—norm on Q%(E) relative to “.H; similarly, we shall have to allow for more than one suitable p—norm on £§(E, 1), relative to H-U. 80 (11.1) Definition: If (E, jJ “.H) is a normed vector lattice, then a p-norm on.2§(E,1), relative to H.", is any monotone norm ”'“P on Q§(E, 1) such that if 2 = Z lxilp in E, then for all u, v e E isI such that u 5_& :_v, uuul/P :11txi. 1111p : “viii/P. Equivalently, o(&)1/P E-HEXi' fl”p 5_u&Hi/P (see (8.11)). In View of the last inequality, one would expect the presence of a p—norm on.Q§(E, 1) to be closely related to the presence of a monotone extension “.H* of the norm H.” onto all of E. Indeed it turns out that to each such extension there corresponds, in a natural way, a p—norm “.H*p on ]§(E, 1). The following definition is motivated by (8.12). Suppose “.H* is a monotone extension of N.“ to E. For each [xi, I] e p(E 1) there exists a uni ue element I I q A A x = ZieIIXiIP e E. If we define, for every [Xi’ I] e Q§(E, 1), 1h, :11., = 1213/13. then , as we shall show in corollary (11.u), “°“*p is a p-norm on Q§(E, 1). This norm will be called the "p—norm associated with H.1*". Note that if we define “[Xi’ I]u*1 = 121*, where x = Ziellxil’ . 1 then “'H*1 is an Q—norm on.€I(E). If E is Dedekind complete, then we write ”'up instead of “11*P. (11.2) Proposition (Holder's Inequality). Let p, q be positive real numbers with %-+ %-= 1, and suppose n.“* is a monotone extension of H.“ to E. If [Xi’ I] e (2(E, 1) and [yi, I] e Q%(E, 1) then 81 [Xiyi’ I] e 2%(E) and Htxiyi. :11.1 :l11xi. 111., 11s,. nihq 3322:. That [Xiyi’ I] a 2%(E) was proved in (10.6). Let x = [Xi’ I] and y = [yi, I]. Assume x ¢ 0 and y # 0, since otherwise the desired inequality is trivially true. Now represent each xi and yi . __ _ ' 1 as functions Xi’ yi E Cm(Q). For each t a Q, let A = 53 Sat) p “yu*p and B = ECUF “X”*p and apply the lemma mentioned in the proof of (10.6). We thus obtain lxiiY1 l 1Xi1 P 1 lyil q 1X“*p Hyu*q p H;W:; + 3' 1T1;; (Vial). Then, letting x = Ziellxi1p and 9 = E. 151lyi1q, we have i- IX-Y-l ieJ i i 1 p 1 q I | + —----- Iieleil : . 1x. 1wa 11y11*q p(ux1|*P)P ”J l qq 1 p121. EieJIXi' + quyu* Eielei1 2 “ . < A + A _' PHX1* qfly1* Thus, taking the norm H.H* of both sides, We obtain 11eJlxyl*_l11 l1X1).p 113711.».q 1’ from which the desired inequality follows immediately. In the proof of the following proposition, the argument is anal— ogous to the usual derivation of Minkowski's inequality from Holder's inequality (see Goffman and Pedrick [2], pp. 4,5). 82 (11.3) Proposition (Minkowski's inequality). Suppose H.u* is a monotone extension of “-“ to E, and let 1 < p < W. Then for all =[Xi911and y = [y1’ I]ianI)(E,1), + < 1x +1 . 1x y1*p __ 1 1*p 1yn*p . -1 Proof. For each i e I, 1xi + yilP §_1xi + yilp lxil + 1xi + -1 -1 . yilp lyil. Note that [1xi + yilp , I] 612%(3, 1), Since (1xi + yi1p_1)q = 1x, + yilp. Thus by the order H6lder inequality (10.6), both [1xi + yi1p—11xi1, I] and [1x. + yilp—llyil, I] belong l to 1%(fi), and in E we have 1 1 zmlxiwilp s Zielflx +y IP 1x11 + Emox +y lp ly l1 Taking the norm H.H* of both sides, and applying Holder's inequality (11.2), we obtain the inequality “2 p p-1 ieIIXi + yil 1* 5-1[1Xi + yi1 ’ 1111*q11[xi. I111...p + 11[lxi + yilp‘l. 1111.q11[yi. 1111,13. That is. 1x + qup :.1P'1u.q. and we thus have xx+n.ph+yfi§ sP1 = 11£xi+ys111§ = 1 1 ”Ixi. 111: + ”[yi, I11? = ”(x /p)p1 + “(y /p)p1 = HxH + 1Y1- Therefore,l1.“ is additive on E+' (11.8) Proposition (Solov'ev [17], theorem 8). If 1.“ is additive on E+, then there exists a monotone extension 1.1* of H.” to E which . .. A+ is additive on E . 85 . . . p (11.9) Corollary. “'"*p is then a p-additive p—norm on QI(E, 1). Section 12. Kothe X—dual spaces Throughout section 12 We continue to assume that E is an Archi— medean vector lattice, embedded in its universal completion E# = Cm(Q), and that 1 is a fixed weak order unit in E#. We may assume that 1 is represented by the constant function, 1(t) = t for all t a Q. We denote the product (relative to 1) of two elements x, y e E# by xy. (12.1) Definition. If AI(E) is an arbitrary subset of wI(E), we define its Kothe X—dual, relative £9_l) to be the set [AI(E)]1 = {[yi, I] e wI(E): V[xi, I] e AI(E), 3 u e E such that ZisJ1Xiyi1 §_u NJ 5 V(I)}. Since E is an ideal we have X _ . 1 A [AI(E)11 — {[yi, I16 wI(E). [Xiyia 1] 6 91(E) \1[xi, I] e AI(E)}. Observe that from (10.6) it follows that if p and q are finite real numbers such that %-+ %-= 1, then Q§(E, 1) E [1%(E, 1)]?. To simplify notation, [AI(E)]1 will often be written A;(E), if the element 1 is understood to be fixed. It must be emphasized that the space [AI(E)]1 depends upon the element 1. In (10.2) we have seen an example of a vector lattice E containing weak order units 1,1' such that E is closed under multiplication relative to 1, but not under multiplication relative to 1'. Using proposition (12.5) below, we see that [1%(m)1¥ ¢ [£%(m)]¥v' We employ another notational simplification: we often write A:X(E) or [11(3)]fx instead of [[AI(E)]f 1. (12.2) Proposition. If AI(E) is an arbitrary subset of wI(E), then A§(E) is an ideal in wI(E). Proof. Let [yi, I], [zi, I] e A§(E) and [Xi’ I] e AI(E). There exist u,v e B such that for every J a V(I), Z Ixiyil §_u and isJ EieJlXizil §_v. By (M4) we may use the distributive law, and thus I [Xi’ I] + [yi, I] e A§(E). That A§(E) is closed under scalar multi— iEJ1xi(yi + zi)1 :_Ei€J1xiyil + ZieJIXiZil §_u + v. Therefore, plication is obvious. Let [vi, I] c wI(E), with 1[vi, I]1 §_1[yi, I]1. Then for all J E V(I)’ ZieJlXiVil = zieJlXiHVil i-EieJlxillyil = ZieJ1Xiyi| f-u' Therefore, A§(E) is an ideal. (12.3) Definition. A subset AI(E) of wI(E) is X—perfect (relative . _ xx to 1) If 11(3) — [AI(E)]] . (12.4) Proposition. If AI(E) is an arbitrary subset of wI(E), then (a) u§(E) Q A;(E) whenever AI(E) E uI(E); xx, (b) AI(E) 5 AI (E), (c) A§(E) is X—perfect; (d) A§X(E) is the smallest X—perfect subset of wI(E) containing AI(E); (e) if AI(E) is X—perfect, then AI(E) is an ideal in wI(E); . X 2 (f) If A (E) = [A (3)] , then A (meg (E, 1). I I 1 I I The proof of (12.4) follows the standard argument. See Kathe [5], §3o. 87 (12.5) Proposition. If I is an infinite set, the following are equivalent: X - Q (a) [¢I(E)]] — wI(E), X (b) 11(11):; [1111(3)]1, (c) E is closed under multiplication (relative to 1); m 1 X (d) 91(E)E[1ZI(E)11» 1 m X (e) QI(E) s [11(3)],. Proof. We shall prove (c)=%(a) =>(b) :(c)=>(d)©(e)=>(c). Clearly, (c) E) (a). That (a) $(b) and (d)=>(e) are seen immediately by using (13.3) — (a), (b). That (b)=%>(c) and (e)=%>(c) are also seen readily, by noting that either (b) or (e) will imply that ¢I(E) ELQ§(E, 1), which, by (10.3) (f), implies (c). Thus it remains only to prove (c)=9(d). Assume (c) and let [xi, I] e Q:(E), [yi, I] e 1%(E). There exist u, w e E such that lxil ion for all i e I and Zieleil :_w for all J e V(I), and hence ZieJ1Xiyi| :ZiEJ u1yil = u Zieleil 3 u w e E. Thus [xi, I] s [11(3)]? (12.6) Proposition. If 1 e I, then (a) [wI(E)])1(<;.mI(E); (b) [Qimfl’fgfimt P_ro_of_. (a) If1eE, then by (10.3) - (d), Tim, 1) mI(E). But [111(5)]? 9.12%(3, 1), (b) Since 1 s E, 1 :_e for some e e E. Let [Xi’ I] e [1:(E)]?. The family [yi, I] defined by yi = e, for i e I, belongs to Q:(E); 88 thus there exists u e E such that 2. ieJIXiyil :_u for all J a V(I). Then for all J e V(I), 1X118 = ZieJlxiyiI iu' EieJIXiI i-EisJ Therefore, [Xi’ I] e fl%(E). By (12.5) (e) and (12.6) we have the following result. (12.7) Corollary. If 1 e E and E is closed under multipli- cation relative to 1, then [Q:(E)]¥ = 1%(E). (Thus, 2:(E) is X-perfect, relative to 1.) (12.8) Proposition. If E has a countable exhausting set (i.e., a set {en: n = 1,2,...} such that for every x e E, 1x1 :_en for some n e N), then [wI(E)]1(C-:¢I(E)' 2393:. Let {enz n = 1,2,...} be a countable exhausting set in E+. Let [yi, I] e [wI(E)]? and Jo = {ieI: yi # 0}. We shall show that JO is finite. Suppose to the contrary that JO is infinite; without loss of generality, assume N E J By the Archimedean prop— 0' erty in E#, for each n e N there exists A e R+ with 1A y21 X e . n n n - n Define [Xi’ I] e wI(E) by Aiyi if i e N, l o if i t N. Let u e E+. Then there exists n0 5 N such that u :_en . If J 1 is any finite subset of N containing no, we have zneJlxnynl EneJlAnyil f_en0; hence, ZneJlxnynl f_u. Therefore, [yi, I] i 1 X [wI(E)]'l. 1 (12.9) Example of a vector lattice E (in this case, Dedekind 89 complete and with 1 e E) such that [§§(E)]?.QEQE(E). Let E = 21, E# = m with the usual vector operations, order and unit. Let [en, N] e wN(E) be given by en = (0,0,...,1,0,...) with 1 in the nth position only. co 1 X Clearly, [en, N] t 1N(E)' But [en, N] e [[N(E)]1. For suppose [xn, N] e Q§(E); say 2 §_u for all J c V(N). Then EiSJIX'e 1 ieJlxil i i f. ZieJIXil : u for all J c V(N). Therefore, [2N(E)]1< $11110“)- (12.10) Proposition. If 1 is a strong order unit for E, and if 1 < p,q < w with %-+ %-= 1, then X [?§(E,1)]1 = 1%(3,1). Thus each 1¥(E, 1) is X—perfect. The only self X—dual subspace of wI(E) is 1§ 1, there exists n0 5 N such that no(p—1) > 2. Then there exists a sequence {Jk} of pairwise disjoint subsets of N such that for each k e N, n0+1 E:118Jklyl “1 =Mk i k 1. For each k e N, let Ak = sup {A: O < A < 1, XMk :_kn0+1 1}. Then n +1 1/ n +1 . AkMk :_k 0 1, but Ak PMk # k 0 1. Define [xi, I] e wI(E) by 1/p ~n0 q—1 . . X 1k k lyil if i e Jk, i 0 if i i U:_1Jk. Then [xi , I] e 2P(E,1), since for each k s N, 2~ xilp iEJle -noP (q-1)p _ -nop q : -nop zieJkAk k 1Yi| ' Ak k ZieJklyil k AkMk k-nop kn0+1 1 = k-no(p-1) + 1 1 §_ 1 since ~nO(p-1) + 1 < —2 + 1 Z Al/p k’nolyilq = = ~1. On the other hand, 2. . ieJk k ieJk lxiyil k—nokl/p Mk % k—no k = k1. Thus there exists no k e N such that for all J a V(I), zieJ This is a contradiction. Therefore, [Q§(E, 1)]?<;.Q%(E, 1). Com— bining this result with (10.6), we see that the desired equality holds. (d) Finally, suppose AI(E) = A§(E). By (12.4) (f), AI(E) g; lxiyil §_k 1. That is, [yi, I] é E(§(E, 1)]f. 1 1 \ 91 2(E, 1). Hence, A (E) = AX(E) :2 L?2(E, 1)]X = Q2(E, 1). Therefore, I I I I 1 I _ 2 AI(E) — 21(3, 1). Section 13. Linear mappings from AI(E) into E. (13.1) Definition. Given vector lattices E and F, a linear mapping f: E + F is said to be (a) positive if X :_0 implies f(x).: 0; (b) strictly positive if x > 0 implies f(x) > O; (c) ord§£_bounded if f is bounded on each order interval of E; (d) sequentially order continuous if f is order bounded and if for sequences {Xn} in E, xn75%o implies f(xn)-£L0; (e) order continuous if f is order bounded and if for arbitrary nets {Xa} in E, xa—3.0 implies f(xa)-3>O. It is clear that strictly positive épositive éorder-bounded. Note that any vector lattice isomorphism is both strictly positive and order continuous. If E and F are arbitrary vector lattices, we let L+(E, F) denote the family of all positive linear maps from E to F. Then L+(E, F) is a cone (not necessarily generating) in the vector space L(E, F) of all linear maps from E into F. Let Lb(E, F) denote the family of all order bounded linear maps from E to F; Lb(E, F) is a linear sub— space of L(E, F), partially ordered by the cone L+(E, F). Suppose that F is Dedekind complete. Then Lb(E, F) is a vector lattice with the order induced by the cone L+(E, F); in fact, Lb(E, F) is Dedekind complete (see Vulikh [18], theorem VIII. 2.1). When F is 92 Dedekind complete, a necessary and sufficient condition for an additive map f: E + F to be order—bounded is that f = g1 — g2, for some g1, g2 e L+(E, F) (Vulikh [18], theorem VIII. 2.2). Moreover, the spaces LO(E, F) and LSO(E, F) of all order continuous and all sequentially order continuous maps f s L(E, F), respectively, are order—closed ideals (bands) in Lb(E, F) (Vulikh [18], theorems VIII. 3.3 and VIII. 4.3). For an arbitrary vector lattice E, any Banach o—limit g 6 8£(E) is an example of a positive (hence order bounded) linear mapping g: m(E) + E, as shown in (5.5). (13.2) Example. For the vector lattice E = m there exists pg sequentially order continuous g E $i(E). To see this let g e $1(E). . (k) . (k) _ m Define x e mN(E) by setting x — {xkn}n=1 where 0 if 1 :_n :_k—1, X = kn 1ifn_>_k. Then x(k) + 0; but for each k, g(x(k)) = 1%m xék) = 1, since x(k) e C(E). Therefore, g.is not sequentially order continuous. (13.3) Proposition. If order convergence of sequences in E implies uniform convergence, then every g c @ilE) is sequentially order continuous. 3322:. Let xn.E;o in E. Then there exists u e E so that for 3 every 6 > 0 there exists n0 8 N such that n :_no implies Ixnl §_6u. 9 Thus n :_no implies |g(xn)| fi_g(1xnl) :_5g(u). Therefore, g(xn)-l;0 in E, which implies g(xn)-£;O in E. 93 . 1 A . (13.4) Example. The linear map f: £I(E) + E given by f([xi, I]) = ZieI xi is order continuous, as shown in (7.3). (13.5) Proposition. Let AI(E) be a vector sublattice of wI(E), and y = [yi, I] e [AI(E)]1' Define the map y*: AI(E) + E by y (x) = EieI Xiyi’ for all x = [Xi’ I] s AI(E). Then (a) 37* e Lb(AI(E), E); (b) {1:0 ify:0; (c) if AI(E) is an ideal, then y* :_0 if and only if yi :_O for every i e I such that AI(E) contains an element [Xi’ I] with xi # 0; (d) if AI(E) is an ideal, then y* is order continuous. 3329:. (a) The results of section 6, along with (M4) and (M5), show that y* is linear. It is clear that y* = (y+)* - (y_) , and that both (y+)* and (y-)* are positive. Therefore, y* c Lb(AI(E), E). (b) If y :_0, then it is clear from the definition of y" that (0) Suppose AI(E) is an ideal and y" :_0. Suppose AI(E) con— tains [Xi’ I] with xi ¢ 0. Then AI(E) contains the element [xi, I], 0 , = . . . , = . , where xi 0 if i # i0 and xi0 lxiol. Since [Xi’ I] > O and y. l0 1o (d) Suppose x(a) = [x§a), I]-g>0 in AI(E). Recalling (3.5), # v = v - > . x. ZieI Xiyi :_O, we must have yi0 __0 (see (M10)) XEGXJ; O in E (hence in E ) for all i e I. Thus for each i, xéa)yi-2>0 in E# (hence in the ideal E) by (9.10). Applying (3.5) (at) again, [x§a)yi, I]-3>O in 9%(E). Therefore, by (7.3) y“(x ) = 94 o . A * . . zieI xi yi_—)O in E. Therefore, y is order—continuous. (13.6) Proposition. Suppose E is an Archimedean vector lattice and 1 is a weak order unit in E#. Then, taking multiplication rela— tive to 1, 1 (a) if p and q are finite real numbers such that %-+ 9-: 1, then for every y e ]§(E, 1), y" is a positive, order continuous linear mapping from Q%(E, 1) into E; (b) if E is closed under multiplication, relative to 1, then for every 1 §_p,q :_m such that %-+ %-= 1 and for every y e Q§(E, 1), y0 is a positive, order continuous linear mapping froml£%(E, 1) into A E. Proof. Both (a) and (b) are corollaries of (10.6), (12.5), and (13.5). Section 14. Convergent families in a vector lattice. In this section we define and discuss a notion of convergence for families [Xi’ I] in wI(E). We shall not assume that I is a directed set, or even a partially ordered set. Thus this notion is not to be confused with that of convergence of a net over I. In addition to abstracting a notion of convergence, this idea leads to an example of a vector sublattice of wI(E) whose Kothe x—dual can 95 be described in terms of the spaces already discussed. (1H.1) Definition. A family [Xi’ I] s wI(E) is said to be zero—convergent if there is some y > O in E such that for every 5 > 0 there exists J e V(I) such that lxil §_6y whenever i t J. We let cg(E) denote the collection of all zero—convergent [Xi’ I] in wI(E). (1H.2) Proposition. (a) c$(E) is an ideal in wI(E), with ¢I(E) 2:. cim 2 mI(E); (b) if I is an infinite set and E is nontrivial, then 03(E) is not a band in wI(E); (c) if E is Archimedean, Dedekind o-complete or Dedekind complete, then c$(E) has the same property; and conversely. Proof. (a) is trivial. Pick e > O in E. For each j e I, define [x§]), I] by setting xij) = 0 if i ¢ j, and xgj) = e if i = j. Then {[xgj), I]: j E I} is a subset of 05(E) having supremum [yi, I] in wI(E), where yi = e for all i e I; since [yi, I] t c$(E), c§(E) is not a band. Finally, (d) is a consequence of (3.1) and (3.10). Let E be an Archimedean vector lattice. Consider an arbitrary [Xi’ I] e mI(E). In E construct the net {2J: J a V(I)} indexed by the directed set V(I) with its usual partial order (inclusion), by defining 2J = supitJlxil’ the supremum being taken in E. Observe that xJ +. The following 96 result is then clear. (14.3) Proposition. If E is an Archimedean vector lattice, a family [Xi’ I] in mI(E) is zero—convergent if and only if xJ 1 O uniformly in E. (14.4) Example. In mR(R) define [Xr’ R] by lifoiril, Xr = Oiflrl>1. Then [Xr’ R] is a family which is not zero—convergent, even though as a net over the directed set R, xfi—2>O. (14.5) Proposition. For any vector lattice E and any set I, u 0 (2 11m) _ c103). Proof. We may assume I is infinite, since for finite I, £§(E) = O u . cI(E) = wI(E). Let [Xi’ I] s QI(E). There eXIst x, y e E such that for all 6 > 0 there exists J6 e V(I) such that J 2 J6 implies 1(EieJ Xi) ..XI (.6— _QY' Let 6 > O. For each i i Jé’ if we let J' = J6 U {i}, then Ixi] 121w Xi ' 215% Xil 5— laieJ' Xi) ‘ XI + IQieJé Xi) ' XI 5_ 6y. Therefore [Xi’ I] e c:(E). (14.6) Definition. A family [Xi’ I] in wI(E) is said to be convergent to an element x e E if [xi — x, I] is zero—convergent. We denote this by xi-f>x. We let cI(E) denote the collection of all convergent families [Xi’ I]. (14.7) Proposition. If [Xi’ I], [yi, I] c cI(E), A e R, and 97 xi -T% x, yi-—E> y, then (a) xi + yi-—f$ x + y; (b) Axi '7) Ax; (c) xiv yi—I> Xv y; (d) XiA yi——I—> XAy. Proof. There exist ul, u2, u c E such that for all 6 > O, A # O, 3 there exist Ji’ J2, J a V(I) such that Ixi — xl §_6ul, ij — yl :_5u2, 3 5 . . and lxk — XI j'fi1u3 whenever i i J j i J2, k t J . Let 5 > O, 1’ 3 J = J1U J2UJ3 and z = 2(ul V u2). Then for all i ¢ J, we have (a) | — | : Ixi — xl + Iyi — 3711611, + 6n, _: dz. Thus xi + yi —ef> x + y. (b) IAxi — Axl = IAIIxi — xl :_5u3. Thus Axi-—E+ Ax. (c) Note that (xi v yi) — (x‘v y) = [xi — (x v y)] V [yi - (x V y)] g(xi - x) V (yi - y) §_%-z, and similarly, (x V y) — (xi\l yi) :_(x - Xi) V (y - yi). Therefore, |(xi\1 yi) — (x V y)| :62. Thus xiv yi _I) xv y. (d) By (b) —xi —f9 —x and —y —Eel-y. Then by (c) —xi V -yi —f* —x V —y; hence, by (b) —(—xi V —yi)-jf> —(-x V —y). That is, xi A yi -E% x A y. (14.8) Proposition. (a) aim) EcI(E) EmI(E); (b) CI(E) is a vector sublattice of wI(E), but can not be an 1 ideal in wI(E) if I is an infinite set and E is nontrivial. Proof. (a) is easy to see. That cI(E) is a vector sublattice 1 follows from (14.7). That cI(E) is not an ideal is seen by the 1 98 following argument. Without loss of generality we assume N E I. Pick any e > 0 in E, and consider the family [Xi’ I] defined by e if i = 2n for some n e N, 0 otherwise. We have 0 i-[Xi’ I] §_[yi, I] e cI(E), where yi = e for all i s I; but [xi, I] i cI(E) while [yi, I] e cI(E). Therefore, cI(E) is not an ideal. (14.9) Proposition. For any Archimedean vector lattice E, (a) if E is closed under multiplication relative to 1, then 9103) E [CI(E)]1 9:— [c$(E)]¥ and C(I)(E) E CI(E)EM1(E)]1' (b) if 1 is a strong order unit for i, then [o$(z)]§ = [cI l-E Ix I i’ I ' isJk i i ieJk k i —'k ieJk i ike.TMs% 0 X eJkIXiyil f_k1, hence, [Xi’ I] i [CI(E)]1' Therefore, [c$(E)]? ELQ%(E). The desired equality then follows with the aid of (a). Section 15. Kothe y—dual spaces. The theory of Kothe y—duality presented here is based on a definition which may not seem as natural as the corresponding definition in the theory of Kathe X—duality. It does, however, have # two major advantages; it is independent of the unit 1 chosen in E , and certain expected duality relationships may be established with apparently weaker hypotheses. Throughout section 15 we shall again assume that E is an Archi— medean vector lattice, embedded in its universal completion E# = Cm(Q), with 1 denoting the constant function 1(t) = 1 Vt s Q. If X, y e E#, xy will denote the product of x and y, relative to 1. then we define its Kothe y—dual to be [A (my = {[y I] e w (E)' [x y I] e Q1(E#) \IEX I] s A (E)} I i’ I ' i i’ I i i’ I ’ where the multiplication is taken relative to 1. For convenience we shall sometimes write A¥(E). (15.2) Remarks. Using the result (9.4) of Rice, we see that the I I (15.1) Definition. If AI(E) is an arbitrary subset of wI(E), definition of [AI(E)]y is not affected by the choice of the unit 1. 100 For if 1' is another unit and multiplication of x and y relative to 1' is denoted x * y, then by (9.4) EieJ IXiYiI j_u implies 2ieal"i""yil = ZieJI(xiyi) 1‘ 1' = (fieJIXiinI 1 311* 1. and similarly, ZieJlxi*yi| §_v implies ZieJlxiyil §_v1'. Moreover, proposition (9.7) shows that the definition of [AI(E)]y is not affected by the choice of universal completions E# of E. Thus the Kothe y—dual of AI(E) is determined intrinsically by AI(E) itself, and the embedding of E into E# is merely instrumental in computing y [AI(E)] . We say that a vector sublattice AI(E) of wI(E) is y-perfect if _ yy 11(3) - [AI] . Corresponding to propositions (12.2) and (12.4) for X-duals, we have the following proposition for y—duals. As the arguments are entirely analogous, we shall omit them. (15.3) Proposition. Suppose AI(E) is an arbitrary subset of wI(E). Then (a) A¥(E) is an ideal in wI(E); (b) AI(E) C_: uI(E) implies Iii/(E) E A¥(E), for any uI(E) E wI(E); (c) AI(E) s 159(3); (d) A¥(E) = Aiyy(E) (A¥(E) is y—perfect); (e) A¥y(E) is the smallest y—perfect vector sublattice of wI(E) containing AI(E); (f) if AI(E) is y—perfect, then AI(E) is an ideal in wI(E). 101 (15.4) Proposition. (a) ¢¥(E) = wI(E); ¢I(E) Eiw¥(E); (b) if E# has a countable exhausting set, then w¥(E) = ¢I(E); (c) if AI(E) is y-perfect, then ¢I(E) SE AI(E). 2323:. (a) is trivial. The proof of (b) uses the same line of thought as the proof of (12.9), and uses (a) as well. To see (c), observe that ¢I(E) §.w¥(E) §:A¥y(E), for any AI(E), since A¥(E) E wI(E). (15.5) Definition. (a) 5&(E) = {[xi, I] s wI(E): :3u# e E# # such that Iin :_u Vi e I}. We also write §:(E) = mi(E). # (b) 1%(E) = {[xi, I] e wI(E):'3u# e E such that VJ e V(I), # ZieJIXiI §_u }' (c) for each 1 < p < 00, I§(E) = {[xi, I] s wI(E): ‘3u# e E# such that VJ e V(I), E #} ieJIXilP j_u ' Note that as in (15.2), the definition of:§¥(E) is independent # of the universal completion E# and the unit 1 chosen in E . (15.6) Proposition. For any infinite set I and any 1 §_p < q < W, P q ‘” we have ¢I(E) ; 11(3) sTIm) $11G). Proof. The arguments used in proving (10.3) may easily be revised to establish these results. We do not include the resulting proofs. It is quite easy to extend the order Holder inequality (10.6) to the s aces IP(E), q(E) for extended real numbers p,q such that P I I + = 1; from this it follows that 'UIH .DIH 102 1§y(E) 2121(3), whenever + %-= 1. It is also quite easy to extend the order 1 p Minkowski inequality, from which it follows that 13(3) is an ideal. (15.7) Proposition. (a) I]:(E)]‘y = ?%(E). # (b) If E has a strong order unit, then for all 1 :_p, q < w i 1- p y_-q sudifith-tq-ih WIG)] -QJEL The proof is completely analogous to (12.7) and 12.10). Finally, since a norm H.“ on E cannot, in general, be extended monotonely to E#, we do not consider l—norms or p—norms for the spaces I%(E) or I§(E)' APPENDIX APPENDIX REPRESENTATION THEORY AND UNIVERSAL COMPLETION In this appendix we present, without proofs, a few pertinent results from the ”representation theory" of vector lattices, as devel— oped in Vulikh [18]. The power of this theory lies in the result that every Archimedean vector lattice E may be represented by a function space; more precisely, E is isomorphic to an order—dense vector sub— lattice of a known space Cw(Q). Moreover, Cw(Q) turns out to be the universal completion of E, and thus the Dedekind completion of E is the ideal generated by E in Cw(Q). For proofs and more details con— cerning these results the reader may consult Chapter V of Vulikh's book [18], which is the most exhaustive presentation of this represen— tation theory available in the English language. (16.1) Definition. A compactum (compact Hausdorff space) is said to be (a) totally disconnected if the open—closed sets form a basis for its topology; (b) extremal if the closure of every open set is open—closed; (c) quasi—extremal if the closure of every open FO—set is open— closed. By a rather simple point—set argument, one can show that every 103 104 I quasi—extremal compactum is totally disconnected. However, the subspace X = {0,i1,r%,r%, ...,r%,...} of the real line R provides a counterexam- ple to the converse; X is totally disconnected but not quasi-extremal. If Q is an extremal or quasi-extremal compactum, then Cw(Q) will denote the collection of all continuous, extended real—valued functions f on Q such that {x s Q: If(x)I = w} is nowhere dense in Q. The assumption that Q is extremal or quasi—extremal allows one to show that Cm(Q) is a vector lattice, under the usual (pointwise) definition of addition, scalar multiplication, and order. Note that any non— negative constant function servesas a weak unit for Cm(Q). (16.2) Proposition. If Q is an extremal (resp. quasi-extremal) compactum, then Cm(Q) is a Dedekind complete (resp. Dedekind o—complete) vector lattice. Moreover, if Q is extremal, Cm(Q) is universally complete. (16.3) Proposition. If X is a Dedekind complete (resp. Dedekind O-complete) vector lattice with a weak order unit 1, then there exists an extremal (resp. quasi-extremal) compactum Q such that X is isomor— phic to an order dense ideal X' of the space Cw(Q). Moreover, the isomorphism can be realized so that C(Q), the set of finite—valued functions f in Cw(Q), is a subset of X', and so that the unit 1 is mapped onto the function which is identically 1 on Q. (16.4) Proposition. Every Dedekind complete vector lattice X is isomorphic to an order—dense ideal X' in Cm(Q) for some extremal compactum Q, which is unique up to homeomorphism. Moreover, X is universally complete if and only if X' = Cm(Q); i.e., if and only if X = Cw(Q). (16.5) Corollapy. A vector lattice E has a universal completion if and only if E is Archimedean; any two universal completions of E are isomorphic. The proof of (16.5) consists of embedding E isomorphically as a vector sublattice of its Dedekind completion E, and then embedding E in Cw(Q) for appropriate Q. 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