Cnnprnn 4 fnfinitesimals in Vector Lattices
BY
E. Yu. Emel'yanov
l6l S.S. Kutateladze (ed.), Nonstandard Analysis andVector Lattices, 16l-230. @ 2000 Kluwer Academic Publishers. Printed in the Netherlands. Infrnitesimals in Vector Lattices 163
Infinitesimal analysis has lavishly contributed to various areas of mathematics since 1961 when A. Robinson published his famous paper [23]. The complete list of applications is very huge even as regards functional analysis. We give below some of them pertinent to the theory of vector lattices. The first natural question is as follows: Where does infinitesimal analysis apply effectively? Clearly the methods of infinitesimal analysis are not a panacea and so they must fail in solving some problems. Furthermore, what new possibilities are open up by infinitesimal analysis when it applies? The book [17] offers a partial answer to this question. Roughly speaking, the nonstandard methods prove fruitful whenever the problem under consideration deals with such concepts as compactness or ultrafilter. On the other hand, these methods are often inapplicable to purely algebraic problems. The main topic of further research is the theory of vector lattices and operators in them. We hope to convince thereader that vector lattice theory is a natural area for applying infinitesimal methods. Vector lattices with some norm or other specific structure were studied by many authors (see, for example, [5, 11], and [18]) in the context of infinitesimal analysis. Our aim is to further pave the infinitesimal approach to vector lattices. In the sequel we proceed in the wake of the articles [6-10] with due modification, considering only real vector lattices. The structure of this chapter is as follows. Section 4.0 is an introduction to Robinsonian infinitesimal analysis and its applications to normed spaces. In the end of the section, we make a short introduction to the theory of lattice normed spaces (for more information on this topic we refer the reader to the recent papers [13-16]). Sections 4.1 and 4.2 deal mainly with an infinitesimal approach to representing vector lattices. It turns out that infinitesimal analysis provides new more natural and subtle possibilities for representing vector lattices as function spaces. We show below that construction of a representing topological space for a vector lattice is possible on using members of the lattice (more precisely, of a nonstandard enlarge- ment of it) rather than ultrafilters or prime ideals (the latter inherent in Robinson's construction). In Sections 4.3-4.1, we deal with infinitesimal interpretations of the basic con- cepts of the theory of vector lattices. We also consider the extension problem for a ,r,-invariant homomorphism over a vector lattice or a Boolean algebra. Some types of elements of a nonstandard enlargement of a vector lattice are defined: limited or finite elements, (r)- and (o)-infinitesimals, prenearstandard and nearstandard elements, etc. We obtain some easily applicable nonstandard criteria for a vector lattice to be Archimedean, Dedekind complete, atomic, etc., and present a nonstan- dard construction of a Dedekind completion of an Archimedean vector lattice. t64 Chapter 4
Using the concepts of Sections 4.3-4.7, we follow the Luxemburg scheme in Sections 4.8-4.11 for defining and studying two nonstandard hulis of a vector lattice: the order regular hulls. An elementary theory of these hulls is given in Sections 4.8 and 4.9. In Section 4.10 we introduce and study the concept of the nonstandard hull of a lattice normed space and that of the space associated with the order hull of a decomposable lattice normed space. Section 4.11 discusses a nonstandard construction of an order completion of a decomposable lattice normed space. The scheme rests on embedding such a space into the associated Banach-Kantorovich space. Throughout the sequel, we use the terminology and notations regarding vector lattices and operators from the books ll, 21, 241, and [28]. Lattice normed spaces are dealt with only in Sections 4.10 and 4.11. Therein we appeal to the terminoi- ogy and notations of [13-16]. In the current chapter, we use some (standard and nonstandard) results on Boolean algebras and measure spaces which can be found in [2, 4], and [25]. For Robinsonian nonstandard analysis and its applications we refer the reader to12, 12,201and [27]. Other explanations are to be made on the route. 4.O. Preliminaries We start with a brief introduction to Robinsonian infinitesimal analysis and recall several facts of it without proofs. The level of formal requirements is chosen so as to avoid overloading the text with unnecessary details. We will mostly foliow the books 12, I2l and [20]. We then recall some well-known results on applications of infinitesimal analysis to the theory of normed spaces and operators in them. For more applications we refer the reader to12,12] and 127J. In the end of the Section, we touch the theory of lattice normed spaces. Our exposition of this part rests on [13-16]. The concept of a quotient of a lattice normed space and Proposition 4.0.14 are new (cf. [9, Lemma 0.5.7]). 4.O.L. Let ,S be a set. A superstructure over ,9 is the set V(S) :: U", V^(S), with Vr(S) defined by recursion:
Yr(S) :: ,S, V"+t(S),:V"(S) u {X : X I Y,(S)}. Superstructures are fragments of the von Neumann universe, providing a basis for various mathematical theories in dependence on the choice of the basic set. For example, superstructures over the reals serve the needs of calculus. When working with the superstructure over some set ,S, we suppose throughout that RgS. We need some formal language -L. The alphabet of -L contains Infinitesimals in Vector Lattices 165
(1) variables: small and capital letters with possible indices; (2) the symbols : and € for equality and membership; (3) symbols for propositional connectives and quantifiers; (4) auxiliary symbols. Atomic formulas of the language are L expressions of the form r - y or r e A. Arbitrary formulas are obtained from atomic formulas by applying propositional connectives and bounded quantifiers for sets (i.e. the prefixes Yr € y and )r e y). Given an d,rbitrary set ,S on which some (partial) operations and relations are defined, we introduce some language Lv@) of the superstructure V(,S). To make the presentation easier, we construct the language Lv6) in a simple particular case. Let ,9 : E U N, with E a lattice. The set -E is equipped with the operations A, V, and the relation the set N of naturals is equipped with the operations -?; *, ', and the relation (. In this case, the language Lv6) is obtained from L by enriching the alphabet with the symbols A, V, *, and . for operations and ? and ( for predicates. The list of atomic formulas extends to include the expressions of the f.orm t1 Y t2 : ts, h.tz : ts, tL I tz, etc., where t1, t2, and f3 are arbitrary terms of .L. Every formula of the language Lv6) is naturally interpreted in the superstructure V(S). For example, the formula V(r, y,z) :: r + A ( z is trueforatriple (a,b,c) of elementsof l/(^9) if andonlyif a,b,c€Nand a*b.--c. It is clear that interpretation of arbitrary formulas of Ly6y involves no difficulty either. In what follows, we choose the basic set depending ^9 on the context. This set will be assumed to contain various objects: real and complex numbers, vector spaces, vector lattices, etc. Our presentation is structured so that it may be trans- lated into the formal language LvG) if need be. Throughout the article, the word "interpretation" means the natural interpretation in the respective superstructure.
4.O.2. Let ,9 be a set equipped with some operations and relations (not neces- *,S sarily defined everywhere). Then there exist an enlargement of ,S and embedding * : Iz(,S) -' V(.,S) satisfying the following principles: *,S Extension Principle. The set is a proper enlargement of S. Moreover, *^9 is equipped with the same set operations of and relations as ^9. In addition, *fr: t for every element r € S. Tbansfer Principle. Let ,h(rt,t2,. . . ,rn) be a formula of Lv6), and het At, Az,. . . , An be elements of the superstructur'e v(s). Then the ass'ertion ,b(Ar,A2,...,A,-) about elements of V (S) is true if and. only if the assertion
,lt(At,*A2,. . .,*A,) 166 Chapter 4 about elements of V(. S) is true. Construction of the enlargement *S and embedding *. : I/(^9) .* V(-S) with the required properties can be found, for example, in [2]. For convenience, we suppose that * is the identical embedding and so Iz(S) g y(.,S). DspINITIoN 1. The superstructure V(. S) is called a nonstandard enlargement of V(S) if the embedding y(S) y(.S) satisfies the transfer and extension prin- -C ciples. Dealing with some superstructure in the sequel, we will not specify the basic set over which it is constructed. This set is chosen to be sufficiently substantive. We will denote the superstructure under considerationby M. DsrtNlrtox 2. Let * M be a nonstandard enlargement of a superstructure M. An element, r €*M is called:
(1) stand,ard if. r : * B f.or some B e M;
(2) i,nternalif r €*Bfor some BeM; (3) erternal it r ( *B for every B e M. Note that every standard set is internal and every element of an internal set is internal too. The following is easy from the transfer principle: Internal Definition Principle. Let ,!(*,*r.,12,. . . ,rn) be a formula of the language Ly, and let A,At,Az,...,An be interna/ sefs. Then the set {r e A: ,b@, At, A2,. . . , A^)\ is internal too. *M It is well known that a nonstandard enlargement of. the superstructure M may be chosen so that the following principle holds: General Saturation Principle. For each family {&}rer of internal sets which has standard cardinality (i.e., card(f) < card(M)) and enjoys the finite intersection property, the condition f1"." X, * 6 is valid. In the sequel, we deal only with nonstandard enlargements satisfying the gen- eral saturation principle. These nonstandard enlargements are called polysaturated. 4.0.3. Let X be an element of a superstructure M. we denoteby g(x) the family of all finite subsets of X . Recall that elements of . ,q (X) are exactly *X the subsets,4 g for which there exist an internal function and element y € *N / *X such that dom(/) : {1, . .. ,u} and im(/) - A. These subsets of are called hyperfini,te and denoted, for example, by {*^}i:, in analogy with finite families. .N Lernrna. Let X e M be an infinite set and / € \ N. Then there exisfs an internal function f such that dom(/) : {1,...,u} and X g i-(/) q *X. In other words, there is a hyperfinite set {r^}i-1 with X e {r^)"n:t e * X. Infinitesimals in Vector Lattices t67
( Let V be the set of all functions r/ such that dom({) I N and irn(/) g X. Given r €.q(X), we assign .V Ao ::{p e : dom(rp) : {1, ...,u} k r ei*(p)}. Since X is infinite, every set An is not empty. By the internal definition principle, the sets Ao are internai. They form a family with the finite intersection property. Since card(9(X)) < card(M),bV the general saturation principle we have / e )ne^(x) Ao for some e *itrr. It is easy to see that / is an internal function with "f *X. dom(/) : {1, ...,u} and X e im(/) q > 4.O.4. Lernrna. Suppose that X e M and X g M. Then, for everyz € *N\N, we have card(X) < card(u). < Consider the sef 9(X) of all subsets of X. Since g(X) € M, by the pre- ceding lemma, g (X) g {r.}i:1, where {*n}i:, is a hyperfinite subset of . I (X). Then card(X) < card(9(X)) < card(rz). > 4.O.5. Let O be a directed set which is an element of the basic superstruc- -O ture M. Denote the set {€ e : (Vr € O) € 2 ri by "O. Elements of oO are called (infi,ni,tely) remote. Lernrna. For every directed set O € M, there is at least one remote element o€oO. < In the case when O is a finite set there is nothing to prove. So, we assume that O is infinite. By Lemma 4.0.3, there is a hyperfinite set ,4 such that O e *O. *O A g Since is an internal directed set, there is an element o € *O satisfying a ) r for all r € A. It is clear that a € "O. > 4.0.6. The main tool for applying nonstandard analysis to normed spaces is the following simple construction discovered by W. A. J. Luxemburg [20]. Let X be a normed space. Consider the two external subspaces Fin(.X) ,: e*X : (3r € re)ll.ll {, S "}, p(.X),: {n e*X: (3r € R)(Vn e N) llnzll S *X. "} in Elements of Fin(*X) are called (norm) Iimi,ted (or finite 'in nonn) and ele- ments of p,(" X) are called 'i.nfinitesi,mal. obviously, p(.X) is a subspace of Fin(.X). Thus we can take the quotient.:pu". X :: Fin(.X)/p(-X) under the norm lltr]ll ,: st(ll xll). In the wake of W. A. J. Luxemburg, w€ cal * the nonstandard hult of X. Defi.ne the mappingfrx : X - X as frx(") ': [*] (r e X). It is easy that i,x is an embedding of X into *. ttr" following is well known: 168 Chapter 4
. Proposition. For every normed space X, the quotient X is a Banach space and the map frx is onto if and only if X is finite-dimensional. 4.O.7. A nonstandard construction of a norm completion of a normed space lies very closely to the construction of the nonstandard hull of the space under tudy. Let X be a normed space. Consider the external subspace
pns(.X) ': {, e*X : (Yn e N)(ls € X) np(x - y) < U . of.*X'
, , Proposition. The quotient normed space pns(. X) I p( X) is a norm comple- tion of X under the embedding p,y. 4.0.8. Let A be a subset in a normed space X. We have a simple and useful criterion for boundedness of A. Proposition. The following are equivalent:
(1) A is a norm bounded set; + (z) *AgFin(-x). 4.0.9. LetX andYbenormedspacesandlet T: X -+ Ybealinearoperator. : Jhe next well-known proposition is immediate from 4.0.8. Proposition. The following are equivalent: (1) T is a bounded operator; (2) *T(Fin(* X)) e Fin(.r); (3) .T(p(.X)) I p(.Y); -T(p(.X)) (4) C Fin(-Y). Thus the operator ? , * -- 7, acting ar f1121; 7lTr] for all r € Fin(*X), is well defined and bounded together with ?. This operator is lhe nonstandard hull of T. Now we briefly present necessary facts from the theory of lattice normed spaces and dominated operators. Our exposition follows [13-16].
4.0.10. A lattice normed space is a triple (x,p,E) with X a vector space, .E a vector iattice, and p a mapping X --,81 such that (r) p("):o <+ n:o; (z) eQ,r): lrb(") (.\ e IR, r e X); Infinitesimals in Vector Lattices 169
(s) p(" + y) < p(r) + p(a) @,a € x). The mapping p is called an E-ualued norm on X. The lattice norm p is called decomposable ((d)-decomposable) if, for all €t,€2 € E+ (for all disjoint €1,€2 e E+) and every r € X, the condition p(r) : er * e2 implies existence of. r1,r2 € X such that rt * tz: r and p(nn) : ak for k : I,2. A lattice normed space with decomposable ((d)-decomposable) norm is called decomposable ((d)-decomposable). We say that a sequence (rr) in (X,p,E) (r)-conuerges to r € X if there exist a sequence (e"r) C lR., e,, | 0, and an element u € E such that p(rn - r) < enu f.or all n € N. By definition, a sequence (zr,) C X is (r)-Cauchy if there is a sequence (rr) g R, €rJ0, such thatp(rp-rrn) 1€n'u, for all k,n'L,n € N such that k, m) n. A lattice normed space X is called (r)-complete tf every (r)-Cauchy sequence in X (r)-converges to some element of. X. The following assertion is a consequence of [13, 1.1.3] and the Freudenthal Spectral Theorem. Proposition. A (d)-decomposable (r)-complete lattice normed space is de- composable.
4.0.11. A net (*o)",rt in a lattice normed space X is called (o)-conuergent to r € X if there exists a decreasing net (er)re r in ,E such that e" J 0 and, for every ? € f, there is a subscript a(7) for which a(ro - z) < e,' whenever o 2 a(l). A net ("o)o.o is called (o)-Cauch,y if the net (ro - rB)@,p)€AxA (o)-converges to zero. A lattice normed space is called (o)-complete if every (o)-Cauchy net in it (o)-converges to an element of the space. A decomposable (o)-complete lattice normed space is said to be a Banach-Kantorouich space. consider a decomposable lattice normed space (x,p, E). Let (rg)g.= be some partition of unity in the Boolean algebra g(E) and let ("e)e.= be a family of elements in X. If there exists an r € X satisfying the condition rlop(r-u€) :0 (€eE), then such an element z is uniquely determined. It is called the mi,ri,ng of (26) by (z'g) and denoted by mix(a-qce)eee or simply mix(n-626). A lattice normed space (X,p,E) is said to be (d)-complete if the mixing mix(z'6re) e X exists for every partition of unity ("e) g g(E) and every norm bounded family ("e) g X.
Proposition [13, Theorem 1.3.2]. A decomposable lattice normed space is (o)-complete if and only if it is (r)- and (d)-complete.
4.O.L2. Let (X, p, E) be a decomposable lattice normed space whose norm lattice E is Dedekind complete. Then there is a unique lattice normed space (X',p',-E) to within isometric isomorphism with the following properties (see [13, Theorem 1.3.8]). fia Chapter 4
(1) (X',p',.E) is a Banach-Kantorovich space; (2) there exists a linear embeddingr,: X -- X'such that n'(z(r)): p(r) for all r e X; (3) X/ is the least (o)-complete lattice normed subspace of X/ that contains ,(X). The space (X',p',E) is an (o)-completi,on of the lattice normed space (X,p,E). Consider some properties of lattice normed spaces connected with decompos- ability and (r)-completeness. 4.0.13. Lernrna. Let (X,p,E) be a decomposable lattice normed space and let I be an ideal in E. Then, for arbitrary elements r,U Q X, e € E+, andrT e I satisfuing the condition p(r - A) S q + q, there is an element A' e X such that (t) p(" - v') < q; (z) p(y - y') € I. < Clearly, p(r - y) < q * n+. By the Riesz decomposition property, there are elements aL,a2 g €such that
0(41 (g, 0 3az1rl+, atla2:p(r-y).
Since p is decomposable, we may find eiements z1 ,22 € X with
p(zr) : p(zz) : att a2, zt * zz : r - U.
It is easy to see that the element A' :: y* zz satisfies (i) and (2). > 4.O.L4. Let (X,p,E) be an arbitrary iattice normed space. Take an ideal 1 in E and consider the quotient space Xt of X by the subspace X(1) 1: {r € X: p(r) € /). Given n€ X (ee E),, assign[r] :: n+X(/) (respectively, [e] :: e*I € EII). Define the mappingpt : Xt-- EII by the rule
p'[r]) :: [p(r)] (r e X).
It is easy that the mapping p'is defined correctly and presents an Ef l-valued norm on the space X/. The so-obtained lattice normed space (X' ,p' , E lI) is called the quotient space of X bE the i,deal I of th,e norn'L latti,ce E. Proposition. Let (X,p,E) be a decomposable (r)-complete lattice normed space, and let I be an ideal of the norm lattice E. Then the quotient space (X' ,p' , E lI) is decomposable and (r)-complete. Infinitesimals in Vector Lattices 171
< Verify that the norm p' is decomposable. Let p'([r]) : [e1] + l"zl, where l"r1,,l"r] € (ElI)+. We may assume that e1,e2) 0. There is an element q € E such that p(r) : et * ez * l. By Lemma 4.0.13, there is an element rt € X such lhaf p(r - r') I er + e2 and p(r') € 1. In particular, P(r-r'):eL+e2+qt (1) for some q' e I. Applying the Riesz decomposition property to the inequality p(r r') < €1 I €2, we find elements € E f.or which - "\, "'z p(r-*'):elr+e!r, 03et1, 1ep (k:I,2). (2)
: instance in Then fe'rl: lepl (k 1,2). Indeed, supposing that for [e1 - "'l ) 0, view of (t) and (2) we obtain a contradiction: A : lP@ - n') - - et - rt'l : ro,'-:,rul; ''t Since the norm p is assumed decomposable, it follows from (2) that there exist Ur,Az € X such that r - r' : At * Az and p(yp) -- uL & : 1,2). Then p'(lgxl) : fp(an)l : l"Ll : lekl for k : 1, 2, and [yt] + larl: l, - ,'l: [r], as required. We now verify that Xtis (r)-complete. Take some (r)-Cauchy sequence ("0) g X/. There exist e € E and (*n@) g (zr), for which p'(r41 - rq,a) < 2-^l"l (3) whenever k, n'r, and n are such that k, n'L > n. Choose elements 7(n e X so that rqn) : lx^1. Then, by (3), there are rlk,m € / with p(xx - ,,.) 3 2-ne * \k,,n (4) for all k,rn,n € N such that k,m 2 n. By induction, construct a sequence (x) e X that satisfies the conditions p@l - ,i"+r) 1 2-n e; (5) t72 Chapter 4 p(xn+t-Nl*r)eI (6) for ali n € N. Assign xl :: %1 arrd assume that the elements Nj are already defined for j < n. From (4) it follows that p@l - nn+r) 1 p(nn - xn+r) - p(Nn - rl,+r) 3 2-ne * \n,n+r * p(rn - r,i,) Since by the induction hypothesis P(nn - ,;) € /, we may apply Lemma 4'0.13 to the elements xl,Nnql e X, 2-ne € E, Tl t:rln,+t *p(Nn-,). In result, we come to an element %[+t satisfying conditions (5) and (0). It follows from (5) that the sequence (Ni,) is (r)-Cauchy in X. Consequently, it (r)-converges to some element uo € X. Then, as is easy to see, the sequence (za1rr;) -- (r"l) (r)- converges in the norm p' to lro) e X/. Since the initial sequence (za) is (r)-Cauchy in the norm pt , we obtain *i !), Vol. > 4.0.L5. Assume that .E and f' are some Dedekind complete vector lattices, while (X,a,-O) and (Y,b,F) are decomposable LNSs. A linear operator T : E -, F is called d,orni,nated if there exists an order bounded linear operator ,S : E -+ F such that lr"l3n(l"l) @ex). The operator ,S is called a domi,nant of.?. The least dominant of an operator ? is denoted by l"l. By M(X,Y) we will denote the vector space of all dominated operators from an LNS X into an LNS Y. The mapping ?'* lTl Q e M(E,F)) satisfies all axioms of an.E-valued norm from 4.0.10. Consequently, M(E,F) is also an LNS with norm lattice the Dedekind complete vector lattice La(E,F) (see, for example, F. [28, Theorem 83.4]) of all order bounded linear operators from E to 4.L. Saturated Sets of Indivisibles Here we deal with lattices with zero and present some elementary facts about them, standard and nonstandard. We prove that a nonstandard enlargement of a lattice with zero contains a saturating family of indivisible elements. 4.L.L. Let L be an ordered set whose order is denoted by 2. We write r > y whenever r > A and r I y. The set -L is called a latt'ice if every two-element subset {*,,y} in .L has the least upper bound r V A :: sup{r, y} and the greatest lower bound r AA::inf{r,y}. If a lattice contains the smallest (largest) element then Infinitesimals in Vector Lattices 173 this element is called zero (uni,ty) and denoted by 0 (respectively, 1). We always assume that each lattice under consideration has some zero. Elements z and y of alattice aredi,sjoi,ntif rAU:A. Alatticeiscalled d,i,stributi,ueif everytriple n,U,z of its elements satisfies * A(aV z) : (" na) V (rAz) and *v (y Az) : (rv y) A(rV z). A distributive iattice ,L with zero 0 and unity 1 is called a Boolean algebra if every element n € L possesses the complement, i.e., a (unique) element r'€ L such that rArt:0andrVrt:I. Let L be a lattice. An element e € L is said to be a weak (order) uni,ty it eAn ) 0 for every r e L, r > A. We call y e L a pseudocomplemenf of an element r € L lf. n AA:0 and rv A is a weak unity. The lattice.L is called a pseudocomplemented latti,ce if, for each r € tr, there is at least one pseudocomplement in ,L. An example of a pseudocomplemented lattice is a Boolean algebra. A less trivial example is the lattice of all nonnegative continuous functions on an arbitrary metric space. Also, we will use the following notation: Given an element x of. a nonstandard *.L, enlargement we denote by U(x) ,: {* € E : r > %} the set of standard upper bounds of. x and by L(x) t: {A € E : N 2 A}, the set of standard lower bounds of. x. 4.L.2. Let L be a distributive lattice. An i,deal of the lattice tr is a nonempty set 1 e .L such that r,A e I implies rV y € -I, and z € I if. z 1u for some u € I. An ideal P of L is called pri,me if, for every r,a € -L, the condition r Ay :0 implies that either r € P or y € P. A prime ideal P is called mi,nimal if, for every prime ideal Pr tr, the condition implies : C h e P Pt P. Every subset ^9 e .L such that 0 I ^9 and r,U € ^9 implies n A A €,S is called a lower sublatti,ce. A lower sublattice S is called mari,mal or an ultraf,lter if for every lower sublattice St e L : the condition ^9 e Sr implies Sr ,S. The following assertion is well-known 121, Theorem 5.4]. Lernrna. Let L be a distributive lattice and let P be some prime ideal in L. Then t\P is alower sublattice of L. Furthermore, L \P rs anultrafilter if and only if P is a minimal prime ideal. 4.1.3. Consider a nonstandard enlargement *L of a lattice L. By the transfer principle, *tr has the same zero element 0 as the initial lattice ,L. We give two important definitions. DprtNtrIox 1. Anelement xe *L iscalled i,ndiuisi,bleitx) 0and,forevery n€L, either n)-N orrA zt:0. DpplNtuoN 2. A subset A of the lattice *,L is called saturating if lt is internal and, for every r € L,,r)0, there is some a €Asuch that n) a>0. The following simple lemma is a key for many of our further results. 174 Chapter 4 Lernma. A nonstandard enlargement of an arbitrary lattice contains a hyper- finite saturating set of disioint indivisible elements. ( Let L be a iattice. We denot eby I the family of all finite subsets of I \ {0} and put *9 9,:-- {X e : (Yy er)(Yr e X)(a) r ot v Ar:0) k (va e r)()r € x) (Y 2 r) k (Yr e X)(Vz e X)(r I z -) r Az: 0)) for all T e ,q. Show that the sets 9n are nonempty. Take an arbitraty t € 9 ' For every element fr €n, there is a set B, such that r e B, e zr", inf B*) 0, and inf(5l" u {y}) : 0 for all y € zr\ ,B". It is easy to verify that the set {inf B" : r € r) belongs to 9o. The fact that the sets 9o are nonempty and the condition -g^ n -91 : gnq implies that the family {9^)^ee enjoys the finite intersection property. All elements of this family are internal sets by construction. Thus, by lh" g"nurul saturation principle, there is a A € )neg- g^-It is easy to see that A is a desired saturating family of disjoint indivisible elements. tr 4.I.4. Let L be a lattice. By Lemma 4.1.3, there exists a saturating set of *L. indivisible elements in the lattice Let A be such a set. Denote A" :: {x € lt r )_ x\ for all r € L. It is easy to see that, in the case when -L possesses a weak unity e, the family {A" , n e L} is an open base for a topology r on A. This topology is called canonical on A. Theorern. Let A be a saturating set of indivisible elements in a nonstandard enlargement of a lattice with weak unity and \et r be the canonical topology on lv. (A, is a compact sPace. Then ") { Assume that (A, r) is not compact. In this case we may extract a subset {A'}rex from the base {A'}r An:: A\ U L" + @. r€r It is easy to verify that {A, : r e An"(X)} is a family of internal sets with the finite intersection property. By the general saturation principle, there is an element x€i{An: r € fln"(X)}. Then x€lv\U".xA', &contradictionwiththefact that {A"}cex is an open covering of A. Thus, (A,r) is a compact space' > Infinitesimals in Vector Lattices 175 4.L.5. We introduce some equivalence relation - in the lattice * L by putting Ltr - t'q whenever the inequalities r ) )a1 and n )- xz are equivalent for al| r e L. Suppose that a lattice -L possesses a weak unity e. Take a saturating set A of indi- visible elements in * L (such a set exists by Lemma 4.1.3). Lel r be the canonical topology on A. The topological space (A, is compact by Theorem 4.1.4. quo- ") Its tient spacg by the equivalence - is a compact ?s-space. We denote this quotient space by A. It is clear that the sets of the type j{.',: {14 e lt : r > N} (r e L) form an open base for the quotient topology (throughout, we denote bV [r] the coset containing an element x € /t). Take a Boolean algebra B as L and consider a saturating set T of indivisible elements in *.B. It is easy to see that T is a totally disconnected compact Hausdorfi space, while the mapping associating with each element b e B the subset Tb of the space t ir a Boolean isomorphism of B onto 1h" slgebra clop(T) of clopen (closed and open) subsets of the compact Hausdorff space T. Thus we obtained the following Theorern. Let B be a Boolean algebra and let T be a saturating set of in- divisible elements in a nonstandard enlargement of B. Then the corresponding topological space T i" the Stone space of B. 4.1.6. The following theorem describes connection between the properties of a lattice and the corresponding topological space. Theorern. Let L be a distributive lattice with weak unity. Then the following are equivalent: (1) L is a pseudocomplemented lattice; (2) The topological space It is totally disconnected. for every saturating set A of indivisible elements in * L; (3) The topotogical space A. satisrges the T1-separation axiom for every saturating set A of indivisible elements in * L; (4) The set {r e L : r A x: 0} is a minimal prime ideal in L for every indivisible element x €*L. < (1)-'(2): It is easy to see that if condition (1) holds, then the base {i' .,er f9r th9 top.ology o_f A, c_onsists of clopen sets. Indeed, given an r € .L, we have A" U As : A and A" n Av : o,, where y is some pseudocomplement of r. (2)--+(3): Obvious. 176 Chapter 4 (3)-'( ): Let condition (3) be satisfied. Take some indivisible element n €*L andconsiderthe setI,-r: {, € L: rAx -0i. Itiseasytoseethat,I,ois a prime ideal in the lattice .L. Indeed, by distributivity, it follows from r, A € I, that (rVy) A % : (r n x)V (t A x): 0, and consequently nY A € I-. If rA A € I. then either r € I. or U € I- (otherwise, since the element N is indivisible, we would have r )_ n and A >- %). It remains to verify that the rdeal I, is minimal. Take an arbitrary prime ideal P e I..Assume that g € 1-\P for some element r € L. Then r Ay ) 0 for every r € L\,I-. Indeed, otherwise it would be valid that n AA - g,.and hence we would have either r € P or y € P, which is impossible. By Lemma4.l.3, there exists a saturating set of indivisible elements in the lattice *L. Let A/ be such a set. Assign A :: L'U {N}. Then A is also a saturating set of indivisible elements, and x € lv. As was mentioned above, r AY > 0 for every n €. L \.I-. Using this observation, it is easy to show that {A"^s}rer,1r- is a system of internal sets with the finite intersection property. Applying the general saturation principle, we find an element d e A such that de n{A"^a: r€r\/-}. The indivisible element d satisfies the condition d S g. At the same time, N AA : .9 because U € 1,. Consequently, 5 / x. By condition (3), the topological space A satisfies the T1-separati.on axiom. Therefore, there is z € L for which lxl e i" and and z A5 :0 are valid. The first of them implies 16l f i". Then the relations x I z z € L\ -[-, which contradicts the second relation. The obtained contradiction shows that I.\P : o. Since the choice of the prime ideal P satisfying the condition P g I, was arbitrary, I- is a minimal prime ideal. ( )-'(1): Suppose now that condition (a) is satisfied. Show that every element of the lattice .L has a pseudocomplement. Take an arbitrary a € L. By Lemma 4.1.3, *,L. there exists some saturating set A of indivisible elements in Consider the topo' logical space (A,T), where r is the canonical topology in A. Let N € A \ 4". By hypothesis, the set r,i.:{reL:r ._ rAx:0} is a minimal prime ideal of ,L. By the choice of x, we have a A N : 0, and so a ( L\1,. Since L\1, is an ultrafilter of the lattice L by Lemma 4.1.2, there exists anelementa@) € r\1- such thaf y(x) Aa:0. In otherwords, , E lvu@) and A" n 6a@) : 0. The family {La@)Y.rA\A" is an open covering of the closed set A\A" in the space (A,"). By Theorem 4.I.4, it contains a finite subcovering {lra@x)1"-r. The element b,:V[=ra@n) satisfies the conditions l\b n \vo : @, Ab u A" : A and, consequently, it is the desired pseudocomplement of a. tr Infinitesimals in Vector Lattices 177 4.L.7. Let L be a distributive lattice. If tr is pseudocomplemented then, by Theorem 4.L.6, for every indivisibie element x € * L, there exists a respective min- imal prime ideai I.:: {r e L: rAx - 0}. The converse is true in a more general setting: Lernrna. Let I be a minimal prime ideal in a distributive lattice L. Then there is an indivisible element z € * L such that I : {r € L : r n n - 0}. { Observe that the subset U :: L / in tr is directed downwards. Lemma \ *U. By 4.0.5, there exists a remote element x € An easy check shows that x is *L an indivisible element in the lattice and I: {r € L: r Ax:0}. > 4.1.8. Let L be a distributive lattice. Denote by "// the set of all minimal prime ideals in L. The set ,.// is equipped with the canonical topology generated by the open base of all sets of the form .z/u:: {Pe .//: uf p} (ueL) (see, for example, [21, Sectio" 7]). Theorern. Let L be a pseudocomplemented distributive lattice. Then, for every saturating set A of indivisible elements of *L, the mapping tpn, defined by the rule pil[x]) ': {" €L: nnzc:0} ([z] e ii), is a homeomorphism of the topological space lt, onto .2il. ( Let A be a saturating set of indivisible elements in the lattice *,L. By The- orem 4.I.6, the mappirg prr ranges in the space '4. The mapping gn is injective. Indeed, take arbitrary elemenls x1,%2 € A such that u1 * ,z.Then U(rr) * U(rr),villz'rl) t' pil[rt]), slnce 9^Q%Ll): \ U(') ' for all N € lv. Show that 96(A):.r//. Take an arbitrary P e,til. As is easy to verify, {A"}'er\p is a system of internal sets with the finite intersection property. Thus, by the general saturation principle we may frnd x € flrer,yrA". Then pillxl):{r€L: rnx:0} : {r e L : N ( L"}: r\ {r e L : N e ["] g,\(,\P):P 178 Chapter 4 and so pillx)) - P, because the ideal P is minimal. It remains to verify that ,pn(ri'):{pr(ln))'r!r): iPe .'//: rf P}:lil'' D 4.L.g. Let A1 and A2 be saturating sets of indivisible elements in a distributive pseudocomplemented lattice ,L. Then, by the preceding tlJeorem,.-the mapping Note that ,b ,: pil o gLt is a homeomorphism of the topological space A1 onto Az. this homeomorphism can be defined explicitly as follows: ,lt(lrr]) ,: {rz € A2 : U(rt) 2 ,z} for every element x1 € l\y. Thus, the topological spac" rl is uniquely determined to within a homeomorphism by ihe distributive pseudocomplemented lattice L and does not depend on the choice of a saturating set A of indivisible elements. 4.2. Representation of Archimedean vector Lattices In this section, we prove some nonstandard variant of the representation the- orem for Archimedean vector lattices. We then give nonstandard proofs for the Brothers KreYn-Kakut ani and Ogasawara-Vulikh representation theorems. Throughout this section we suppose that -E is an Archimedean vector lat- tice. The positive cone .81 of ,E is a distributive lattice with zero. Therefore, .by *E-.,'-. Lemma 4.I.3, there exists a saturating set of indivisible elements in Here we fix such a set and denote it by A up to the end of this section. 4.2.L. Let e € E and N e I\ be such that e 2 x. Given f e E, define the element f "(r) of IR' as f"(r):: inf{} € IR : (}" - f)+2 n)' (1) Granted f e E, let 9(f) stand for the subset {x e [ : lf "@)l < *] of A. properties of the mapping We establish some f '--+ f "(')' Lemrna. For every f ,g € E and evely o € lR., the following hold: IR' (}" (1) f "(n): suP{tr € : - f)- 2 ',}; (2) ("f)"(n) : af "(x); (3) (f +s)"(r): f"(n)+ g"(r) fot alt xe 9(f)n9(g); Infinitesima,ls in Vector Lattices 179 (4) (/ n g)" (n) : min{/" (n), g"(r)} and (f v g)"(r) : max{/"(n), g"(r)}. < (1)' Denote the right side of the equality under proof by /-(:a). We consider only the case in which both /"(x) and f-(r) are finite. Let a > f"(r). Then (o"- /)+ > N, and so (oe - f)- Ax:0. consequently, a> f-(z), which implies f "(r) > f - (z), since the choice of the number a > f "(r) is arbitrary. Conversely, suppose that o > f-(r). Then (o"- f)- I N, and hence (o"- /)_ A ,4:0, because x is an indivisible element. At the same time, since e) %, we have ((" + tln)e - /)+ * (ae - f)- 2 (tln)e> x for all natural n. Hence (("+ lln)e- /)+ 2 u and a * Ifn> f"(r) for every a> f-(u) and n € N, which is possible oniy if f-(n)> f"(n). (2)' Omitting easy verification of the relation (af)"(r) : af "(n) with 0 < d ( €, we show only that (-/)"(r): -f"(r). Indeed, the required condition follows from the equalities (-/)" (r) : inf {) : (.\e - /)+ >_ z} : -sup{p, (-fre+/)+ 2 r}: -sup{p , (0e- f)-> x}: -f"(r). The last equality is valid by assertion (1) proven above. (3): Let x e 9(f) n 9(g) observe that the conditions (I" - /)+ > N and, @"- g)+ > Nimp|y ((,x + B)e - (/ +e;;* : ((.\e - /) + @" - g)+) 2 ((lr f) (0" -g))+ : ()" /)+ n (0"- Z x. - ^ - il+ The following inequality is easy from this remark: f"(n) + g"(r): inf{A: ()e - /)+ > ,€} +int{8, (0e- g)+2 r} 2 inf {7 , (te - (f + g))+ > ,} : (f + d"1n1. Replacing f bV - f and g by -g and applying (2), we obtain the reverse inequality. Thus we have f " (n) + g" (n) : (/ + g)" (n), as required. ( ): It suffices to prove that (/ A 9)" @) : min{/" (r), g"(r)}. Since ()r- (/^e))_ :t((f ((/-)e) n(g-,\r))+ ^s) -.\e)*: : (/ - k)+ n (g -.\e)a : ()" - /)_ (\" g)_, ^ - thecondition()e-(/ng))- ) xis validif andonlyif ()"-/)_ ) z-and (Ar - g)- > x. The required assertion follows now from (1). > 180 Chapter 4 4.2.2. Let n € A and e€ E be such that e 2 x. Consider the mapping h-: E -' lR assigning to each f e E the element "f"(:r) defined by (1). By the preceding lemma, the restriction of the mapping h- onto the vector sublattice E. :: {r € E : lh.(r)l < oo) of the lattice E is an lR-valued Riesz homomorphism on 8,. Let h be an arbitrary lR.-valued Riesz homomorphism on E. By the general saturation principle, there exists an elemenl x €. A satisfuing the condition x I r for every r e Ea whenever h(z) > 0. Take an arbitraty e €.81 with h(e) : 1. Clearly, h: h.. In other words, each real-valued Riesz homomorphism on the vec- tor iattice E may be represented as h-' 4.2.3. Take some maximal family ("o)ort of disjoint nonzero positive elements in a vector lattice ,8. Put oA': {Nelt: (3o e S) uleo} (2) and consider the family of subsets oA'': {Nelt: xSr) (reEs) of the set 0A, where ,Es is the union of the order intervals Io : l},eo] for all o € ,S. 0A. It is easy to see that {oA"}re1s is a base for some topology r on Throughout this section, we denote by (oA,r) the corresponding topological space. It is easy to verify that the condition f "(r):: inf {tr € lR : (\"o(.) - /)+ 2 ,} (3) soundly defines an lR-valued function on (0A,r). We now state the main result of the section: Theorem, [Jnd,er the above assumptions, the function f " belongs to C."(04) for every f e E. Moreover, the mapping assigning to each element f e E the func- n tion f is a Riesz isomorphism of the vector lattice E onto the vector sublattice f "(E) of the space C""(oA). { We show that the functions defined by (3) are continuous in the topology of 0A. Take an arbitrary f € E. If suffices to establish the continuity of the function subspaces 0A"' (o of 0A. Fix some o e. S and let e:: €o. Consider "f" on the e ^9) the sets P1 :: {% €oL': (,\e - /)+ } ,}, N1 :: {% eoL': (,\e - f)-2 r} Infinitesimals in Vector Lattices 181 for all ,\ e lR. Then {Pr}ren is an increasing family of open subsets of 0A", while {l[r].1.n is a decreasing family; moreover, P1fl N.\ : A f.or all ,\. In addition, since (r" - /)+ + (t" -/)+ > (s - t)e for arbitrary s,t € lR, s ) t, we have P" U lft :0A". Hence, oA" clPt I \ Nr q P" : intP" (" > ,). (4) By the definition of f", the following holds for all u e0l\": f "(r): inf {) € IR' : , E Px}. (5) 0A", It is easy to verify that conditions (4) and (5) imply continuity of "f" on as required. Now we show that functions of the form /" are finite on dense subsets of the space 04. As in the proof of continuity, we confine exposition to considering the functions on the subspaces oAt, where e is some element of the family ("o)ors. Thus, we must prove that the set Q(f) is dense in 0A" for every f e E. Take an arbitrary f € E (we may assume that / > 0) and suppose that an element u € 8,0 < u ( e, satisfies /"(n): x (x € oA"). Then u:0. Indeed, the condition (,\e - /)+ ) :a fails for all % eoL and u <-u. Since elements of the set 0A" are indivisible; therefore, oA', x A(),e- /)+:0 for all x € .\ e R.. (6) The set A is saturating, and so (6) implies that the elements u and ()e - /)+ of the lattice E are disjoint for all .\ e 1R.. Consequently, u A(e- (tl")f)+ : 0 (n e N). It follows that g, u Ae : Lr, Asup{(e - (tl")f)+ : n€ N} : because the lattice ,E is Archimedean. At the same time, u I e. Hence, u : 0. Thus, the set {.f" : oo} does not contain any nonempty open subset of the space A. is a Riesz homomorphism of the vector By Lemma 4.2.1, the mapping "f ,. f " lattice .E onto the vector sublattice f "(E) of the space C."(oA). To complete the proof of the theorem, it remains to establish that this mapping is injective. To this end, it suffices to verify that the conditions f € E+ and /" : g L82 Chapter 4 04. imply.f :0. Let an element f e E+ satisfy f"(n):0 for all x € Choose an arbitrary o € S. Then inf {) : (),eo - /)+ )- x} : g (n S eo), andso (f -(tl")eo),rtrx :0for alln € Nand x 1eo. Sincetheset Ais saturating, it follows that (f -(tl")"")+Aeo:g (n eN). The vector lattice E is Archimedean. Therefore, the last relation implies eo A f : €o Asup{(/ - (tln)e")1 : n e N} : suP{(/ - (Iln)e")+ A eo : n €N} : 0. was the element is disjoint from every Thus, since the choice of o € ^9 arbitrary, / element of the family ("o)ors, which is possible (since this family is maximal) only if / - 6. So, the mapping f - f" is injective. The proof of the theorem is complete. D 4.2.4. Define some equivalence relation I on A as follows: utQxz means that f"(n): fn(zt2) for every f e E.By Theorem4.l.4, A is a compact topological space. It follows immediately that the quotient space hs of A by Q is compacl too. This quotient space is Hausdorff by construction. Given x € lt, denote bV @) the coset oL xin the space ltg2. It is easy to see that the formula eff)((r)),: f"(n) U e n, x e L) (7) soundly defines the mapping p : E --+ C*(lys), where C*(lys) is the space of extended continuous functions on the compact Hausdorff space lvs. The following lemma is a consequence of Theorem 4.2.3 and the definition of tp. Lernma. The mapping g is a Riesz r'somorphr'sm of the vector lattice E onto the vector sublattice g@) of C*(Ls). Furthermore, g(E) separates points of ltse, and g maps the element e to the identically one function. 4.2.5. Let E be a relatively uniformly complete Archimedean vector lattice with a strong unity e. Then, by the preceding lemma, .E is Riesz isomorphic to the vector sublattice g@) of the space C(Ls) of continuous functions on the com- pact Hausdorff space l\g; moreover, g@) separates points of. lts7 and contains all constant functions. Since E is relatively uniformly complete, the sublattice g(E) is uniformly closed in C(hs). Applying the Stone theorem, we obtain 9@) : C(lvs). Thus, we have Infinitesimals in Vector Lattices 183 Theorem (S. Kakutani; M. G. Krein and S. G. Krein). For every relatively uniformly complete Archimedean vector lattice E with a strong unity e, there ex- ists a compact Hausdorff space Q such that E is Riesz isomorphic to the vector lattice C@). Moreover, such an isomorphism may be constructed so as to send the element e to the identically one function. 4.2.6. We also give a sketch of a nonstandard proof of the Ogasawara-Vulikh Theorem. Theorern (T. Ogasawara; B. Z. Vulikh). For every Dedekind complete vector lattice E with unity e, there is an extremally disconnected compact Hausdorfr. space Q such that E is Riesz isomorphic to an order dense ideal Et of the Dedekind complete vector lattice C*(Q). Moreover, Eome isomorphism may be constructed so that c@) e Et and the identically one function corrcsponds to e. ( Let E be a Dedekind complete vector lattice with unity e. Take Ivs *s the compact Hausdorff space Q. We first verify that l\sa is extremally disconnected. It suffices to show that the closure of the union of every family of sets in some base for the topology on Lss is open. Consider the base {Lh}*en+ of the topology of the space Asz constituted by the sets I\fo::{(r)elres: u 4.2.7. In conclusion, we show that in the case when .E is a lattice with the prin- cipal projection property, the equivalence relation I can be described in a simpler manner. Lernma. Suppose that a vector lattice E has the principal projection property. Then for every %L, L/2 € A we have x19x2 if and only if x1 and x2 have the same standard uppq bounds in E. < Assume that the elements 2ft and x2 have the same sets of standard upper bounds. Then x19x2 follows immediately from the definitton ol fl. Conversely, assumethatthesets{/ €E: f }-rt}and {f eE: f 2 nz} aredistinct. For instance, take r € Ea so that r ) x1 and r l rz. Then r A x : 0 because the element x is indivisible. Consider the band projection prr(e) of the unity e of the lattice ,E onto the principal band generated by r. It is easy to see that 0@) : 1 and i@) : 0. Consequently, (xr, xz) * n. > 4.3. Order, Relative Uniform Convergence, and the Archimedes Principle We now introduce some types of infinitesimal elements in a nonstandard en- largement of a vector lattice and use them for a nonstandard description of various kinds of convergence. Also we obtain a nonstandard criterion for a vector lattice to be Archimedean. 4.3.L. Let E be avector lattice. Given x €*E, we consider the setU(n):: {r e E: r} x} of standard upper bounds of. x and the set L(r) t: {Y e E: x2 y\ of standard lower bounds of. x. Define the following external subsets of some * nonstandard enlargement E of. the vector lattice E: fin(.E) ,: {n e*E: U(lxl) # s}, *E: a'pns(..E) :-- {x e in|(U(x) - f@)):0}, ' q(. E) ,: {, € * E : inf. U(lxl) : 0}, X(.E) ,: {n e*E t (fa e E)(Vn e N) In:al It is easy to see that fin(*-O), o-pns(.E), q(.8), and.\(*,8) are vector lattices with respect to the lattice operations, addition, and multiplication by scalars in IR. Infinitesimals in Vector Lattices 185 which are inherited from the standard vector lattice *-8. The elements of fin(.-E) we call finite or li,mi,ted; the elements of o-pns(.E) we call (o)-prenearstand,ard; the elements of n(E) we call (o)-i,nfini,tesimal; the elements of )(.8) we call (r)- i,nfinitesi,mal. The elements of E * q(. E) (of E + .\(.,E)) we call (o)-nearstandard, (respectively, (r) -nearstand,ard,). Note some simple properties: (1) .O is a vector sublattice in o-pns(*E), while o-pns(.-E) is a vector sublattice in fin(-^E); (2) n( E) is an ideal both in fin(.,E) and o-pns(.E); (3) Enq(.E): {0h (4) A(-E) is an ideal in fin(.E). 4.3.2. There are simple nonstandard conditions for a monotone net to be order convergent. Let (r")",ee be a decreasing or increasing net in a vector lattice E. Then the following axe equivalent: (1) the net (r..) converges to 0; (2) rB e r7( E) for all 0 e "E; (3) for some t rp €q(*E) I e"E. < Wu consider only the case of a decreasing net. (1)--+(2): Assume that ro J 0. Then every remote element I e "E satisfies no ) rB > 0 for all a € E. Consequently, inf6 U(I"BD: 0 and rp € q(. E). (2)-+(3): This is an immediate consequence of Lemma 4.0.5. oE (3)-*(1): Take an element B e for which rB €q(E). Since ro|, we have @d- 2 (2,)- ) 0 for all o € E. Inviewof rB €q(E), *" have (zo)-:0 for all o € E. Consequently, rol ) 0. Let U € E+ be an arbitrary element such that *E rol ) A. By the transfer principle, each c € satisfies ro ) A. In particular, rB )- 3r. This is possible only if U :0. Hence ro 10. tr It is easy to see that (1)--+(2) and (2)-(3) are true for an arbitrary (not necessarily monotone) net ("o)o.= e E. But the implication (3)-+(1) may be false without the monotonicity condition. Indeed, Iet E :: Lt [0, 1]. For each n € N, we take elements /f , fT,. . . , fT^ € ,E such that ff is the equivalence class containing the characteristic function of the interval lf,#l Arrange these elements in the sequence ! fl,f;, f?,f3,f?,f?, ..., fi,ftr,...,fl- Obviously (2) and (3) hold for this sequence, but it does not cenverge in order to any element of .8. 186 Chapter 4 4.9.3. We now give nonstandard conditions under which a monotone sequence converges relativelY uniformlY. Let (rn) be a decreasing or increasing sequence of elements in a vector lattice E. Then the following a're equivalent: (1) ,* 9 0; .N (2) r, e ),(* E) for everY z € \ N; .N (3) r, € )'(* E) for some z e \ N. { We verify only (2)-+(1) in the case of a decreasing sequence. Let t, € ,\(-,8) for some z e *N\N. It is clear that nn ) 0 for all n € N. Assume that the condition 9 0 is false. Then, for every d, e E, there is a number n(d') e N such that *n *N. d for all /c e N. By the transfer principle, n(d,)'rx I d for all ,h e "(d,).rn I In particular, n(d,) 'r, fd, contradicting to r, € ,\(.^E). So, r,, I O' > As in 4.3.2, observe that (1)---(2) and (2)-+(3) hold for every sequence (r^) g E. At the same time, the implication (S)--+(t) can be false without the monotonicity condition. This may be checked by considering the example in 4.3.2. Indeed, it is easy to see that the constructed sequence satisfies conditions (2) and (3) but not (1). 4.g.4. We now give nonstandard conditions for a vector lattice to be Archi- medean. We start with proving one auxiliary assertion. Lemma. Let u be an element of a vector lattice E and let u €.N \ N. Then either u:0 or uu (. o-pns(. E). { Let u * 0. Take arbitrary r e L(luul) and g e U(luul). Then n < luul < v' By the transfer principle, we find rn. € N such that r < lmul < y. Comparing this inequality with the previous, we obtain l"l S lrul - l*ul < A - t. Since the choice of the elements r in L(luul) and y in U(luul) is arbitrary, we have U(luul) - L(luul) ) l"l > 0. Thus, luul and, consequently, uu do not belong to a.pns(.E). > 4.g.5, Theorem. For every vector lattice E, the following arc equivalent: (1) E is an Archimedean vector lattice; (2) ^(.,E)nE:{o}; (3) 9rt(E); ^(.E) (4) ,\(-E) e o-pns(* E); (S) the set o-pns(* E) is a relatively uniformly closed vector sublattice of fin(-E); Infinitesimals in Vector Lattices 187 (6) rl(-E) is a relatively uniformly closed ideal of fin(-,O). { We will prove the theorem by the following scheme: (1) * (2) - (3) - (4) --+ (1) and (1) * (5) - (6) -- (1). (1)--+(2): This is obvious. (2)-+(3): Let x € I(.^U)\q(.E). Thenthereexists aA e Esuch thatlnxl L(rn- ene) 1 x 3U(nn* ene). (1) Givenn€N,assign En :: U(nn * ene) - L(rn - ene). The inclusion (nn) e a'pns(*.E) implies, by (1), that t# t": 2€ne (n e N). (2) Since E is Archimedean, it follows from (2) that inf s ULt En : 0, and hence inf.B(U(n) - f1u71 : g. We have used the inclusion ULr En e U@) - L(r) which ensues from (1). Thus, x € o-pns(.E). Consequently, o-pns(-E) is relatively uniformly closed in fin(-,8). (5)-'(6): Let xn e rt(E) and. zn !\ n e fin(.E). Then x € o-pns(-E), by hypothesis. Check that x e q(. E).We may assume that xn 9 , d-uniformly for some d e E. This means that lnn - xl l end for all rz € N and some appropriate sequence (ur) g IR., errf 0. Assume that x f ,l(E). Take an arbitrary element 188 Chapter 4 a € E satisfying U(lrD 2 a2 0. For each n, € N, choose an arbitrary u* eU(lx*l). It is obvious that un * end 2 lr"l + end 2 lxl. We thus have U (ln.l) * end e U (lnl) and U (lxnl) + e^d 2 a. Now it follows from inf6U(ln*D:0 that end2 a. This inequality is true for all the sequence (err) decreasing to zero. Therefore, d > ka f'ot every natural rz and *N. k e N. Applying the transfer principle, we obtain d 2 ka for all k e Take some z € *N \ N. It is easy to see that the sequence (ko)Pt of elements of .E converges d-uniformly to the element ua of. the vector lattice fin(.,O). Since, by hypothesis, apns(.E) is a relatively uniformly closed vector sublattice of fin(.E), we have ,i e *pns(-E). By Lemma 4.8.4, this implies a : 0. Thus, int BU(lul) : 0, and so u € n( E), as required' (6)-'(1): Take arbitrary u,,t) € E satisfying 0 ( nu 1u for all n € N. Show *N\N. that u : 0. Let lz € It is easy to see that a sequence (r') with nn:0 for all n, € N converges u-uniformly to an element uu' By hypothesis, the ideal 4(-'O) is relatively uniformly closed in fin(-E) i so t,'tl e rt(E). Therefore, by Lemma 4.3.4, we have u:0. > 4.g.G. Theorem. For a vector lattice E the following are equivalent: (1) E is an order separable Archimedean vector lattice in which or- der convergence and relative uniform convergence coincide for any sequence; (2) q(.8) : I(-E)' < (1)-*(2): .E satisfies the inclusion ,\(..E) e n(E) by Theorem 4.3.5. Prove the reverse inclusion. Take an arbitrary x €q(-E)-Then intsU(lxl) : O' Since (J :: U(lrD is a downwards-directed set such that t/f 0 and since E is an ordel separable vector lattice, there is a sequence (un) e U with un I0.The condition (1) .N implies un 9), g. Then, by 4.3.3, u, e ),(* E) for all z e \ N. Consequently, *N. zt e ),(* E), because lrl < un fot all n, € (2)*(l): .E is Archimedean by Theorem 4.3.5. Show that order convergence and relative uniform convergence coincide for every sequence in .8. It is sufficient to prove that un|0 implie, un 9 0. Take an arbitrary sequence (u,r) e E such *N that unJ 0. Then , by 4.3.2, u, e q(* E) for all y € \ N. Hence u, € )(.,8) for g' all u €*N \ N. Now we see from 4.3.3 that ,r, $ It remains to verify that the vector lattice -E is order separable. Take an ar- bitrary net (rg)ge E such that 26f 0. By Lemma 4.1.5, there exists remote = e *E' element r in the standard directed set Then, by 4.3-2, we have t, e r7(* E), and Infinitesimals in Vector Lattices 189 so nr € In this case, there is d e ,E satisfying nnr <. d for all n € N. Assume that (16)^(.^E). does not contain a subsequence convergent relatively uniformly Lo zero. Then there is a numblt rls € N such thal nsrq d, for all e E. By the transfer *E. I { principle, nofr€ I d for every € e The contradiction with n6r, I d ensures existence of a subsequence (*e^) e ("e ) with 26. lI O. Since .E is Archimedean, , (o) we have re^ 3 0. Thus, -E is order separable. tr 4.4. Conditional Completion and Atomicity In this section, we give a nonstandard construction of a Dedekind completion of an Archimedean vector lattice. Also, we give an infinitesimal interpretation for the property of a vector lattice to be atomic. 4.4.L. Let E be a vector lattice. Consider the quotient vector lattice E ,: o-pns(..E) lq( E) and denote by f'the mapping r ,- [n], where r € E and [r] e E is the coset containing r. Theorern. For every Archimedean vector lattice E, the following hold: (1) E it Dud"kind complete; (2) fi is a Riesz isomorphism of the vector lattice E into the vector lattice E; (3) for every r € E z:sup{e e0@): a In other words, the vector lattice E it u Dedekind completion of E. We prove the theorem in four steps: Step 1. Let E be an arbitrary vector lattice. Then, for every 0 < r e E, there exists an element e € E such that 0 < f'(e) g r. < Take an elementr € E,r) 0. Let x €o-pns(-E) be such that N>A and *: lr). Then there exists an e € E f.or which 0 < e { x. Indeed, in the other case, sup B L(u) : g. Consequently, infB U(r) : 0, since x € o-pns(*E). This contradicfs [zt]: * # 0. So, e is a sought element. D Step 2. Given r e E, assign C@) ': {s €O(E) : y ) r}; 9@ ,: {, €i@) : r }- z}. Then we have r: itf C@) sup g@). EE - Chapter 4 190 {Letx€olpns(.E)and'letu:[{fop'."*l:claimitsufficesto qt is equal to z' Take check that every element y e E satisfying epl ( g ( @) 0' Since the an arbitrary y e.0 such in* 91") ( y C(4 - 9@) )- lr - al >- fr(e) > o. (3) It is easy to see that iV@)) sfi(d andfi(L(xD s 9@) Now, the inequalitY (3) imPlies fi(u(,)-L(,))>i@)>0, to x € and consequently u(r) - L(r) 2 e > 0. we obtained a contradiction apns(.E). Hence, l" - al: 0, and a : r' > - subset D in step 3. Given an A.rchimedean vectot lattice E, evety nonempty fi(E) bounded' above has a least uppet bound in E ' subset I :: < Let D I fr@) is nonempty and bounded above. Then the all upper bounds fr t(D) of .E is bounded above in,o. Denote bv u(9) the set of of.9 in,E. Since -E is Archimedean, we have int(U(s) - e) : o' (4) * that Applying the general saturation principle, find an element 5 e E such e < 6 that infB(u(6) r!ol) : 0. Hence d e a'pns(*'E)' The From ( ) and Q), it follows - of the set D :i@).Show that [d] : suPiD' element [6] e ,0 i, upper bound "r, (5) we have Let y e -0 be some upper bound of D such that [d] 2 A.By 0<[6] -y(iV@)-e). (6) Therefore, (4) and According to step 1, the vector lattice f'(,8) is minorant \n fr' (6) imply y: [6]. Thus, [d] - tuPA D' > Infinitesimals in Vector Lattices 191 Step 4. Pnoop oF THE THponnu. { Assertion (2) is obvious. Assertion (3) is valid in view of Step 2. It is interesting to note that (2) and (3) hold in an arbitrary vector lattice. Verify the condition (1). Take an arbitrary nonempty subset A in -0 bounded above. Denote d :: {r e E : (3o € A)i@)( o}. According to Step 3, the set fi(d) has a least upper bound in E. Assign a :: sup6.i@). It is easy to see that a - sup6A. Thus, every nonempty subset in,E bounded above has a least upper bound, as required. > 4.4.2. The preceding theorem easily implies the following assertion to which we prefer to give a simpler and more direct proof: Theorern. For every Archimedean vector lattice E, the following are equiva- lent: (1) The vector lattice E is Dedekind complete; (2) o-pns(*,E) : E * q(. E). < (1)-*(2): Obviously, E +q(.8) e o-pns(.E). Show the reverse inclusion. Take an arbitrary % e o-pns(.E). Then x e fin(*E). So, U(x) is nonempty. Therefore, L(n) is bounded above. Since E is Dedekind complete, L(x) has a least upper bound. Assign o :: supg L(r). It is easy to see that L(x) 1 a I U(r). Hence l, - ol 3 U (r) - L(r). Since x € o-pns(. E), the last inequality implies that inf.BU(lx- al) : O. We have tc : a + (r- a) with a € E and x - a € q(.8). Consequently u e E + rt( E). (2)--r(1): It suffices to show that every net (ug)66e e E such that ue t < d e E is order convergent. Assume that u6J < d e E. It is well known (see, for example, [21, Theorem 22.5]) that the following condition holds in an Archimedean vector lattice ,E: i{{a-ue: €e E, ue E, uet<9}:0. (7) aE. Fix a remote element r e It is easy to see that {g €. E : u6t S a} : U(u"). Moreover, ("e) q L(u"). Thus, (7) implies inf6{t/(u,) - L("")): 0 or, in other words, ur € o-pns(.,B). Then u, € E +q(.8) Uy (Z). Let u € -E be such that 'ttrr -'u e q(.8).Thus, 4.3.2 implies that the net (uE) converges in order to u. tr 4.4.3. Now we consider the property of a vector lattice to be atomic. Recall that avector lattice Eis atomi,cif E isArchimedean and for every 0 < r e .E there exists an atom a € E such that 0 < a ( r. Also, we recall that, for every atom a in an Archimedean vector lattice and for each element 0 I r (a, there is a real o such that r: ea. We start with the following 4 r92 Chapter Letnrna. Let E be an atomic vector lattice. Then fin(.E) : o-pns(*E)' satisfies x € { It suffices to verify that every element N e fin(*E), ,2 0, apns(..E). Let u be anarbitrary positive element of fin(-E). Assume that u(x) - f(A'>'r ) 0. Then, by hypothesis, there exists an atom a € E such that U\rl f@) 2 a > 0. Take an element u e U(n). Since .E is an Archimedean - The element a is ,ru.to1. latiice, there exists a number n, € N for which na { u' a, g an atom; so we have u Ana: oo and xAna: Ba for appropriate e*[0,t]' Assign It :: st(0 - Ll}) 'o and LL' i: u - st(a - B - ll3)'a' Then u' e tJ (x) and I e L(x) where st is the taking of the standard part of a real' ' ' L(n)): 0, and so but u' -t' I a; a contradiction. Consequently, infs(U(n) - u e o-pns(.E). > 4.4.4.Theconditionfin(*E):o'pns(*E)isnotonlynecessarybutalsosuf- to introduce ficient for a vector lattice E to be atomic. To prove this, we need the concept of punch of a positive element of a vector lattice. Let E be a vector lattice and let e € Ea' An element x of Dnr,rutrtON. *E a nonstandard enlargement of..E is said to be an e-punchlf. (1) 03x3e; (2) inf6{Y€E:Y2x}:e; (3) supp{z €E: x2z):Q' We recall that an element e of the vector lattice ,E is called nonatornic if : 0 for any atom a € E. Below we will need the following easy f ef n a RpUeRX. For every nonatomic element e € E, e ) 0, and every natural 0 < ep I e number n, there is a family {"u}T:, e E of disjoint elements satisfying for all k:1,...,r1. Lemrna. Let E be an arbitrary Archimedean vector lattice. Then, for every y € *N and every nonatomic element e € E, e ) 0, there exisfs a family {"xlt:r,9 * E of disioint e-Punches. *N, in .E' { Take an arbitrary u e and let e 2 0 be some nonatomic element e > 0' De- Since the assertion of the lemma is obvious for e :0, we suppose that generated by e in E' note by .L the set of positive elements of the principal ideal -8" a hyper- It is olvious that .L is a lattice with zero. By Lemma 4.1.3, there exists in a nonstandard finite saturating family {rn}H:, of disjoint indivisible elements *L Applying enlargement of L. Ciearly, every n: L,...,u) satisfies 0 < xn ! e' proof, we can easily see the transfer principle and remark before the lemma under lnfinitesimals in Vector Lattices 193 *E that, for each n - 1, . ..,a, there exists a hyperfinite family {l*}"r!l C such that 0 < i* I xn (k: 1, ...,L/ + 1); l* nf,-: O (k * p). Applying the transfer principle once again and using the fact that the vector lat- tice E is Archimedean, we find a hyperfinite family {oh}i:r,i=, e *lR with the fol- lowing properties: (1) 0 xn for all n : l,...,u) and /c :1,...,u, < "hl*,1 (2) the condition o > af, implies that *lR. o.1*,{ unfor all n:1,..., tr, k:1,..., u, ar'rd a € Put e;, :: V|=r oh| for all k: I,...,u. It is easy to verify that {e3}[:r is the desired family of z disjoint e-punches. D 4.4.5. Theorern. For every vector lattice E, the following are equivalent: (1) E is an atomic vector lattice; (2) fin(.8) : o-Pns(.8). < (1)'-+(2): This is established in Lemma 4.4.3. (2)+(1): Let fin(..E) : *pttt(.E). In particular, o-pns(..p) is an (r)-closed vector sublattice of fin(..E). Therefore, E is Archimedean by Theorem 4.3.5. Ver- ify that E is atomic. It is sufficient to show that .E has no nonzero nonatomic elements. Take an arbitrary nonatomic element e € E. We.may suppose that *-8. e 2 0. By Lemma 4.4.4, there exists an e-punch x € The element :a satisfies inf.B(U(x) - t'1u11 : At the same time, :a is finite and, by hypothesis, x is an (o)-prenearstandard ".element of *-8. So, e:0. > 4.4.6. As an application of the last theorem, we establish some useful non- standard criterion for a vector lattice to be atomic and Dedekind complete. Theorern. For every vector lattice E, the following are equivalent: (L) E is a Dedekind complete atomic vector lattice; (2) fin(.-E):E+n(E). { Observe that E +q(*E) e o-pns(.E) q fin(-E) and use Theorems 4.4.5 and 4.4.2. > We point out that, in the proof of Theorem 4.4.5, we used only the fact that there exists one e-punch for a nonatomic element e e Ea. We will need the Lemm a 4.4.4 in full strength below, in the proof of a criterion for a vector lattice E to be isomorphic to the order hull of ,8. 194 Chapter 4 4.5. Normed Vector Lattices In this section, we consider normed vector lattices and study some of their infi.n- itesimal interpretations. Throughout the section we assume (E,p) fo be a normed vector lattice. 4.5.L. It is well known that, in a nonstandard enlargement of. E, together with fin(.E), a.pns(*E) , n( E), and ,\(*E), we may also consider the following subsets: Fin(..B),:{re*E: p(z) e fin(.R)}; pns(.E) ,: {, €*E: (Vn e N)(ly e E) np(x - s) S th p(.8),: {, e*E: p(x) = 0}. It is easy to see that these are vector lattices over lR under the operations inherited from *E. Furthermore, pns(*E) is a vector sublattice of Fin(.E), while p(-E) is an ideal in pns(*E) as well as in Fin(.E). 4.5.2. Let E be a vector lattice. If there exists a strong unity e €. E then we may introduce the Riesz norm ll ' ll" o" E by the well known formula llrll" ': inf{,\ € IR : lrl < ,\e} (r e E). We prove the next Theorem. Let (8, ll ' ll) b. a normed vector lattice. Then the following are equivalent: (1) -E possesses a strong unity e, and the norm ll'll" i" equivalent to ll'll; (2) Fin(. E) : fin(*E); (3) tt( E) e fin(.,E); (4) P(. E): )(.E); (5) p(.8) e q(.8); (6) Fin(.E) : fin(*E) + p(.8)' { First of all, we prove the equivalence of conditions (2)-(5). To this end, it suffices to show that (2) -* (3) -' (4) -' (5) -* (3) and (4)-'(2). The implications (2)-'(3), (a)+(5), and (5)--(3) do not require checking. (B)-'(a): Let p,(*E) e fin(..E). To prove the implication, it is sufficient to verify iheinclusion 1t(.8) g I(.E'). Take an arbitrary x € p'(.8).Then llaxllx ,-, 0 with o: llrll-L/z, and consequently au e p(.8) e fin(.^E). Thus, the2-is' Infinitesimals in Vector Lattices 195 an element A € E for which lonl < y. Then lnnl for y € Eu and ra € N. Since lrl{(n+l)y for every y € E+ and every n € N, we have-n+ 1 € Af,, and all these sets are nonempty. The family {A?}1.'}* possesses the finite intersection property since Aiff{",*} gA;nAT. By the general saturation principle, there is some r € *lR. satisfying r en{A;: y € E'r, n € N}. Then r is an infinite positive number such that lrl { ry for all y € .8-.,.. However, (Llr)x e p(.8), since ll(tlr)ull : Ifr x A. By hypothesis, p(.8): tr(*E'), therefore, l(tlr)xl I z for some z € E'4, and so lrl < rz, which contradictsr e A!. Thus, the equivalence of conditions (2)-(5) is established. In order to finish the proof, we show that (1) -* (2) -* (6) --+ (1). The impli cations (1)-+(2) and (2)--+(6) are obviously true. (6)-r(1): Let Fin(.8): fin(.E) + p(.8). First, we prove that the unit ball B::{reE:ll"ll the contrary. Take an arbitrary r e.81. There is a y e E+ such that llyll :1and ylr.Consider z:U-yAc. Then 0 tz An 1y (t e R-.). (8) Lett € lR+. We represent r as (r -rAy)+(rAg) and assign u,:tz,u: r-rAA, and'tr.' : r AA. It is clear that u, u,u) €.Eu and tz Ar: u" A(a +u). The easy relations uA(u+.) -uAu1ut, u A (u + r) - u Au 1u A (u * w) < u imply the inequality uA(u *tr,r) < uAu*uAw. (e) 196 Chapter 4 The elements u : tz and'.) : r - r Ay ate disjoint because : z Au : (A - r Ay)A (r - r AA) : Y At - nAg 0. Hence, bY (9)' we have tz A r : u A(u + tu) ( r.r.r A u -- tzA r A y 3 y. Inequality (S) is proven. Consider the element s : (Zlllzll)z of the vector lattice -8. It is clear that llsll :2. Since sAr ly,we have llsAzll < llyll :1' Consequently, the internal set *Ea A,;: {s € : llsll :2 k s Ar € B} is nonempty for all r € Ea. Since Anv! g A, fr Ao (r,a e ,8..), the family { Ar}*E+ possesses the finite intersection property. By the general saturation i * : principle, there exists Ao € E+ such that ' Ys€n{A":t€E+}' l : In particular, gt6 e Fin(.E). By assumption (6), yo e : ' It is clear that llyell 2. ; fr.r(.p) + tt(E). Therefore, there are elements u6 € X1 and hQ p(.8), for which ',; n At the same time, : 2 and uo 3 ro I h. obviously, llyo "oll = llgoll. llyoll The obtained contradiction shows that the unit ball : llyo n*oll < 1, since Uo € Aro. i of E is order bounded. : Choose ane€Esothat l*l 3" forall r€8. Then i lrl S llrll"'e (r e E). This implies that e is a strong order unity of E. Moreover, ll"ll" 3 ll"ll @ e E)- At the same time, ll"ll < cllrll" @ e E) for c : ll"ll-t. Consequently, the norms . . (6)--+(1) is established. The proof of ll ll" and ll ll are equivalent. The implication the theorem is comPlete. tr 4.b.8. Now we give a nonstandard condition for a norm to be order continuous. Theorern. The norm p of a normed vector lattice (8, p) is order continuous if and only if q(. E) e p(. E). ' SinceU(lxl) is directed downwards and inf BU(lNl):0, order continuity of the norm ' p imPlies inf {p(u) : u €u(l:rl)} :0. Infinitesimals in Vector Lattices 197 Then p(r<) = 0. Since the choice of. u € rl( E) is arbitrary, it follows that q(. E) g p(. E). Now, Ietq(.8) e p(.8).Assume that p is not order continuous. In this case there are a net (zg)g6o I E, rg 10, and a number 0 < a € IR such that p(r) >_ a for all € e O. Take some remote element 0 e "@. Then, by 4.3.2, xB € q(E). Thus p(rfi x 0. On the other hand, by the transfer principle, p(r) 2 a for all .O. € e The contradiction shows that the norm p is order continuous. D As an example of applying Theorem 4.5.2, we propose a nonstandard proof for the following well-known assertion: Let a Banach lattice E have an order continuous norm. Then E is order separable and Dedekind complete. Moreover, order convergence in E coincides with relative uniform convergence. < In view of 4.4.2 and 4.3.6, it suffices to verify the relations apns(*,B) g E + q(. E) and 4(.,8) g )(.^E). Let x € o-pns(*.E). Using the fact that the norm is order continuous, it is easy to see that zt e pns(*E). According to Proposition 4.0.7, the Banach lattice (E,p) satisfies pns(.E) : E* p(.8).So, there is an r e E such that %-n e p(-E). Obviously, L(n) < r 3U(r).Since x € o-pns(.E), we have % - r €q(.8).Thus, x€E+q(.8). Verify that r1(E) g )(-E). Let x e q(.8). Then U(lul) I O. By order continuity of. p, for every n € N, there is an un € U (lnD with p(un) < 2-^ . The sum u ,: Dp, u' exists in the Banach lattice E. Obviously, lnxl ( u for all n € N. Thus x e ),(. E). > 4.5.4. Concluding this section, we will establish a nonstandard criterion for a normed vector lattice to be finite'dimensional. Theorern. A normed vector lattice (8, p) is finitedimensiona,l. if and only il q(. E) : p,(* E). { Necessity is obvious. To prove sufficiency, we let ,l(E): p,(.8).By The. orem 4.5.2, E possesses a strong unity e. Moreover, the norm ll . ll" is equivalent to the initial norm p. According to Theorem 4.5.3, p is order continuous. Con- sequently, ll .ll, is order continuous. Next, by Theorem 4.5.2, q(.8): )(.8). Applv'ng Theorem 4.3.6, we conclude that ,E is order separable. Assume that dim E : oo. It is easy to see that in this case there exists an infinite disjoint order basis Ae E+ such that a€ A implies lloll": ]. Since E is order separable, the set A is at most countable, because it is order bounded in E by some element e. Thus, we may suppose A: {o,.}T:t. For each natural n, 198 Chapter 4 define the element Un:: e - ("Ioo) n" . It is easy to see thal un | 0. Since the norm ll ll. it order continuous, it follows that ll,rrll" --+ 0. On the other hand, by the construction of the sequence (un) we have ll',r'll" 2 lla,+rll" : 1; a contradiction' Hence, dim'E < oo' > 4.6. Linear Operators Between Vector Lattices In this section, we establish nonstandard criteria for linear operators in vector lattices to be order continuous and order bounded. These criteria are similar to those in 4.0.9. Below, the symbols .E and F denote some vector lattices and 7 : E - F is a linear operator. 4.6.L. We first prove one useful auxiliary assertion (see also 4.0.8): Lernrna. For every nonempty subsef D of a vector lattice E, the following are equivalent: (1) D is order bounded; (2) *D g fin(-.E). < We need to prove only the implication (2)-'(1) . Let * D e fin(-^O). Assume that D is not contained in any order interval. Then, for every u € Ea- there is du e D satisfying (d," u)+ > 0. By the general saturation principle, there exists - .D\fin(-E). some de*D suchthat (d-")+ ) 0forallu €.81. Thendsatisfiesde This contradiction with *D e fin(-E) shows that D is order bounded. > 4.6.2. Theorern. Let E and F be vector lattices, and let T : E --, F be a linear operator. Then the following arc equivalent: (1) is an order bounded operator; " (2) .?(fin(.E)) e fin(.F'); (3) .?(l('.^E)) e )(.r); (4) ."(^(.,8)) e fin(.r). < (1)-'(2): Obvious. *T(fin(-E)) (2)-+(3): Let e fin(-F). Take an arbitrary % e ,\(-E). Then, for some de E, the conditionlnxl ( d holds for all n, € N simultaneously. It is easy *N to see that in this case there is a y € \ N such that luVl < d. ConsequentlS ux € fin(.,E) and, by hypothesis, u*T(x) : *T(vn) e fin(.F). This implies Infinitesimals in Vector Lattices 199 *Tyt € l(.tr.). The latter means that condition (3) is valid, since the choice of the element x € X.E) was arbitrarY. (3)-*( Obvious. ): *Tx (a)-.(2): Assume that f fin(-F) for some x € fin(* E). For every rz € N and every I e F, assign An,f i:{k e -N : k 2 n k (l.T(k-t r)l - /)+ > 0}. The sets An,1 are nonempty by the choice of. x. By construction, they are internal, comprising a system with the finite intersection property since Amax(n,p),.up("f,g) C An,f A Ao,s for arbitrary n,p € N and € F. Applying the general saturation principle, f ,g *N\N, we find a u € It is obvious that u € and so lu-Lxl € ,\(.,8). )n,yAn,y.-7(.\(.^E)) By assumption, e fin(.f'). Therefore, there is a y € F such that (l.T(u-rx)l -y)+:0, which is impossible since u € Ay,o. The obtained contra- diction means that .?(fin(-E)) e fin(.F). (2)+(1): Take an arbitrary u €. 81. By condition (2), .(?([-r,"])) : *T(*l-r,ul) q fin(-F)' Hence, by Lemma 4.6.1, the set ?([-u,u]) is order bounded. > 4.6.3. Before stating a nonstandard criterion for a linear operator to be order continuous, we find a connection between (o)-infinitesimal elements of an Archime- dean vector lattice F and those of a Dedekind compietion of F. Lernlrrra. Let F be an Archimedean vector lattice and let Fy be a Dedekind *F completion of F. Then q(. F): n 4(-F'1). < Given x €*Fy, assign Ur(n),: {* € F: r) x}, Urr(r),: {* € .F1 : r> %}. Let x € ?(-F). Then inJp(Jp(lxl) : O and, since Fr is a Dedekind completion of F, we have infp, Up(l ,l) :0. Furthermore, Ur(lrD e Upr(lrD, which implies inf.pr(Jpr(lrD:0. Hence, u eq(*Fl). At the same time, x e*F. ConsequentlS x €*F n q(.Fr). Conversely, let x €*F n t7(.F1). Then infp' [/r, (lnD:0. Since .t[ is a Dedekind completion of F, it is easy to verify that infp, Ue(lrD: 0' This immediately implies inf pUp(lxl) : O. Thus, u e 11( F). > Chapter 4 4.6.4. Theoretn. Let E and F be Archimedean vector lattices, let Ft be a Dedekind completion of F, and let T : E "'+ F be a linear operator. Then tl'e following are equivalent: (1) is an order continuous opetator; " (2) .T(q(. E)) e q(-rr); (3) .T(q(- E)) e '?(.F). < (1)---i(2): Let ? be an order continuous operator in L(E,F). Then it is easy to veriff that ? is an order continuous operator in L(8,F1). Since Fr is Dedekind complete; l"l is defined, presenting an order continuous operator from E into Fr. In view of the inequality lT"l < lf l(l"l) @ e E), to verify the required impli- cation it is sufficient to show .lTl(q(E)) I 'r(.Fr). Take an arbitrary x €rl(E).Then, since l"l is order continuous, infsU(lxl):0 implies infp, l7l(U(lr): 0' But lrl(u (lxl)) s u(l." @)0, *Tx so infp, Ufff fl: 0 and, consequently, € 4(.,111). (2)+(3): This follows readily from Lemma 4.6.3 since,.T(.8) q *f'. (3)-'(1): Let.T(q(-E)) g ?(.F'). Since ,E is Archimedean, I(.8) e ri(.E) by Theorem 4.3.5. Consequentl5 ."(^(-E)) e rl(.F) e fin(-F). By Theorem 4.6.2,, this implies ? € Lr(E,F). To verify order continuity, it remains to prove that inf p lT"el : 0 for every net rg l0 in ,8. Take an arbitrary net ("g)e.= e .E such that rg J 0. Assume that, for some element f e F, .f > 0' the condition l?zgl > / holds for all € e E simultaneously. Then, by the transfer *E. principle, lT"el > / for all { e Let B be some remote element of the directed set *E element exists by Lemma 4.0.5). According to the criterion established (such an *T*B in 4.3.2, we have zB e q(.F). Then, by condition (3), e n(F), which contradictsl.TrBl > /. Thus, inf lf"el:0 for every net (16) decreasing to zero, and so the operator 7 is order continuous. F Infinitesimals in Vector Lattices 201 4.7. x-Invariant llomomorphisms One of the important facts of nonstandard analysis is the assertion that each limited internal real number o € fin(-lR) is infinitely close to a unique standard real number st(o) called the standard part of a. The operation st of taking the standard part of a real is a Riesz homomorphism of the external vector space fin(*lR) into jR. such that st(a) : a for all a € lR. and st(o1) : st(oz) whenever 04 N o2. This leads to the problem whether or not we may take the standard part of an element in . nonstandard enlargement of a vector lattice or a Boolean algebra. In other words: What conditions will guarantee existence of a Riesz or Boolean homomor- phism keeping standard elements invariant and not distinguishing infinitely close elements? In this section, we discuss this question for nonstandard enlargements of vector lattices and Boolean algebras and establish that such an inuariant ho- momorphism exists if and only if the vector lattice (Boolean algebra) in question is Dedekind complete (complete). In the end, we consider the structure of invari- ant homomorphisms on nonstandard enlargements of complete normed Boolean algebras and establish that, for atomless complete normed Boolean algebras, every invariant homomorphism is almost singular with respect to the measure, in the sense that the carrier of the homomorphism is contained in an internal set whose measure is arbitrarily small but nonzero. The considerations in this section rest on [10]. ..8 4.7.L. Let E be a vector lattice and let be a nonstandard enlargement of E. *E. We may assume that .E is a vector sublattice of DnrrNIttoN. A mapping $ : fin(*E) - E is called a *-i,nuariant R'iesz homo- morph'i,sm if / is a Riesz homomorphism such that ',/,@) : r f.or r e E. Henceforth we abbreviate a *-invariant Riesz homomorphism to a *-IRH. It is easy to see that the inequalities sup{r € E : r < x} S rltQ) S iBf{y € E : y 2. x} (10) E are valid for every *-IRH tlt and every y € fin(*E) provided that the supremum and infimum exist on the right and left sides. In particular this implies @-il €q(E) -rb(x):, (r e E,1e fin(-E)). (1 1) Theoretn. Let E be a vector lattice. There exists a *-invariant Riesz homo- morphism,,h on fin(.E) if and only if the vector lattice E is Dedekind complete. If E is atomic and Dedekind complete, then a *-IRH on fin(*E) is uniquely defined by lhT): sup{r € E:r sup{z €B:rSF} for all F e *B provided that the supremum on the left side and the infimum on the right side both exist' This implies in particular that h(B) : 0 for every element *B F e such that inf 6{b e B : b 2 0} :0. *B Theorem. There exists a *-invariant Boolean homomorphism B if and *B -+ only if the Boolean algebra B is complete. Moreover, a *-IBH h : --+ B is defined uniquety if the complete Boolean algebra B is atomic. In this case h(P) --sup{z € B : * < P} : tBt{, € B : Y > B} B for all 0 e*8. Infinitesimals in Vector Lattices 203 Before proving the theorem, we present one nonstandard characterization of an atomic complete Boolean algebra which is due to H. Conshor. For this we need some notations. Let B be a Boolean algebra. Given x e. * B, consider the set U(n) ,: {* € B : r 2 u} of standard upper bounds of. x and the set L(n) ,: {A Q B: u > y) of standard lower bounds ol x. Define the external subsets of.*B: o-pns(*B) ,- {, e*B: inf(U(z) - t'(r)):0}, q(. B) t: {n e * B : inf U(lztl) : 0}. *.B, It can be shown that o.pns(-B) is a Boolean subalgebra of and q(. B) is an ideal in o-pns(*B) and the quotient o-pns(*B) lq( B) is Boolean isomorphic to B (see [4, Theorem 4.1]). From this and from [4, Theorem 4.3] we have immediately the next Lernrna (H. Conshor). For every Boolean algebra B, the following are equiv- alent: (1) B is an atomic complete Boolean algebra; (2) *B: B +q(.8). Pnoop oF THE TttpoRnu: { Let B be a complete Boolean algebra. We apply Sikorski's Extension Theo. rem (see, for example, [25, Theorem 33.1]) to the triple (8,*B,B) and the identical Boolean homomorphism z : B -. B. We then obtain a Boolean homomorphism -B). h : *B --+ B that extends z. Obviously, h is a *-IBH on We now show that the existence of a +-IBH h : *B --+ B implies completeness *B of B. Let h: -+ B be a 'I-IBH. Take an upwards-directed nonempty set I g B. By the general saturation principle, there exists an element d in * I satisfying d > d for all d e 9. Then, as is easy to verify, h(d) : sup.a 9. Since the upwards-directed nonempty set I e B was chosen arbitrarily, this implies that B is a complete Boolean algebra. Let E be atomic and Dedekind complete. Then the proof of uniqueness of *B a *-IBH h : ---+ B is similar to the proof of uniqueness of a 'r-IRH in 4.7.1. We must use the preceding lemma instead of Theorem 4.4.6 only. D Note that uniqueness of a *-IBH on a complete Boolean algebra B implies that B is atomic. For a proof of this assertion we refer the reader to [10, Theorem 1.1]. 4.7.3. Let B be a complete Boolean algebra. For convenience, given a family (o") g B,we denote supror by @"a" whenever the elements ar are disjoint. A partiti.on of anelement b e B isafamily (b") G B suchihat b: @rb,. Let p: B * IR+ be a mapping on B satisfying the following conditions: Chapter 4 (1) p(b)>0++b>0; (2) The equality p(o;:r a?,) : DL, p@.) is valid for every sequence ar,a2,... of disjoint elements of B' measure and Recall that a mapping p with the above properties is a o-additi'ue normed Boolean algebra' the ^rctpair (B,ti ts a cornplete *B be a complete normed Boolean algebra and let h : -+ B be ia,'p) B a *-IBH. Represent B as a direct sum of atomic and atomless components: - B"@8". Then (8", p) and (8", p,) arecomplete normed Booiean algebras' The re' ,triJioo of h to *Bo is a *-invariant Boolean homomorphism preserving the measure *Bo. of lz to p in the sense thal p(h(o)) : st(-p(o)) for all a € The restriction following *8" with respect to the *""rur" p has an opposite behavior. Namely, the holds: Theorern. Let (8, ti be an atomless complete normed Boolean algebta and for every real e > 0, Iet h : *B --+ B be a *-invariant Boolean homomorphism. Then, h(b AX€) fot all there exists X" €*8,*t(X) 1e, satisfying the conditionh(b): b e*8. Before proving the theorem, we establish one elementary property of atomless complete normed Boolean algebras' Letnma. For every atomless complete normed Boolean algebra (8, p') and * p(nA d) : every natural n, there exists a partition (Xt)T=t e B of L'e such that *p@) for aII d e B, i:1, " ' ,Tl" { Take an arbitrary hyperfinite partition (e7,)[:1 of 1'6 in the Boolean al- gebra *B which is refined into each finite standard partition. Existence of such a and partition is easy on using the general saturation principle. since B is atomless that the measure p is o-additive, there exist partitions ek : @T=, ef such . p@f) : l. u@o) *B for all k : L,... ,t/ and i - 1,. ...n. Put yi ': @il=r ef . ttre family (X)T=r 9 is a required Partition of unitY. tr Pnoop oF THE TgPoRPIt't: { Take an n € N such that }A(1) < e. According to the preceding lemma, there exists a partition (Xi)T=t g*B of 1"6 satisfying the condition * d) : rr(xnn lu@) for arbitrary d, e B, i : 1, . . ., n' In particular, * t (xon h(x-)) : L P(h(x^D Infinitesimals in Vector Lattices 205 for k, m e I,... ,T1,. Consider the element Xe t:(b ,- n h(xn). k=1 Then Ttl1fL *p(x): *p(xn n h(xr)) : )p(h(xi) : I )--p(n(x*)) tHrL .fl- 4 : k=I /c=1 :|'(0r'(xo)) : : : < r. \ lc:1 *,(,(9"-)) |u@$.")) lu!u) At the same time h(L-n \ x") : h(P,-$ xn nn(Y*1) :6Onkonh(x-)) :0, rn:L k*m since h(xo nh(xd): h(xx) ^h'(x*) : h(xx) A h(x,) : h(Xn A x,;: h(0) : 0 for k I rn. Thus, h(b) : h(b n1.a) : h((b n x,) o (bA (1.6 \ x,))) : h(b x,) o (h(b) A h(t-B \ x,)) -- h(b A X,) ^ for all b e*8. > 4.7.4. Consider the real-valued mappings sto*p and poh defined on a Boolean *,B, *;.1 algebra where h is a 'r-IBH. Obviously the mappings st o and p, o h are finitely additive measures on *.B. Moreover, these mappings are o-additive, since *B the condition b : @L, b' imposed on elements of implies b : @T=rbn f.or some rn € N. Thus, sto*;,l and poh extend to o-additive measures p and *Bo *.B. 1t1, on the o-completion of the Boolean algebra Observe that 1t is the Loeb measure corresponding to the initial measure p. Theorern. Let (B, p) be an atomless complete normed Boolean algebra and Iet h : *B - B be a *-invariant Boolean homomorphism. Then the following hold: Chapter 4 *Bo, an element e p(xn) : 0, such that the equality (1) Thereexists xn *B h(b) :0 holds for every b e satisfying the condition b A Xru : 0; (2) The carriers of the me&Sules 1l' and p,7, are disjoint. *B < (1), By Theorem 4.7.3, for every n € N, there exists an element Xn e suchthat ir1*)i j h@):otorali be*B,bAxn:0. Put Xh: h,@-rx^' *Bo""a b *B' b Axn:0' It is clear that xi tt and' fu(y') :0' Take an arbitrary e :0, as required' Then there exists an n € N such that b AXn:0. Thus, h(b) p, arc disjoint (2) ensues from (1). Indeed, the carriers of the measures and ir,6 erements Xn and " ;:* ::::ilJ1i"-1"" ?*,*o In this section, we define the order hull of a vector lattice. Some properties (r)- and (o)- of order hulls are established. In particular, the question about their found for the order completeness is studied to some extent. some conditions are lattice -E and (if E hull of a vector lattice -E to be isomorphic with the initial vector the nonstandard is a normed lattice) for the order hutl of ,E to be isomorphic with hull of ,E regarded as a normed vector space' of 4.g.1. Let E be a vector lattice. As it was mentioned in 4.3.1, the set fin(-E) (o)-infinitesimal limited elements in* E is a vector lattice too, while the set q(. E) of *.8 lattice elements in is an ideal in fin(.E). Consider the quotient vector (o)-E:: fin(*E) ln( E)' (o)-E we call (o)-E the ord,er hull of. E and, denote bv lnl the coset x*rl(.8) e Define the mapping E "-+ (o)-E by that contains x € fin(.E). 0r " in@) '- ["] (r e E). bv if this clearly, in , E -+ (o)-E is a Riesz homomorphism. It will be denoted 0 does not lead to ambiguitY. (o)-E- 4.g.2. Theorem. The set fi(E) is a complete vector sublattice of of Before proving, we give some explanations. Let L be a vector sublattice of M if, for every a vector lattice M. Recall that .L is a com,plete aector sublattice a' nonempty D 9 L and every a € L, the condition inf; D: a implies inf vD: only if, for every It is easy to see that L is a complete vector sublattice of' M if and nonemptyDgL,theconditioninf;D:0impliesinfTaD:A' { Let D g E such that inf6D:0. show that inflo>Tfi(D):0. Assume the contrary. Then for some x e fin(* E) we have fr(D)>lnl>0. Infinitesimals in Vector Lattices Since , # rl( E), there is an a € E such that U(n))a)0. Take an arbitrary d e D. Then i@) > fx] and, consequently, (x - d)+ e rt( E). Thus, inf sqU :0, where 8/:: U((, - d)+). Given u € Q/, note d+u>d+(x-d)a2x. Hence, d+% 9U(r), and so d+% 2 a. Therefore, *-) inf G -t %\ > a. - E Since the last inequality is valid for all d e D and inf6 D:0, we have a: 0. inf A contradiction shows that 1o)_E0@) - 0. > 4.8.3. Theorern. The order huII of a vector lattice E is Archimedean if and only if E is Archimedean. { Necessity follows from the fact that each vector lattice may be embedded as a vector sublattice into its order hull. To prove sufficiency, we consider an Archi- medean vector lattice .8. By Theorem 4.3.5, ,l( E) is a relatively uniform closed ideal in fin(.,B). Then, according to the well-known theorem by A. I. Vekster [26] (see also [21, Theorem 60.2]), the quotient (o)-,8 is Archimedean. D 4.8.4. Theorem. The order hull of a vector lattice is relatively uniformly complete. { Let E be a vector lattice. Since every quotient vector lattice of a relatively uniformly complete vector lattice is relatively uniform complete too (see, for exam- ple, [21, Theorem 59.4]), it is enough to establish relative uniform completeness of fin(.E). Take a relatively uniformly Cauchy sequence (n*)T=, e fin(..E). Then, there exist a sequence (rr) g R, €r | 0, and an element d e fin(*,B) such that lnrn - xnl 3 e,,,6 for all m,k, n € N whenever m,k ) n. We extend (z"r)",eN to an internal sequence (."r)",a.* 9*E and associate with each natural /c the internal set: *N Ip:: {* e : lra- - ,rl < e,n|}. It is easy to see that the family UrlE, has the finite intersection property. By the general saturation principle, there is a u e flt-I;.. Then every k e N satisfies lnt - x) < e1d. This implies x, € fin(*E) and ,n 9 %u. D Chapter 4 4.g.b. The matter with Dedekind completeness of the order hull of a lattice differs from that with relative uniform completeness. We show that the order hull of a Dedekind complete vector lattice containing nonatomic elements is not necessarily Dedekind complete (Theorem 4.8.7 establishes that the order hull of an atomic Dedekind complete vector lattice is Dedekind complete). Recall that a Dedekind complete vector lattice E is called regular (see, for example, [29]) if the following hold: (1) Order convergence and relative uniform convergence coincide for every sequence in E; (Z) Each ideal with a countable order basis in -E is contained in some princiPal ideal; (3) E is order seParable. As examples of regular vector lattices, we may take Banach lattices with order continuous norm and .Lo([O,1]) with 0 < p < 1' Theorern. The order hull of a nonatomic regular vector lattice is not Dedekind complete. lattice. Then there is a nonator ric { Let E be a nonatomic regular-N\N vector element e € E, e ) 0. Letu € be some illimited natural number. By Lemm a 4.4.4, there exists a family {"n|!n:, of disjoint e-punches. Assign D :-- {[rr]];1r. Then D is a nonempty and bounded above (for example, by the eleme"f Irt) subset of (o)-8. We show that this subset fails to have a least upper bound in (o)-E. By way of contradiction, assume lrl: sup D for some x efrn(*E)' (o)-E Then for all lc € N, we have ("r - %)+ e q(. E). By Theorem 4.3.6, ,E has the property q(-E): )(-E) (here properties (1) and (3) from the definition of a regular vector lattice are used). Hence, ("r - %)+ € l('.8) holds for all ,k e N. By using (2), it is easy to see that there exists a d e E fot which ("r- %)+ 3m-!d (k,* € N). *N Applying the general saturation principle, we find c.r,7 € \ N such that u 1 uand (e, - %)+ < ^Y-td^ Infinitesimals in Vector Lattices Then (", - %)+ e X-E) : q( E) and, consequently, [rr] < [x]. At the same time, [",] > 0 (because w I u while e, is an e-punch) and [tr]n [t,'] : 0 for all ,b e N (since e, and ek are disjoint). Hence lrl > lnl - 1",) > l"rl for every k € N, which contradicts the assumption lx] : sup(o)-ED. Thus, the order hull (o)-E of E is not Dedekind complete. tr 4.8.6. We establish one more property of order hulls concerning cardinality. We denote by card(,A) the cardinality of a set .4. Letnrna. Assume that a vector lattice E is not Archimedean or not atomic. Then card(E) < card((o)-E). -N\N. { Take an arbitrary u e According to Lemma 4.0.4, card(E) < card(z). Therefore it is sufficient to establish that the order hull of ,U contains z distinct elements. Assume first that ,O is not Archimedean. Then there are elements u, u € E such that 0 < nu ( u for all natural n. By the transfer principle, 0 < nu ( u holds for all n € *N. In particular,nu € fin(.,E) for all n € *N. The inequality 0 4.8.7. Theorern. For every vector lattice E, the following are equivalent: (1) E is Riesz isomorphic to (o)-E; (2) E is an atomic Dedekind complete vector lattice; 4 2n Chapter (e)feisaRieszisomorphismofEontotheorderhul]ofE. by Lemma 4'8'6' < (1)-+(2): The vector lattice.E is Archimedean and atomic Then' (o)-E:9-pT.(l E)lq(E)' and So, fin(*,g) : *pr,r(.E) bv Tl:ot-"* 4'4'5' E is Dedekind complete. by 4.4.1(1), the iattice (o)-E and, consequently, ";;1;. : for eve-ry u e (o)-E' (2)+(3): Since by Theorem 4'4'6 fin(-E) E *q(.8)' particular, the image of the Riesz ho- there is an r € -E such that u : n*q(. E)'In wiitr (o)-f. Thus, is a Riesz isomorphism' momorphism f ; B --+ (o)-E coincides f The implication (3)-*(t) is obvious' tr with Proposition 4'0'6' It is interesting tL compare this theorem 4.8.8.Let(E,p)beanormedvectorlattice.Recallthat,accordingto4'0'6' we may arrange the quotient vector lattice E ,: F\n(. E) I p,( E) nonstandard hult of. (E,p)' with the respective quotient norm. E is carle_d rhe onlv on but also ;;;;r; ,e i, a Banach lattice. Note that 'E depen'ds not 'E of an element :a e Fin(.{).ti on the choice norm.p. Denote the coset "itt" the mapping p -+ E (cf' the quotient vector lattice E av (:a)) and consider 0 " is easy to see that p is a Riesz 4.0.6) such that fu(r) ,: ((z)) for i"".V r € E. It monomorPhism. the following are Theoretn. Let (E,p) be a notmed vector lattice' Then equivalent: onto E such that (1) Therc exists a Riesz isomorphi sm T of (o)-E Ton: p; (2) E is finitedimensional' that zr oi: < (1)-'(2): Let n : (o)-E E be a Riesz isomorphism such .--, and, furthermore' fr(E) : ideal, e"*rut"a Ui a(B), coincides with (o)-E F,,The with E. Consequently, n(i@));therefore, the ideal, generated bv F(E), coincides Fin(.E) : fin(.E) + p( E), to Theorem 4'5'4' it implies (E) 9 rt(E) by Theorem 4'5'2' According which t t: U(l:al) is remains to establish the reverse inclusion. Let x eq(.8).Th1n Y this 0 in (o)-E by Theorem 4'8'2' directed downward,s and t/ 10. In .c-ase f'(u) | Hence,rofi(U)f0inthevectorlatticeE'Inotherwords' o : 0' i\rj F(U) - in! r fi(U) (:a)) : 0 u19' consequent\v' x e p(. E)' Thus' Since F,g) > (@)l ) 0, we have p; by 4.5.4. ,c-p4' : ,;( ,','urrd ,o -E is finite-dimensional (2)-+(1): This is obvious' tr Infinitesimals in Vector Lattices 2TT 4.9. Regular Hulls of Vector Lattices Here we define and study the regular hull of a vector lattice. We will obtain a criterion for a vector lattice to be isomorphic to its regular hull. Some related questions concerning regular hulls are discussed. 4.9.I. Let E be a vector lattice. Like in the preceding section, we consider the quotient (r)-,8:: fin(*E)l^(E) and call it the regular hult of..E. We denote the coset x*),(. E) where zc e frn(* E) bV \r), and define a mapping ie : E - (r)-E as follows' t B(n) :: (nl (r e E). Obviously io is a Riesz homomorphism. We denoted it by i if tnir does not lead to ambiguity. We present a criterion for a vector lattice to coincide with its regular hull. Recall that a vector lattice .O is called almost regular if E is Dedekind complete and order separable, and if order convergence and relative uniform convergence are equivalent for every sequence in E. Theorem. For every vector lattice E, the following are equivalent: (1) i, E -* (r)-E is a Riesz isomorphism of E onto (r)-E; (2) E is atomic and almost regular. < (1)+(2): Assume that I is a Riesz homomorphism of -E onto (r)-8. In particular,.\ is injective. Thus, obviously, -E is Archimedean, so grl(E), by Theorem 4.3.5. Since fin(.E) : E * X.E), we have ^(.8) fin(..E):E*q(.8). (1) Therefore .E is an atomic Dedekind complete vector lattice by Theorem 4.4.6. In or- der to complete the proof of the implication (1)-+(2), in accordance with Theo. rem 4.3.6, it remains to check the inclusion 4(-.O) g )(.8). Take an arbitrary element x € r7( E). Since q(. E) g E + )(.^E) holds by (1), the element x can be written as x:e*%tt where e€E and x1€ Then ^(.,E'). e : 2c - 24 € rl( E) - )(-E) g rt( E). Consequently e € En q(.8) - {0}, and hence e:0. Finally, t1 : ,tL €,\(.,O). (2)-'(1): Let E be an almost regular vector lattice. Then, by Theorems 4.4.6 and 4.3.6, fin(.,E) : E * q(. E) : E +^(.8), which immediately implies that i is onto. Moreover, since .E is Archimedean, I is an injection. Thus, ) is a Riesz isomorphism of ,E onto (r)-8.> 2r2 Chapter 4 4.g.2. According to Theorem 4.3.6, the regular hull (r)-E of an Archimedean order separable vector lattice .E in which order convergence and relative uniform (o)-'E' convergence are equivalent for every sequence coincides with the order hull property' We show that there are no other types of vector lattices with this Theorem. For an atbitrary vector lattice E, the following are equivalent: (1) There exists a Riesz isomorphism r of (o)-E onto (r)-E such that r ofi -- ),' (2) q(. E) : )(.E) ; (B) E is an order separable Archimedean vector lattice in which order convergence and relative uniform convetgence are equivalent for every sequence. < By Theorem 4'3.6 it suffices to verify (1)-+(2)' : Take Lel r z (o)-E --- (r)-E be a Riesz isomorphism such that z"o f .\. ( is easy to see elements u,u'€' E^satisfying the condition 0 I nu o for all n e N. It Since.f is injection, the relation u:0 that a- "fi(u): i(r), and hence i@):0. ,E is Archimedean. The inclusion ,\(*.8) C t7(-E) follows now' holds. Thus, g To complete the proof, it remains to establish the reverse inclusion: n( E) l(.B). Assume that there is a N e\(.8) \ I(.E)..We may suppose x20. Then implies inf-BU(x): 0' There- G) ,0. At the same time, the condition x e rt( E) io*, u".ording to Theorem 4.8.2, infloy-E i(U(n)): 0. Since zr is an isomorphism of (o)-E onto (r)-E, we have inf.-i(u(x)) : inf:zr ofi(U(N)) - o, U)-E (o)-E q(. g which contradicts the condition i(t/( ,)) > (r) > 0. Thus, we have E) ,\(.8)' The proof of the theorem is complete' > 4.g.g. We now discuss interrelation between the regular hull (r)-E and the condition for nonstandard hull E of a normed vector lattice .E' NamelY, w€ find a (r)-E to coincide with E. We use the notation and terminology of 4.4.2 and 4'8.8' Theorem. Let (E,p) be a normed vector lattice. Then the following are equivalent: p equivalent (1) .E possess es a strong order unity e such that the norm is to the norm ll ' ll"; (2) (r)-E : E; Infinitesimals in Vector Lattices 213 (3) There exists a Riesz isomorphism g of (r)-E onto E such that 9o\:fi' < (1)+(2): This is immediate from Theorem 4.5.2. (2)--*(3): Obvious. (3)-*(1): Letg z (r)-E --. E be aRiesz isomorphismsuch thatpol: F, By Theorem 4.5.2, we need to establish the relation Fin(.^E) : fin(*E) * p(. E). The inclusion fin(*,E)+ p(. E) e Fin(..O) is obvious. For proving the reverse inclusion, take an arbitrary :a € Fin(.E). Then ((nD : p(@)) for some ut Q fin(.E). Lel r € E+ satisfy lrtl < z. Hence l(r)l :lp(rt))l: p({,'l)) S p(r): e"i1"1 -F@): ((")). The inequality l(.))l < (r)) implies that the element N can be written in the form x:€t*€2, where l€rl Sz and €ze p(.8). Thus, xefin(*E)+p(.E).> 4.g.4. In contrast to 4.8.2, the image of the vector lattice .B under I is not necessarily a complete vector sublattice of (r)-E. Indeed, consider the vector lattice l- of all bounded sequences in IR., and let D be a subset of l- consisting of all sequences with the property that all but finitely many coordinates are equal to 1. *N\N, Then inf 6D:0, but XD) > l""l ) 0 for aIl u € where e, is the internal *lR sequence in in which the only nonzero coordinate has index v and equals 1. 4.9.5. Exactly as in the proof of Theorem 4.8.4 in which relative uniform completeness of order hulls was established, we can show that the regular hull of an arbitrary vector lattice is relatively uniformly complete. At the same time, since, by Theorem 4.3.6, the regular hull of a regular vector lattice coincides with its order hull, Theorem 4.8.5 shows that the regular hull of a nonatomic regular vector lattice is not Dedekind complete. 4.9.6. It follows from Theorems 4.3.6 and 4.3.3 that the regular hull of an or- der separable Archimedean vector lattice in which order convergence and relative uniform convergence are equivalent for every sequence is Archimedean too. Another case is described by the following Theorern. Let E be a vector lattice in which for every sequence (r) g E* , there exists a sequence ()r) of strictly positive reals such that the set {),rz,r} is order bounded. Then (r)-E is Archimedean. ( We need to show that ,\(*E) is a relatively uniformly closed ideal in fin(.,B), i by Theorem 4.3.5. To this, consider 0 1un f and u- !\ u, where u' e ,\(.E). It I I is sufficient to prove that u € ^(.^E). I i I I t_ j t F L I E 214 Chapter 4 Since ,n 9 u, there exists a sequence (er) q R+, €n + 0, and an element d, e E+ such that lun - ul 3 end for all n € N. Since un € there exists un € E for which 01kun l wn simultaneously for all k e N. By^(.8), hypothesis, take 0 ( tr", € IR. and w € E such that \nun 3 r.r.' for all n € N. Consequently, ( w) lul S lu, - ul + lul S e*d'* max{e,, ,lf n}\nw- max{e^,Lln}(d + for every n € N. Therefore, we have u € .\(.-g), by using €", -r 0' D Corollary. The regula,r hull of a Banach lattice is Archimedean. There are non-Archimedean vector lattices whose regular hulls are non-Archi- medean either. To see this, we consider an example of the vector lattice L by T. Nakayama (see [21, Example 62.2]). The ideal /o(^L) ,: {* € L: (ly e L)(Vn e N) lnrl < y} is not relatively uniformly closed in -L. Hence, there are a sequence (*^) g Io(L), 01rn t, and an element t e ,L such that *n9r/Is(L). Since IoQ): l(.tr) (\L,we have that the ideal )(.I) is not relatively uniformly closed in fin(-.L). Thus, by Veksler's Theorem (see [21, Theorem 60.2])' (r)-L is non- Archimedean. The question remains open whether the regular hull of an arbitrary Archimedean vector lattice is Archimedean. 4.10. Order and Regular Hulls of Lattice Normed Spaces In this section we define and begin studying the order and regular hulls of lattice normed spaces' *.O. 4.L0.1. Let (% ,a,* E) be some internal LNS normed by a standard lattice Consider the following external subspaces of the internal vector space %: fin(%),: {* € % :a(r) e fin(.E)}, q(%) t: {* € % : a(r) e q(. E)}, {* € % : a(r) e}(.,E)}. ^(%),: The vector spaces ,l@) and ),(%) are subspaces of fin(,%). Therefore' we may arrange the following quotients: (o)-T :: frn(fl ) lq(%), (r)-T ,: fin(%) I x@). Infinitesimals in Vector Lattices 215 We denote by [r] the coset r+q(%) in (o)- ,% and by (") the coset r+),(%) in (r)- %, where r € fin(%). Given r € frn(%), assign a(["]) :: a(r) + rt( E), ar'l((")) :: a(n) + ,\(.,8). see the mappings : (o)-T (o)-E and 41"; (r)- It is easy to that a -, ' T -. (r)-E are well defined. DnrrNrtroN. We call the LNS ((o)-7,a,(d-n) (((r)-T,a6,Q)-E)) the order hull (regular huII) of an internal LNS (.f , a,* E). 4.LO.2. Theoretn. Let (%,o,*E) be an internal decomposable LNS with a standard norm lattice * E. Then its order hull and regular hull are decomposable and (r)-complete LNS. < Consider the external LNS (fin( %),a,fin(.E)). The proof of (r)-complete- ness of this LNS is almost the same as the proof of relative uniform completeness of the vector lattice fin(-.E) in 4.8.4 (it suffices to replace fin(.^E) by fin(.ff) and the modulus by the norm o). Clearly, the norm o is decomposable in frn(,%). Since the order hull (regular hull) of (,% , d,* E) is the quotient of. (fin(tr ), a, fin(*E)) by the ideal n( E) (respectively, the ideal .\(.E)) of fin(.E), *" complete the proof by using Proposition 4.0.14. > 4.10.3. Let (E,ll . ll) be a normed vector lattice. A Dedekind completion E : i@) of .E is a normed vector lattice under the norm llrll :: inf{llell : e € E k iG) > l"l}. (2) Now, the LNS (o)-E is a normed vector lattice under the norm lrl :: llp@)ll. W" have the direct expression for l'l as follows lrl :: inf {llell : e € E e ik) >- l*l} (r e (o)-,8-) which extends the norm (2) from E to @)-E. Note that the embeddings f : . . . g (8, ll ll) - (t, ll ll) and (8, ll ll) ((o)-E,l l) are isometric. Recall that a normed vector lattice (E,ll .ll) satisfies the weak Riesz-Fi,sher cond,iti,on if every sequence (r") 9.8 with the property DLr llr'll < oo is order bounded. Theorern. The normed lattice ((o)-E,l l) it a Banach lattice if and only if . (,E, ll ll) satisfies the weak Riesz-Fisher condition. 4 2t6 Chapter ,,";i#il"ffi J,l:'"f.J,P"'i":*"?fi ,Iifi H:';'H'ilfi i''H*,ni"tfl ((o)-E,p,E) with Theorem +.r.zl,we obtain from (r)-completeness of the LNS p(*) :ilt{f(t) : e € E e ik) >- l*l}, E that ((o)-E,l'l) it a Banach lattice' take an arbitrary conversely, suppose that ((o)-E,l.l) is a Banach lattice and Then sequence (r') q -E such that lflr llu'll < oo' oooo Ili'(|,'l)l : t ll,'ll < *. n=I n:! ConsequentlY, there is an u e (o)-E, @ u: (o)-f atlr"l) e (o)-8. n=L element u € E such that ) since f'(E) is cofinal in (o)-E, there exists an fi(u) the weak Riesz-Fisher u. Obviously, (r,") q [-r,r]. Thus, (8, ll 'll) satisfies ondition. > the quo' 4.LO.4.In the sequel, we assume that .E is Archimedean. we consider tient . E :: o-pns(. E)lrl( E) E by Theorem and recall that the vector lattice .0 ir u Dedekind completion of 4.4.t. we need some preliminary work. we start with a few lemmata: Lemma. Let Y e o-Pns(*E)' Then [y] : i4 i(u(aD. E following holds: < since r(y) < a { u (y) and ,0 is Dedekind complete, the supfi'(I(s)) < [y] ( i4 i(u(Y)). EE ConsequentlY, i*r r(v))' 0 ( i4 iV@) - [v] < i4f'(n(v)) -'lp fiQ@) < i(u(a) - EEEE Thus Since v € apns(.E), we have infs(U(A) - f(v)):0' iryti(U(Y) - L(a)) :0' 'in i Therefore, the above- I because .0 i, u Dedekind completion of the sublattice i@)' . established inequality implies lgl: tn|6f(U(v))' > Infinitesimals in Vector Lattices 2I7 4.10.5. Lemrna. Each nonempty order boundedsubsef g g Epossesses some supremum and, infimum in (o)-E. Moreover, (1) infloy-Eg:tnf79; sup,, : supi . (2) )-E I I < (1) Suppose that I e E and I * s.It is sufficient to show that inf 69 : 0 implies inflo;-E I : 0. Take a x € fin(.E) such that 0 { x and lxl < 9. To complete the proof, it remains to establish that lnl : 0 or, in other words, to verify the condition infr U(x) : g. Assume that an element a € E satisfies the inequality 0 ( a {U(z) (3) and take an arbitrary d e L Then d : l5l for some d € a-pns(.E). It is obvious that 2t: ztld+ (r-6)+ < t/(6) +U((x -d)+). Consequently, Y(d) +u((,-d)+) e u@). (4) Flom (3) and (4), it ensues that 0(o Using Lemma 4.1.2 and, Dedekind completeness of fr, *" obtain from (5) that o < 4'(") ( iqf i(u(5)) +i4i(u((,-d)*)) : [d] +irylfi(u((x (6) EEE -6)+)). At the same time, l@ - d)*] : (rl - d)+ : 0. Hence, inf BU ((x - 6)+) : 0. Thus, we have inf 6i(U((n - 5)+)) : 0. Now, (6) implies 0 < 0(o) ( [6] : d. (7) Since d e I is arbitrary; therefore, in view of inf.6I : 0, we deduce from (7) that a: 0, as required. Assertion (2) ensues immediately from (u). > 4.10.6. Let r e (o)-E. Assign % (") ,: {" € E : fi(e) >- r}, A@) ,: {a e E : u }- n}. It is clear that %(r) and O@) are nonempty order bounded subsets of E and, E respectively. 2t8 Chapter 4 Lernrna. For every r e (o)-8, the following hold: (1) i*6c@) e c@); : inf (")) (2) ifi 6 o@) fi. iQr ; (3) If an element x e fin(.E') satisfies 7 : ln] then irgC@) iqr iQt (")) i4 i(v(,)). AEE - - < (1)' By Lemma 4.10.5, 0@) has an infrmum in (o)-E, and inflo;-EO@): it6fi@). Hence, from C@) r, it follows that inf ) r. Then ) 1o 1-zO{"1 i*6O@) ) r, as required. (2): Assign ts i:tnt6fi@). From (1) it ensues that Q/("0) g %(").Estab lish the reverse inclusion. Let z e %(r). Then iQ) > r, and hence iQ) e C@). Consequently, fi(z) 2 ro which is equivalent to z € Q/ (rs). To complete the proof, it remains to see that the relation ro:inf7Q@@o)) is valid since E is an order completion of f'(E). (3): Let zt € fin(*E) and * : lrl. Assign n6 :: inf 6% (z). According to (1), 16 )- r. Take an element y efin(.E) such that *o : lal and gt 2 x. Then we have from Lemma 4.L0.4 ro: lyl - iqf i(u(y)) > iqr i(u(,)). (8) EE Next, the obvious inclusion fi(U(r)) e C11ry implies iryf fip(x)) ) i4 CMD ro. (e) EE - From (8), (9), and (1) we now obtain the required result. > 4.LO.7. Define the mapping p : (o)-E -* 6 * follows: p(r) iryt Ofu) @ e (o)-E). (10) ': E We list some properties of this mapping. Theorern. The mapping p is an E-vatued norm on (o)-E such that, for all r,a € @)-8, the following hold: Infinitesimals in Vector Lattices 2t9 (r) P(") : inf f i@ (l"D; (z) p(") 2lrl; (3) l"l >- lvl implies p(r) 2 p(a). Moreover, an arbitrary sequence (nn) g @)-E (r)-converges in the norm p to an element rs € (o)-E (is (r)-gauchy in the norm p) if and only if it (r)-converges to rs in the vector lattice (o)-E (is (r)-Cauchy in (o)-E). The lattice normed space ((o)-8, p, E) is (r)-complete and. d.ecomposable. { From (10) it is straightforward that the mapping p satisfies conditions 4.0.10 (2), and 4.0.10 (3), and item (3) of the theorem. Item (2) of the theorem follows from 4.10.6 (1). Now, conditions 4.0.10 (1) ensue from (2). Hence, p is an .0-valued norm on (o)-E. Condition (1) is a particular instance of 4.10.6 (2). ('), If. rn ro in the norm p with regulator e € E then, in view of item (2) of ('), ('), the theorem) rn *o in (o)-E with the same regulator. Conversely,if. rn ro @), in (o)-E- with regulator d e (o)-E then 2,, ,oin the norm p with regulator p(d,), in accordance with iiem (3) of the theorem. For (r)-Cauchy sequences the proof is essentially the same. The quotient (o)-E is relatively uniformly complete in view of Theorem 4.8.4. So, as we showed, ((o)-E, p, E) is (r)-complete. Now, to verifu decomposability of p, it is sufficient to establish (d)-decomposability of p by Proposition 4.0.10. Let x e (o)-E and let at,€2e 6 be such that p(t) : et * ezand e1 A e2 :0. Assign frr i-- t+ Ae1- r- Aell 12 t: r+ Aa2 - it- A€2. It is easy to see that p(21) : €!, p(r2) : a2, and r : tL * tz. D 4.10.8. Consider the mappittgp "a: (o)-T -- E, where d': (o)-T -- (o)-E is the (o)-^E-valued norm defined in 4.10.1. Theorern. The triple ((o)-T,p o a, D it a d.ecomposable (r)-comptete LNS. < It is easy to see that p o o is an -0-valued norm in (o)-7. Take an arbitrary sequence (*^) I @)-T that is (r)-Cauchy in the norm pod with regulator e e .8. In view of Theorem 4.I0.7 (item (2)), it is (r)-Cauchy in the norm d with the same regulator. Consequently, by Theorem 4.10.2, there is an element re e (o)-T such ('), that, rn ro (e) in the norm a. Now, from Theorem 4.10.7 (item (3)) it ensues U), that rn ro in the norm poa with regulator p(e): e. Thus, every sequence @^) e @)-7, that is (r)-Cauchy in the norm pod, is (r)-convergent in (o)-T with the same regulator. Thus, (o)-T is (r)-complete in the norm p o d. Chapter 4 o d, In view of (r)-completeness, to estabiish decomposability of the norm P 4.0'10' Let r € it is sufficient to verify its (d)-decomposability, by Proposition O. and let €t,€z e E be such that poA(r): et* e2 and Tl"1 @)-T "t l:'-: such that decomposability of th" norm p implies that there &r€ o1 ,d2 € @)-E 4'10'7 (item (2))' d.(r) : 6,1* azt p@) : €!,, and p(o2) - €2' In view of Theorem az 0 and we have or ( i.,i'or( ez. Hence, from the conditions at * -a(n) ) ", 0 and remains to use decomposability of ey Ae2:0 it ensues that nr ) or2 -}J^-f ixr ltz : r, a(x't) : ot' the norm d for finding elements fr1,,n2 e (o)-T such that * : e1 and p od(r2) : e2' D and o(r2) - c,2.It is clear that p od(r1) A.LL.AssociatedBanach_KantorovichSpaces decomposable we give a nonstandard construction of an order completion of a Banach-Kantoro- LNS. The scheme rests on embedding the LNS into the associated internal dominated ,ri"t, ,pu"" (BKS). We study extensions onto associated BKSs of Throughout the section operators admitting standard (o)-continuous dominants' which the norm *" ,rppore that (i ,o,E) and'(g,b,F) are decomposable LNS in lattices E and F are Dedekind complete' is called A.L:..L. The lattice normed space ((o)-T,P od,.0) aennea in 4'10'8 LNS (X, a, E)' We establish associated, with the order hull ((o)-T,-a, (o)-E) of the that this LNS is a Banach-Kantorovich space' we have E E Since the vector lattice E is a Dedekind completion of -O, ? '+E -t E is a Riesz under our assumptions. Jo be more precise, the mappi"g, j E defined by isomorphism of E o*o E. Consider the mapping pn : @)-E -+ the rule pn@):: igf {e € E :i@) > l"l} (r e (o)-E)' We have the following -' the Lemma. The mapping pn is connected. with the norm p: (o)-E E by o fot every r e fin(-E) we have relation PE : i-r p. Moreovet, ' PIMD:i#te €E:"21'l\' p p6, and < The first part of the lemma ensues from the definitions of and the second from the relation p"(lnl) - igf u(ll,1l) : (llnllD : i*' ixf i(v(lrD) Tt l.tfi(%EE Infinitesimals in Vector Lattices 22I : i3f U(lrD : igf{e € E : e 2 lnl}, in which the second and the fourth equalities are valid because f is a Riesz isomor- phism, and the third in view of item (3) of Lemma 4.10.6. > Theorem 4.10.8 and Lemma 4.II.I imply the following Corollary. The triple ((o)-7, pn od, E) is a decomposable (r)-complete lat- tice normed space. Moreover, for each r € fin(.E), we have pBoa((x)):inf{e e E:e>-a(x)}. (11) 4.LL.2. Denote by 9(E) the family of all band projections in E. Note that for each internal band projection r e *9(E), there exists a unique band projection h(r) in 9{' satisfying the condition a(h(r)x) : ra(n) (x e %). This property is easily obtainable from decomposability of the internal norm a : ,% --. *E. Lernrna. For all T € g(E) and x e fin(%), we have r o pn oa((xl) : pp od((h(.tr)x)). ( Let r e frn(% ). Show that, for every r e g(E), the inequality r o pE oa((ul) 2 pn od((h(.tr)x)) (12) holds. To this end, take an e € E, e 2 a(r). Then re > .r(a(x)) : a(h(.r)u). Applying (11), obtain n(e)>-tgt{l €E: f ) a(h(.n)r)}: psod((h(.n)r)). Since e € E, e ) a(x), is taken arbitrarily, we obtain from order continuity of z- and (11) that T o pa oa((x)) : r inf{e €. E : e ) a(x)} : i#{ne : e € E k e ) a(r)} ) pn oa((h(.tr)x)). Inequality (12) is established. Chapter 4 222 T e g(E) and denote zrr the com- consider an arbitrary band projection .bv (12) to zr and zrr' we have plementary projection to zr' Then' applying p6 od(\x)) : n o pE oa((t)) + nL o PE oa((t)) )z pEo a((h(-n) ,")) + ps od;((h(-t't)'o)) *tL o(:a)]) : pE oa((x))' >- pn(l*n o a(x) + " Hence, It o pE oa((x)) + rL o PE oa((')) : pnoa((h(.zr) ,4) + pB od((h(.nt)n))' > o oa((n)) : o d((h(-zr) x))' as required' consequently, in view of (L2), r pE 9E space ((o)-T'pB odtE) is A.LL.3. Lemma. The associated' Iattice normed' disjointlYcomPlete' , r r (re)e.= g g(E)^,a\ and a family-:, ("e)cee e { Take an arbitrary partition of unity od' @)Z bounded in the norm Pn SuPPose that, for e € E, we have p6od(rs)(t (€€E). (13) the : z€ for all E' Using the definition of Choose a xq € % suchthat (:rg) {^e item (z) of Theorem 4'10'7, we rewrite norm d, the ,"t"tJ;;;'l Atl'; p, uid for a suitable rtE e q(.8)' we have inequality (13) as lo(ne)l< a(r): consequentlv, Hence, according to Lemma 4'0'13' there the inequality o(JJti; i;; 1q € E) are element s /, e fin(%) for which a(xc-rL)en(E), a(x'){e (€e e)' (14) *N g the set of alr internar mappings from Fix some y € \ N and denote by *linto%.Letcardrtanafortheinternalcardinality'Given€€E'definethe internal subset Aq ol I as follows: q & : x'e Aq :: {p e I : ao g(*E) [-","] e(€) *E: & Card({€ e P(€) l0}) E z}' property' Conse- the family (Ag)gee has.the intersection It is easy to see that fnife : principle, there is an element 9e € n{Aq quently, in view of the general ,utoriii-o1 { e E}' Put *E o :: t€ € : po(€) + oI. Infinitesimals in Vector Lattices 223 Since Card(O) ( z, the set O is hyperfinite. Furthermore, sgog*E and a(po(€)) (e (€e o). For convenience, we let x! :: po(€) whenever € e O. This does not cause difficul- ties, since eo(€) : /€ for all ( e E by the choice of po. Thus, the family (/)qe= extends to a hyperfinite family (/)eee 9 ,% such that a(x') {e (€ e O) (15) - Let (r6)EE'= :: ((te )e.=) be the nonstandard enlargement of the partition (re)e.= of unity. Then ("g)e..= is an internal partition of unity in.9(E). Furthermore, *zr4 re : for all { e E. The hyperfinite strm ,c :: Dgeo h(rq)r/, is an element of the internal vector space 9t' , where h(rg)'s are the band projections defined in 4.L7.2. Disjointness of the set of projections 16, together with (15), implies lzl ( e. In particular, x e frn(9{). For every €o € E, consider the following chain of equalities: T€o o pn o o(rEo - @l) : Teo o pE oa((rt, - n)) : pE o d((h(-z'6 )@t, - ,))) : pE..((nf"r,l( I nf"e),t))) : o €e o\{€o} The first equality hoids due to the choice of the elements xq and (14). Validity of the second is ensured by Lemma 4.11.2. The third equality is valid by the choice *zr€o of. x and the equality r€o - mentioned above. Finally the last equality holds since zg is disjoint. Therefore, n6 o pn od(*e - @)) :0 for every € e E, and hence (r): mix(a-6rE)E6s. Thus, for every partition of unity (re)E.= g g(E) and every bounded in the norm pp o a family (re)e.=, the mixing mix(zr6z q) e (o)-T exists. The proof of the lemma is complete. tr 4.LL.4. We are in a position to state the main result of this section. Theorern. The associated lattice normed space ((o)-T , pBodtE) r's a Banach -Kantorovich space. < It is follorvs immediately from Corollary 4.11.1 and Lemma 4.11.3 by using Proposition 4.0.11. > Note that, whenever we take an internal normed space ff as an internal de. composable LNS, the associated space coincides with the classical nonstandard hull Chapter 4 g. Furthermore, from the above.established theorem, the well-known assertion ensues that the nonstand.ard hull of an internal normed space is a Banach space. If we consid.er an internal lattice normed space (.E, I ' l,.E) then the associated -' E space is the LNS ((o)-8, pn,E). the deflnition of the mapping Pn : @)-E ryo* ( pe(v)' ii is clear that, for a]l r',y e (o)-8, the condition lzl ( lgl implies p6(r) so, we obtain that the associated LNS ((o)-E,Pn,E) is a Banach-Kantorovich lattice. 4.IL.5. It is known that a norm completion of an normed space X can be obtained by taking the closure of the space in the nonstandard hull of X. Similarly, as we show below, an (o)-completion of a decomposable LNS can be constructed on embedding it into the associated BKS' For simplicity, denote by ((o)-T, pn od,E) the BKS associated with the order hull of (X,;,E). Consider the mapping 0 t X -* (o)-T such that fi(r) :: (r) (r e X). (16) LNS (X, a, E) It is easy to see that f is an isometrically isomorphic embedding of the limits of all od, convergent nets into ((o)-X , pE oA, E). Denote bV f the set of Pn of elements of f'(X). Lemma. For every element r e (o)X, the following are equivalent: (1) rert; (2) od,(r : 0. Jt"r" - 0(s)) < (1)+(2): This is immediate from the definition of -f' (2)--'(1): Let an element r e (o)-T satisfy condition (2). Show that r € rt. Define the relation < on X as follows: y < z * pn od,(r -i@) 2 pn o a(r -iQD. The set X is directed upwards with respect to {. Indeed, for all U,z € X, we g(E) that satisfies have y, z { h(tr)y + h(nl)2, where T € is the band projection the condition r o pp od,(n - i@) * rt o PE od(* - fi(")) : pE od,(r - i@) A pn od(, - iQD, and h(zr) and h(zrr) are the corresponding band projections in (x, a,E). Consider the net'(4'(y))r.tx,rl. From the definition of (x, <), in view of the condition to z in infeex fi.i1i: atgll : 0, it follows that the net (f'(g))s€(X,<) converges (o)-7. Since the net is constituted by elements of f'(X), we have t € X - > Infinitesimals in Vector Lattices 225 4.LL.6. Theorem. The triple (rt, pu o a, E) is an (o)-comptetion of the de- composable LNS (X,a, E). < It is sufficient to verify properties 4.AJ2(1)-(3). Clearly, (rt,p, o a,.E) is a lattice normed space. Show that this space is (o)-complete. Take an arbitrary (o)-Cauchy net (16). Then, by (o)-completeness of the associated LNS, there is an element r € (o)-X such that , : (o)-lim(26). Show that z e -?. tne conditions 14 e * and r : (o)- lim(rg) imply that p, od,(rE fr(aD : O, ,tt o" od(n r€) : 0. (r7) Jtf" - - Next, from (17) we obtain 0 < pB od,(r u?l - i(a)) ( inf od(r r) pn od,(rq ;t @" - * - 4'(y))) ( inf ps o _ re inf od(nq _ : a(r ) * ;tLo, f(y)) 0. Thereby intyex ps o a(r -i@) : 0 and, in view of Lemma 4.1I.5, we have n € ft . Thus, every (o)-Cauchy net (rg) g -f is (o)-convergent. Hen * is (o)-complete "", in the norm pn o a. It is easy to verify that the norm pn o a is (d)-decomposable on X. Consequently, taking (o)-completeness and Proposition 4.0.10 into account, we obtain decomposability of the norm pn o a on f . Property 4.0.12(1) is established. Property 4.0.L2(2) is obvious for the embeddin1i: X -- ft. To verify 4.0.12(3), take some r' e * and e € E+.Assign E :: {n e g(E) : ro pEoa(rt -f'(g)) < u for some n e XI. Since r' e rt, we have o : o. itf*0" a(r'- f(")) Hence, the set d is dense in the band g"(E) generated by the band projection pre. By the exhaustion principle, there exists a partition (or)reo I E of pru. In accordance with the definition of E, there is a family (r'r)1ea e X for which otopooa(r' -0("1,)) < r (z e Cl). (18) Takea1'16f A andassignf ::OU{fo}, o.ro:: pr},and r'ro.=0. ForeachT€f, due to decomposability of (X, a,E), there exists a unique band projection r, in 226 Chapter 4 the space X which satisfies aor^r - droo. Define a family (rr)rer e X so that n^t t: r.rr'^, for every ? € f . From (18) it follows that : a(rtr'^) : o-,t o 01 o oa (i@-r)) a(r) "(r'r) -- Pn ( or o pn od (*') * o, o pn o a (*' -qp) (2po oa (rt) * e for each ? € f. Thus, the family (zz)"rer is bounded with respect to the norm a. By (o)-completeness of (X, pn od,,E), the mixing mix(6.y,f(rr))rer e -f exists. Using (18) again, we see that pr" o ps o a(rt - mix(o, f'(rr))) ( e. Since the choice of the elements r' e fr and e € E+ is arbitrary, property 4.0.12 (3) is established. The proof of the theorem is complete. D 4.LL.7. Denote the space of all regular (respectively, order continuous) oper- ators from.E into ]? by L,(E,F) (L"(E,F)), and denoteby.,//(,%,U) the set of all internal linear operators from % into % whtch admit some standard dominant *Q, L.(E,F) (see 4.0.15). Next, let //^(%,%)be the set of all operators in Q € .s 2(r ,%) each of which admits a dominant of the form with s e L*(E, F). Lernrna. For every internal linear operator T : ,% -- U , the following hold: (1) T e -,// (,%,%) - r$n(9{ )) e frn(U); (2) Te //*(%,U)'r(q@))e q@)' { We verify only (1), since (2) is established similarly. According to the con- dition T e.,//(%,U), there is an operator Q e Lr(E,F) for which g!n) <.Qo(n) (x e e['). (1e) Take an arbitra ry x e frn(%). Then a(r) ( e for some e e E. By (19), we have gQ4 < Q(t) and, consequently, Tx € frn(U). > Throughout the sequel, the vector lattice Ln(E,F) is denoted by L. 4.11_.8. suppose that ? € r//.(%,u). According to Lemma 4.Lt.7, the map ping 7 : (o)-T -- (o)-U acting as T((r)) p (rul (N e fin(%)) (20) is soundly defined. Infinitesimals in Vector Lattices Theorem. The mappingT is a dominated linear operator from the associated ((o)-T,pE od,E) the associated, g,F). BKS to BKS ((o)-U,pe " Fbrthermore, (7D < pr([((r))]). Before proving, we make necessary clarifications. ev ((7)) (Uy ((fl)) we denote the least (least internal) dominant of 7 (of By pr we denote the .L- valued norm in the LNS ((o)-T, pr,, L). "). < It is sufficient to establish that the operator pi,([((")]) e L,(E,F) is a dominant for 7. Take an arbitrary operator ,S e L that satisfies the condition .S > ("))). Then, using relation (19), for every x €fin(%), we obtain pF.BF\r)) :tpt{r € F : f > 0Qr)} < tFt{f € F : f > ((g})a(n)} ( inf{^Se : e € E : e) a(x)} :,Silf{e € E : e ) a(x)} :,S o pB oa((x)). Consequently, ,9 > (7D. Using Lemma 4.11.1, we find pr([((r))]) : inf {s € L:.s > (")))} > ((70, as required. > 4.11.9. Denote by M^(%,U) the set of all linear operators from % into % which admit (o)-continuous dominants. It is clear that the conditions ? e *T Mn(%,U) and € Mn(*%,V) are equivalent. Take T e M^(fl' ,%). Then, according to (20), there exists a mapping T : (o)-T -- (o)-T such that T((r)) : (Txl (.ur e fin(.9)). Theorem. For every T e M^(J[',U), the mappingT is a dominated ]inear operator from the BKS ((o)-7,d,, E) to the BKS ((o)-@,6, p). Furthermore, (t) T(frr(")) : is€") (r e %); (2) (7D : (("))), where is , % --+ (o)-T and fisr : % + (o)-T are the canonical embeddings defined by (16). < The mapping 7 is finear by construction. Equality (1) ensues immediately from the definitions of. fiey, fisr, and 7. tfre fact that the operator 7 is dominated, as well as the inequality ((70 < (")), is established in Theorem 5.2. It remains to verify the reverse inequality. Chapter 4 Let r e %. Taking property (1) into account, as well as the fact that fis and fis are isometrically isomorphic embeddings, we obtain bQ@\ : pF "6(fis(rr)) : p" o6(T(fis(r))) < (7D (p" "a(is@))) : ((?))a(z). since the element r € % is chosen arbitrary, it foliows that ((")) < (7D > Let F and, O be the (o)-completions of % and U, constructed in Theorem 4.11.6. From the previous theorem we obtain the following assertion (see [19, 14, Theorem 2.3.3]). Corollary (A. G.Kusraevl-V. Z. StrizhevskiY). Fot everyT e M*(%,9), there exisfs a unique operator i e M*(f,A) extending T in the sense that t@r(")):fip(Tr) for all n € g'. Furthermore, ((f))) : (("))). < It is sufficient to take u, f th. restriction of the onSrg!5rr 7 onto f. Vniqu"' ness of extension ensues from the requirement f e M*(% ,%) and the construction ofF.> References 1. Aliprantis C. D. and Burkinshaw O., Positive Operators, Academic Press, New York and London (1985). 2. Albeverio S., Fenstad J. 8., et al., Nonstandard Methods in Stochastic Anal- ysis and Mathematical Physics, Academic Press, Orlando etc. (1990)' g. Bernau S. J., "Sums and extensions of vector lattice homomorphisms," Acta Appl. Math., 27, 33-45 (1992). 4. Conshor H., "Enlargements contain various kinds of completions," in: Victo' ria Symposium of Nonstandard Analysis, Springer-Verlag, Berlin etc., 1974, pp. 60-79. 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