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1 Mth-Net.Ru All Russin mthemticl portl L. Kntorovitch, Liner opertions in semi-ordered spces. I, Rec. Mth. [Mt. Sbornik] N.S., 1940, Volume 7(49), Number 2, Use of the ll-russin mthemticl portl Mth-Net.Ru implies tht you hve red nd greed to these terms of use Downlod detils: IP: Mrch 11, 2018, 15:11:56

2 1940 МАТЕМАТИЧЕСКИЙ СБОРНИК Т.7 (49), N.2 RECUEIL MATHEMATIQUE Liner opertions in semi-ordered spces. I L. Kntorovitch (Leningrd) In my Memoir entitled Linere hlbgeordnete Rume" (Mtem. Sbornik, 2 (44), N. 1, *) theory of liner semi-ordered spces ws developed. This clss of spces possesses severl importnt properties not possessed (generlly) by other clsses of bstrct spces. Their importnce is due to the fct tht in these spces some notions cn be defined (e. g., those of positive element", the upper limit of sequence" etc.), which ply n importnt prt in the theory of severl concrete spces, while they cn not be introduced into usul clsses of bstrct spces. The Introduction of semi-ordered spces mkes it possible to tret more fully nd in n bstrct form different questions of the theory of spces which could not be treted before. These remrks pply in prticulr to the theory of liner opertions in semi-ordered spces. This theory contins some fcts nd conceptions (e. g., tht of positive functionl") which re importnt for concrete spces, while they re not met with in (nd cn not be introduced into) the usul theory of liner opertions in Bnch spces. The present pper contins full exposition of this theory; the most prt of the results hve been nnounced (sometimes with proofs) in previous publictions by the present uthor**. In the first chpter the principl clsses of opertions re introduced nd their properties re investigted. Owing to the existence of two types of convergence in semi-ordered spces n (o)-convergence nd (t)-convergence or ()-convergence, the notion of continuous opertion cn be defined in three essentilly different wys nd ccordingly three clsses of continuous opertions will be considered, viz. the clsses H\, H nd // where H\z>H*42H. The most importnt of these three clsses is Щ the clss of (t)-continuous liner opertions which in mny importnt cses coincides with the Bnch clss. The clss // of (o)-continuous opertions is closely connected with the clss of positive opertions, viz. this clss is in mny cses the clss of opertions which re differences of two positive opertions. In the first chpter questions s: wht conditions must be stisfied in order tht n opertion should belong to one of the bove clsses, the questions of the interreltions between the clsses etc. re considered. * This memoir will be cited in the sequel s H. R. ** L. Kntorovitch p], [8], [4], [6], [6], [7] f [8] (seethe Literture t the end of this pper nd in С. г., 206, (1938), 833; С. г. Ас. Sci. U. R. S. S., XIV, ',1937), 531, 537). 1 Математический сборник, т. 7 (49), N. 2.

3 210 L. Kntorovitch The second chpter dels with nlyticl representtions of liner opertions trnsforming clssicl concrete spces into bstrct semi-ordered spces. The representtions given there re, in generl, nlogous to the representtions of functionls. The generl theorems of this chpter involve s prticulr cses gret mny theorems pplying to different spces; of these only frction is indicted. The question on the representtion of opertions, belonging to the clss H\ hs been considered by I. Gelfnd, N. Dunford nd others. The other clsses hve been prtly considered by G. Fichtenholz, B. Vulich nd myself. The third chpter * dels with the questions concerning the possibility.of n extension of given opertion. The principl results of this chpter re formulted in theorem stting tht for opertions of the clss # the conditions of the possibility of n extension re the sme s for functionls, nd in theorem on the extension of opertions of the clss // involving s prticulr cses the theorem on the extension of dditive functions of sets nd other theorems of the theory of functions. We point out lso theorem contining probbly the! simplest proof of the possibility of pssing to limit in integrls of Riemnni Stieltjes. In the fourth chpter the question of the convergence of sequences of liner opertions is considered. Here we point out theorem extending to the cse of (o)-convergence the well known theorem of Bnch-Steinhus tht sequence of opertions bounded in every point nd convergent in n everywhere dense set converges everywhere in the spce. This theorem comprises s prticulr cse the theorem of Bnch nd Sks on the convergence of sequence of functionls depending on rel prmeter. The present Memoir is the second prt of the course of lectures on the functionl nlysis bsed on the theory of semi-ordered spces" red by me in the Leningrd Stte University in The first prt of this course is the lredy mentioned H. R. nd the third, dedicted to functionl equtions, hs been published in Act Mthemtic, 71 (see L. Kntorovitch [ 9 ]). INTRODUCTION. LINEAR SEMI-ORDERED SPACES A liner (vector) set Y= {y} is clled liner semi-ordered spce ( spce of the type /Q, if in this set the reltion J>>0 is defined, i. e. if for every y Y we hve either y^>0 or у is not ^>0 nd if this reltion stisfies the following five xioms: I. 0 is not >0. II, If y x > 0 nd y 2 > 0 then y x -\-y 2 > 0. III. Whtever be у Y there lwys exists such у г Y tht j^^ond Уг У>0. IV. If \ is positive rel number (X>0) nd у is positive element of F(y>0) then Xy>0. These four xioms enble us to define such expressions s у г ^>у 2 (which mens by definition: у г.у 2 >0)> Уг <^У2 etc s wel1 s tne notions - of tne * The chpters III nd IV will pper shortly in this Journl s the second prt of this Memoir.

4 Liner opertions in semi-ordered spces 2^1 upper nd the lower bound of set Ec Y (defined s in the spce of rel numbers) nd of the exct upper (lower) bound of E\ these lst elements we shll denote respectively by supz: nd inf E. We cn now stte the fifth xiom. V. Every set E bounded from bove possesses n exct upper bound (sup ). If in spce Y xiom V holds for countble sets only, then we shll sy tht Y is spce of the type AT", nd if this xiom is true only for finite sets then Y is spce of the type K±. From xioms I HI follows tht every finite set is bounded nd therefore in every spce of the type /Q (nd consequently in every spce of the types K 5 or К ъ ) there lwys exist the elements sup^,..., y n ) nd inf (y v.. -,y n )- We shll denote (see H. R., p. 125): y + =sup(0, y), y_ = sup(0, у), \у = =У++У- = ыр(у, у) (H. R., Theorem 11, e), p. 127). In the spces ЛГ 5 ~~ the notion of limit of sequence cn be introduced in the following wy: We set liin^-infisup^,^!,.. )); l}my n = sup(lnf(y n, y n+1,...)) If (for certin sequence y n ) hmy n = ^ny n =j/, then we cll у the (ordinry) limit of the sequence^ nd write у = \lm.y n or y n *y(o). The limit thus defined possesses ll most importnt properties of the ordinry limit of sequence of rel numbers, nd, in prticulr, Cuchy's criterium of convergence of sequence is pplicble to semi-ordered spces (see H. R., p. 134, Theorem 20, b)). We shll cll regulr" (type KQ) every spce Y of the type K 5, which stisfies the following xiom: VI. If E n cy for л=1, 2,... nd if sup tends to finite or n infinite limit y, then there exist such finite subsets E' n of the sets E n tht limsup =y. Such spces possess the folowing property: if y n > у (о) in F, then there exists such n y Q Y tht whtever be rel number г > 0 there lwys exists n integer N e such tht \y n У\<СцУо> provided tht n^n e (see H. R., p. 142, Theorem 27). A spce Y of the type K^~ will be sid to be of the type K 6 if in this spce the following theorem (see H. R., p. 140, Theorem 25* ) holds: In order tht set Ec Y be bounded, it is necessry nd sufficient tht whtever be y n E (#=1, 2,...) nd Х л -^0 (rel numbers) we should lwys hve l n y n + 0 (o). A spce of the type AT 5, in which Theorem 25 holds, shll be clled spce of the type K. Insted of sying tht spce Y belongs to the type K v or K b etc. we shll write lso Y K A, Y K b etc. Besides the bove defined ordinry convergence nother kind of convergence topologi#l (or (t)-)convergence cn be introduced in the spce Y; it is defined s follows: sequence y n converges topologiclly towrds у in symbols y n *y(fy * The proof of this theorem in H. R., ppels to the non-proper xiom VI й, which we do not here suppose to be fulfilled. 1*

5 212 L. Kntorovitch or y = (t) \imy n7 if every subsequence: y n^ у щ,... of the sequence {y n \ contins nother subsequence y n, у,... such tht Уп к.~+у (о) for i *- oo. A set E is clled (t)-bounded if whtever be the elements y n g E nd rel numbers * л *0 we lwys hve: l n y n -+0 (t) (H. R., pp. 143 nd 145, deff. 16 nd 17). The spces with metric function (the types /? 3, /? 4, /? 5 see H. R., 8) shll be but seldom mentioned in the sequel, so tht they need not detin us now. Let К be spce K~ nd let norm" \\y\\ be defined for elements of У, i. e. suppose tht to every j/^f rel number \\y is coorrelted such tht the following conditions re fulfilled: ) 01 =0, b) ILyi+^IKIWI + ILM. c) P.y = m- Lyl, d) LyiKbl im P lies ЫК1Ы1, e) If {y n } is monotonous sequence tending to zero or infinity {y n >-0 or y n *-oo) then \\y n \\ lso tends to 0 or oo respectively (.yj <-0, resp. Then we shll sy tht К is spce ot the type B 2 (H. R., p. 153). Such spce (of the type B 2 ) will lwys belong to the types К б, В (i. e. it will be Bnch spce, see H. R., p. 154, Theorem 40, e)) nd V (the complex normed spces of B. Vulich*), if we define \\y x,...,y n \\ s equl to sup (\у г,..., \y n ) [ (see ibid.). Besides in this spce the (t)-convergence coincides with the (^-convergence (convergence by norm) nd the (o)-convergence with the (k)-convergence (1. e. y n -+0 (o) is equivlent to: 11m \\y n, y n+1 >...,y m \\ = 0). If Y is of the type /Q nd norm \\y\\ is defined in K, stisfies the conditions ) c) nd the following condition d') l^i I < 1^2 I im P lies \\Уг \\ < 11^2 II» tnen ^ Js са^е( * spce of the type B b The spce of the type В г belongs to the types В nd V, so tht in such spce (b)-convergence nd (k)-convergence cn be defined. A liner normed spce, i. e. liner spce in which norm stisfying the conditions ) c) bove is defined, we shll cll spce of the type B~. If such spce is complete, we shll cll it spce of the type В (а Вапясп soce). A spce of the type V is liner set, in which norm \\y x,...,у п \\ of group of elements is defined stisfying ll tne xioms of B. Vulich (see [*]). We shll enumerte now some concrete spces which will be mentioned in the sequel. LP{p^\) is the spce of ll functions cp (t) defined on bounded set E nd integrble (in the sense of Lebesgue) together with their pth power Equivlent functions (i. e. functions coinciding lmost everywhere) re considered s the sme element of LP. This spce is B 2 -spce with cp^o if cp (x) ^0 -most r I everywhere nd with cp =(\ <f(f)\pdi)p. The (t)-convergence in LP is Ihe * See B. Vulich [Ц,

6 Liner opertions in semi-ordered spces 213 convergence in the men of /?th order nd the (o)-convergence is the convergence lmost everywhere of bounded sequence, so tht ср л > cp (o) mens tht ср л (t) > cp (t) lmost everywhere nd tht there exists such cp 0 Z/ tht whtever be n, the inequlity ср л (t) ^ cp 0 (^) holds lmost everywhere in E. We shll denote by M the spce of bounded functions cp, considered s normed spce with the norm Ile Ij = sup cp (t). This spce is of the type В nd the (b)-convergence in this spce is the uniform convergence in the ordinry sense. The sme spce, but considered s semi-ordered spce will be denoted by if (with cp^o, if in every point t E we hve cp(^)^o). The spce M is of the type К, the (t)-convergence in M coincides with the (o)-convergence nd with the ordinry convergence everywhere with the supplementry condition tht ll functions of convergent sequence must be uniformly bounded. УМ* is the normed spce of bounded mesurble functions, where two equivlent functions re not considered s different elements of M * nd where the norm is defined s cp = vrisup cp (t). The spce M* consists of the sme elements, but is semi-ordered (in the sme wy s LP); this spce lso belongs to the type Къ \ the (o)-convergence in M* is the bounded convergence lmost everywhere, the (t)-convergence is the bounded convergence by mesure. С is the spce of continuous functions, defined in the intervl (; b) with cp ^ 0 if cp (t) ^5 0 (everywhere) nd with [ cp = sup cp (t). This spce is of the type Bj nd the (b)-convergence in С is the uniform convergence. V is the spce of functions of bounded vrition defined in the intervl (; b). This spce is of the type B 2, if we set: 1) /(*)>() if f()^0 nd / never decreses nd 2) / = \f() I - - vr /W- The (t)-convergence is the converge gence by vrition. Л is the spce of ll bsolutely continuous functions; it is closed subset of V. A*(p>\) is the spce of such functions f A tht/' ZA All these spces (A nd A?) re of the type B 2 nd re isomorphic to tne spces L nd LP respectively. IP(p^l) is the spce of ll sequences y = (r^\ 7j (2 >,... ) such tht Х1ч (л) 1 /, <! Tnese spces re of the type B if we set: \)y^0 whenever ll V<)^0 nd 2) ЫНЩИ^!*]' CHAPTER I FUNDAMENTAL CLASSES OF OPERATIONS AND THEIR PROPERTIES 1. Fundmentl definitions 1. The definition of the clsses H. Let X nd Y be two liner spces nd consider n opertion y = U(x) trnsforming X into Y. Such n opertion is clled dditive, if for ny x x nd x % (х г X y x 2 X) we hve: U(*i + *,) = U{*i) + UfrJ-

7 214 L. Kntorovitch If n dditive opertion U is continuous in ny sense, it is clled liner {in the sme sense). An opertion is continuous, if U(x n ) * U(x) whenever x n >x; this definition depends on the definition of convergence in the spces X nd Y. Let X nd Y be liner semi-ordered spces of the type K^. In ech of the spces X, Y two kinds of convergence re defined: the ordinry convergence ((o)-convergence) nd the topologicl convergence ((t)-convergence). Accordingly we my consider four clsses of liner opertions in semi-ordered spces which we shll denote resp. by #, #, H\, //, where Я, е. g., is the clss of ll dditive nd ( )-continuous opertions, i. e. n dditive opertion U belongs to H v if x n -+x(t) in X implies U(x n )-+U{x) {o) in Y. The other three clsses re defined similrly. We shll prove presently tht these four clsses my be, in fct, reduced to three. 2. Elementry theorems on clsses of liner opertions. We shll begin with theorem on the interreltion of these clsses. Theorem 1. ) #* = #'. b) Я ся с:#{. c) If the (i)-convergence in X coincides with the {^-convergence, then d) // the (t)-convergence in Y coincides with the (o)-convergence, then ll three clsses of liner opertions coincide (i. е. Щ = Н о =Н. Proof, ) The inclusion Н\аН { 0 is evident; it remins to prove tht Н\аН\. Suppose, in fct, tht / #*; we shll prove tht U H\, 1. e. tht if x n -+x(t), then U(x n )-+U(x){t) in Y. Let, in fct, x n +x(t) nd let n v n 2,... be ny sequence of positive integers such tht ^<«2 <... By the definition of (t)-convergence we cn find subsequence n u, n k,... (i t <0" 2 <C ) f tne sequence n x, n 2,... such tht x m ~> лг(о). Consequently (opertion U belonging to the clss //*) U(x ni ) +U(x)(t), whence follows tht there exists such subsequence x n.^, x ni,... of the sequence {х П[ } tht U(*n lkl )-+U(x)(o). We hve proved tht every subsequence {U(x ni )} of the sequence {(J(x n )} contins subsequence {U(x ni )} such tht U(x nik ) +U(x)(o). In other words we hve proved tht U(x n ) >U(x){t), q. e. d. The proposition ) being proved, ll other ssertions of Theorem 1 become evident. Remrk 1. The opertions of the clss H t we shll sometimes cll strongly-continuous opertions", those of the clss # we shll cll (o)-continuous nd those of the clss H\ (t)-continuous opertions. Remrk 2. If Y is the spce of rel numbers then (by Theorem 1, d)) ll three clsses H v H o, H\ coincide. In this cse. liner opertion Is clled simply liner functionl.

8 Liner opertions in semi-ordered spces 215 Theorem 2. If n dditive opertion U is continuous (in ny of the bove defined senses) in point x 0 X, then it is continuous in ll points of X in the sme sense. Proof. Consider the following identity U(x n ) U(x) = U(x 0 + (x n -x)) U(x 0 ). (1) Suppose, e. g., tht the opertion U is (^-continuous in the point x 0 ; if x n +x(t), then x Q -\-(x n x) +x 0 (t) nd, consequently, the right hnd side of the identity (l)(o)-tends to 0; it follows tht U(x n ) U(x) -* 0 (o) or U(x n )-+U(x) (o), q. e. d. The other cses re proved exctly in the sme wy. Theorem 3. Every liner opertion is homogeneous y rel number X, we hve lwys i. e. whtever be U(kx) = \U(x). (2) Proof. Consider the most generl cse when U H\. If X is rtionl number, the bove equlity holds for ny dditive opertion U. If X is irrtionl, let {\ n ) be sequence of rtionl numbers converging to X. We hve then \ n x +\x y whence U(kx) = (t) iim U(\ n x) = Urn (k n U(x)) = liml n -U(x) = \U(лг), я-*оо я-э-оо n-+co q. e. d. Let now X, Y, Z be three semi-ordered spces nd U nd V two opertions, one of which (U) trnsforms X into Y nd the other, I/, trnsforms Y into Z. Consider the compound opertion W= VU trnsforming X into Z [so tht W(x)= V(U(x))]. It is evident tht if opertions U nd V re both dditive, then the opertion W is lso dditive; but we cn sy more thn tht, viz. we cn stte the following Theorem 4. If both opertions U nd V re liner then their compound W= VU is lso liner opertion, more precisely: 1) W H\, if U H\ nd V H\\ 2) We//;, if U H Q nd V H 0 ; 3) W H V if U^H\ nd V H Q, or U H\ nd V H V The proof is evident. 3. Opertions in spces of other types. If X nd К re spces of other types, viz. B~, B, B v V, then in these spces the (b)-convergence nd lso (in the cse of spces B x nd V) the (k)-convergence is defined. Accordingly we shll obtin such clsses of liner opertions s H\, H^ etc. Their definition is quite nlogous to the definition of the clsses #?, Я 0, H l (e. g., supposing X B nd Y B V we shll sy tht U H\, if U is liner opertion such tht х п -»х{ъ) Implies U(x n )-+U(x) (k)). Just s in Theorem 1 we hve

9 216 L. Kntorovitch Theorems 2 4 cn lso be pplied (muttis mutndis) to these clsses of opertions. The proofs of Theorems 1 4 cn be pplied without ny ltertions to these new clsses. Finlly, if one of the spces X, Y is semi-ordered nd the other Is normed (one of the types B", B, B v V), then we cn consider the clsses //, H\, etc. To these clsses Theorems 1 4 lso pply. The clss // hs been systemticlly treted in Bnch's work (see S. Bnch [1]); opertions of the clsses // nd // hve been considered by B. Vu- Jich (Vulich, [1] [3]). If one of the spces X, Y is of the type B 2, then in this spce ll four kinds of convergence mentioned bove (i. е. (о)-, (t)-, (b)- nd (k)-convergence) re defined, though only two re distinct, becuse the (b)-convergence in B 2 -spces coincides with the (t)-convergence nd the (k)-convergence with the ^-convergence. Therefore, if X nd Y re both B 2 -spces, we hve gin the sme three clsses of liner opertions, viz. # t (=# = # = #*), H 0 ( = H = = H* = H$ nd Н\{=Н1 = Н1 = Н\). In the following sections we shll consider more closely the three clsses of liner opertions in semi-ordered spces, i. e. the clsses //, H Q nd H\, while the other clsses (such s // k, etc.) rising when one of the spces X, Y (or both; is normed spce, will be considered only occsionlly. 2. Opertions of the clss H\ 1. Criteri of the continuity of n dditive opertion. Theorem 5. In order tht n dditive opertion U trnsforming regulr semi-ordred spce X (of the type KQ) into semi-ordered soce Y of the type К~ ъ should belong to the clss H\ it is necessry nd sufficient tht the imq;e of every {^-bounded set E X should be {^-bounded in Y. Proof. A. Necessity. Let U be n opertion of the clss H\ nd E (t)-bounded subset of X. We must prove tht the set U(E) is (t)-bounded in Y. Let {y n } be sequence of elements of U(E) nd {l n } sequence of rel numbers tending to 0. Since y n U(E) (n= 1, 2,...), there exist such x n E tht y n =U(x n ); but then, by Theorem 3, l n y n =U(l n x n ). The set E being (t)-bounded, we hve l n x n -+ 0(t), whence by the definition of the clss //*, U(l n x n )=l n y n * 0 (t), which proves tht the set U(E) is (t)-bounded. B. Sufficiency. Suppose tht U in n dditive opertion stisfying the condition of our theorem. We must prove tht x n *0(t) implies U(x n ) * 0 (t). Suppose tht x n > 0 (t) nd tke n rbitrry sequence n l9 # 2,... (n x <^n 2 <^ <л 3 <С---) f positive integers. The sequence x ni, x n^... contins subsequence x nh, x nh,... 0(o). In H. R. we hve proved (Theorem 26, p. 141) tht sequence of positive integers {X ; } cn be found such tht X ; - ^-f-oo, while ljx n. *0(o). The set of ll elements of the form \/x n. m (j=\, 2,...) is (o)-bounded nd therefore lso (t)-bounded, whence follows (by our supposition)

10 tht the set of ll U(k J x ni.) = lju(x ni.)* sequence {U(x m,)} Liner opertions in semi-ordered spces 217 is (t)-bounded. Consequently the = { y-* У 7 ( *> ) } (t)-converges to 0 (for-^- +0) nd we cn select subsequence of this sequence (o)-converging to 0. But since our originl sequence n v /z 2,... is rbitrry, it follows tht the sequence {U(x n )} (t)-converges to zero, q. e. d. Theorem 5. If X nd Y re spces of the type /? 5 ** then in order tht n dditive opertion U should belong to the clss Ht it is necessry nd sufficient tht sup p ( /(*))<+ oo. Proof. A. Necessity. The set {-K}p(*)<ci of ll such points xg^ytht р(дг)<1 is evidently (t)-bounded in X (see H. R., p. 153, Theorem 39) nd, consequently (if U H\) t the set {U{x)}?^X)^i is (t)-bounded in Y (since.a"nd y^.s~, see H. R., Theorem 34) nd hence by the theorem cited bove (H. R., Theorem 39) the set {p (U(x))} P ( X )<^i is bounded, q. e. d. B. Sufficiency. Suppose tht sup p ( /(*))< oo. We shll prove tht P (x) ^ 1 U H\. By Theorem 5 it is sufficient to prove tht if set E с X is (t)-bounded then the set U(E) is (t)-bounded in Y. In fct let {y n \ = {U(x n )} be ny sequence of elements of U(E) (where x n E for # = 1,2,...) nd {l n } ny sequence of rel numbers tending to 0; we must prove tht \ n y n *0(t). Denote by k the integrl prt of === We hve then: H ^ V\l n \ 1) Kl<72 < 3 > \-x n -± 2) JL[j(x n y= u( -x n \ (cf. (2) nd the footnote on p. 14); but since R n K n \R n / 0(t) (E being (t)-bounded set), it follows tht p (i - x n ) *0 (H. R., Theorem 37), whence р(г- в лг )^1 for sufficiently gret /z, nd therefore (by our supposition) the set p \U\Y " x n) ) } Abounded; hence the set \U[r- x ) \ is (t)-bounded, which implies tht the sequence iff - x n \ = 2 U(x n ) > 0 (t); but then, since, by (3), l n U(x n ) < 2 [ U(x n ), we conclude tht l n U(x n ) -+ 0 (t), q. e. d. Finlly, if X nd Y re normed spces, the corresponding criterium (tht U H\) hs been given by S. Bnch (Bnch [*]). Since we shll more thn once pply this criterium we shll stte it here. It is * By (2), which is true for ny dditive U nd rtionl X. ** It is sufficient to suppose tht X nd Y re spces of the type /? 4 in which ny set is (t)-bounded if nd only if the set {p(y)\y Eis bounded,

11 218 L. Kntorovitch Theorem 5b (S. Bnch). If X nd Y re spces of the type B~, then in order tht n dditive opertion U should belong to // it is necessry nd sufficient tht A= sup \\U(x)\\<+oo. (4) luil^i Remrk 1. The constnt A is clled fter Bnch the norm of the opertion U. In the sequel It will be convenient to denote this norm by / l{\ It my be esily verified tht </ ъ= sup 'i^i. х Х,хфО И* II Hence we obtin the fundmentl inequlity D iitf(*)ii<ii /Hi;-ii*ii. (5) Remrk 2. In the cse when X nd Y B 2 the clss //{J coincides, s we know, with the clss H\. In this cse the Theorems 5 nd 5b give the sme criterium of U с Н\. 3. Regulr opertions nd opertions of the clss # 1 Fundmentl definitions. In this n. X nd Y will be two spces of the type A" 4 nd U n dditive opertion trnsforming X into Y. We shll begin with the following remrk: n dditive opertion U is completely determined if we know its vlue for positive rguments. In fct if x is rbitrry we cn write U(x) = U((x) + ) U((x)_) y (6) where (x) + nd (x)_ re, of course, positive. Conversely, if U is defined for positive rguments only nd is dditive for such rguments, then defining U(x) for rbitrry x by (6), we obtin n dditive opertion. In fct, let х г nd x 2 be two rbitrry elements of X; then U(x 1 + x 2 ) = U((x 1 +x 2 ) + ) U((x 1 + x 2 )_) = = [U((x 1 ) + )^U((x 2 ) + ) U((x 1 ) + -\-(x 2 ) + (x 1 -\-x 2 ) + )] [U((x 1 )J + U((x 2 )J-U((x 1 )_ + (x 2 )_ (x 1 + x 2 )_)l becuse the quntities nd (*i) + + (* 2 ) + (*i+* 2 ) + (*i)- + (* 2 )- (*i + *2)- re positive (see H. R., Theorem 3,e)). But we hve evidently (x 1 ) + + (x 2 ) + (x x -f x 2 ) + = (*!)_ + (x 2 )_ (л^ 4-x 2 )_, whence follows (in conjunction with the bove formul) tht и(х г + х 2 ) = и(х г ) + и(х 2 ), ч. е. d. Definition. We shll cll n dditive opertion U positive nd write 1/^=0, if for ny лг^о (x X) we hve U(x)^0. If U is positive opertion nd is not identiclly 0 for ll vlues of x, then we shll write />>0.

12 Liner opertions in semi-ordered spces 219 Definition. We shll cll n dditive opertion U regulr, if it cn be mjorized by positive opertion, i. e. if there exists n dditive opertion U x such tht 1) U x^0 nd 2) U x U^0. The clss of ll regulr opertions trnsforming X into Y we shll denote by H v Theorem 3'. If Y is of the type / r, then every regulr opertion is homogeneous, i. е., if U is regulr opertion, then whtever be x X nd rel number X we hve lwys U(kx)=W(x). (2) Proof. It is sufficient to prove this identity for x nd X positive. Then it will be estblished for ny x nd X. In fct, for rbitrry x nd positive X (supposing the theorem proved in the cse when x^o) we hve U(lx) = U((lx) + Y U((kx)J = = U(k(x) + ) U(\(x)J = \[U((x) + ) U((x)J]=W(x) t nd if X is negtive, then U(lx) = U( \l\x) = U(\l\x)== \l\u(x) = lu(x). On the other hnd, in the cse when X is rtionl number the theorem follows immeditely from the fct tht U is n dditive opertion (cf. Theorem 3). It remins to prove our ssertion in the cse when лг^о nd X is positive irrtionl number. Let U x be mjornt opertion for U. Then for ny x^o we hve U(x) ^ < U x (x) nd U(x)^ U x (x) U (x); hence, setting V= U x -f (U x /), we hve U(x)^ V(x) nd U(x)^ V(x), nd therefore U(x) = sup (U(x), U(x)) < V(x) for ny x^ 0. Denote now by r x nd r 2 two rtionl numbers such tht r x <^\<^r 2. U(lx) W(x) < U(he) U(r x x) +1 r x U(x) \U(x) = = \U((l r x )x)\-\-\(r x l)u(x)\^v((l Гг)х) + (к r x )V(x)^ < V{{r 2 -r x )x)-^(r 2 -r x ) V(x) = 2(r 2 -r 1 )V(x); Then r 2 nd r x being rbitrry rtionl numbers stisfying only the condition r x << X << r 2, we cn choose them such tht 2 (r 2 r x ) = -r, where & is n rbitrry positive integer. We hve then or whtever be &, which cn only be if I /(X*)_M/(*) <-± K(x), 0 < & U (kx) W(x)\^V (x), \U(\x) W(x)\ = 0 (see H. R., Theorem 15, f), p. 130) or q. e. d. U(kx)=W(x),

13 220 L. Kntorovitch 2. The spce of regulr opertions. Regulr opertions form liner set. In fct, if U t nd U 2 re two regulr opertions nd V t nd V 2 re is their positive mjornts, then V x -j- V 2 positive mjornt for U x -f- U 2 nd therefore / 2 -(-f/ 2 is regulr opertion. Similrly, if U is regulr opertion nd V its mjornt constructed s in Theorem 3' (i. e. such tht V^U nd V ^5 U) then \U hs X V for its positive mjornt nd is therefore regulr opertion. Thus regulr opertions trnsforming spce X K± into nother spce F /C 4 form liner spce in which the conception of positive element ( />0) is defined. We shll prove now tht in the cse when Y K b this conception stisfies ll the xiom (Axioms I V) of semi-ordered spce. Theorem 6. // X K A, Y K b, then the set of ll regulr opertions trnsforming X into Y forms semi-ordered spce of the type K b *. Proof. Axiom I. 0 Is not >0 by definition (of>0). Axiom II. If ^>0 nd / 2 >0 then /7 1 + ^/ 2 >0 (evidently). Axiom III. Whtever be U there lwys exists such U x tht U 1^U nd i/^o (by the definition of regulr opertion). Axiom IV. If />0 nd i>0 then U/>0 (evidently). Axiom V. In order to prove this xiom we shll estblish first the following Lemm. If x x'-\-x"\ x'^0, л/'^=0, nd if, on the other hnd, x = х г х л' х^о (7=1,..., n) then we cn represent ech x. s sum of two positive elements (x. = x'.-\-x".; x.^0, x'^^o) in such wy tht x == x \ \ \~ x n> x == x i ~r \ x n' Proof. It is sufficient to prove tht if x x n ^ x 1 ^ 0 nd x. ^ 0 for i = 1,..., n then we cn find such elements x\ (i = 1,..., n) tht 0 ^ ^x'^x. nd x\ -\-... -\-x' =x! (becuse then we cn set x'! = x. x' for * * 1. 7* i l l i = l,..., n). This lst proposition we shll prove by induction. For n=\ it is evident; supposing it to be proved for n = n 0 we shll prove it for n = n 0 -j- 1. Set x' nrhl = lni(x lh+v x'); then *!+ +^ 0 = SU P( > (*i + + x no+i) x n 0 +i) > ^sup(0, x' x no+,)=x' inf(x\ х По+1 )=х' х' щ^^о. nd But by the inductive hypothesis there exist such elements x' v..., x' tht or, which is the sme, O^x'.^x. (i = 1,..., n Q ) x i + + х 'щ = x ' ^n-i x \ i i х я 0 -и == x» since дг'^+i is lso ^0 nd ^л: Ло+1, the lemm is proved. * The ckss of regulr functionl nd the spce formed by these functionls hve been considered by F. R i e s z [ 3 ] in the prticulr cse when X is the spce of functions. As to the generl cse see L. Kntorovitch [ 4 ].

14 Liner opertions in semi-ordered spces 221 We cn prove now Axiom V. Let {6^} e sbe bounded set of regulr opertions nd U Q its upper bound (i. е. Ц^ U 0 for ny SgS). We shll set for x^o so tht U(x)= sup [^(^) ^Wl- ( 7 ) Xi+...+X n =X *i^q,..., ^n^o 6 5 «s First of ll this sup Is finite nd does not surpss U 0 (x), becuse Ц, (*j) < t/ 0 (Jfj),..., Ц (* ) < U 0 (x n ), On the other hnd if we tke n = 1 nd ^ = S 3, then U^ {xj -f-... -f- ~h^n С*/») ^'ЦМ» so tn * U^(x)^U(x) for ny лг^о. We shll prove now tht the opertion U which is defined for positive x is dditive for such x. In fct let x = x' -f- У; x' ^ 0, лг" ^ 0 nd suppose tht x' = x[-{-... -\~x' n - 9 j?'= x'{-{-...-\-x" m, where л:,, X". re ll positive (1. e. ^0). We hve then (fore;,..., s;, $;,..., C^s) [ц, (*;) u %, (x' n )}+[ц, Ц') щ K,)] < t/(*> 1 n 1 m (by the definition of /(*), becuse x = x'^-\-... Н-^ + ^Ч" ~b x m ll summnds re ^0). This being true for every subdivision of x' nd x" into positive summnds, we conclude tht nd U{x')i-U(x")^U(x). (8) If, conversely, x x x -}-... -f- x n, x^o (/ = 1,..., n) then ccording to the bove Lemm we cn find such x' v \..., x' n nd x" v..., лг^ tht 2) х / = У / +^ for /=1,...,я; 3) *'=*;+...+<; *" = *{+...+*. Then Щ*1)+- +Цп (* ) = [Ц>1>+ +^n «)] + + [^ (xj) U in (xl)\ < U(x>) + ЩлГ). This being true for ny representtion of x s sum of positive elements it follows tht /(*)< /(*') +1/(*"). This inequlity together with the Inequlity (8) proves tht U(x) = U{x') + U{x?') (where x x' -\- У; л:'^ 0; x"^0), q. e. d. Thus U is n dditive opertion defined for x^0. As we lredy know, we cn define (nd in one wy only) U for ll x so tht it remins dditive. Since

15 222 L. Kntorovitch for x^so we hve U(x)^U 0 (x) 1 U is regulr opertion; besides, U^U^ioi ny E nd U^U Q, where U 0 is ny upper bound of the set Щ}. Consequently U is sup (Щ}), which proves Axiom V. We hve thus proved tht regulr opertions trnsforming spce X(zK± into nother spce Y(zK b form semi-ordered spce. We shll show now the mening (in this spce) of the symbols U +J \U\ 9 U n +U(o). Remrk 1* The opertion U + =s\\p(u, 0) my be defined for positive x (in ccordnce with (7)) s follows n U + (x) = sup _ 2 /(x.)=sup U(x'). Remrk 2. The opertion Lr = / = sup(t/, U) = U + + U_ my be defined for x^o s follows U*(x)= sup [t/^) /(* 2 )] = sup lu(x') U(x x')]. (8 bis) In fct We shll prove now the following Importnt inequlity: \U(x)\^U*(\x\). (9) U(x) = U(x + ) U(xJ^U + (x + )^U^(xJ^U*(x^) + U*(x_) = U*{\x\) nd in the sme wy U(x)^U*(\x\) 9 whence I /(*)! = sup ( /(*), U{x))^U*(\x\), q. e. d. Remrk 3. The opertion U* my be defined lso in severl other wys; for instnce, it my be defined s the lest dditive opertion U t stisfying the following inequlity \U(x)\^U 1 (\x\). (9') ( We hve seen lredy tht for U x = U* this inequlity holds. It remins to prove tht if U t stisfies (9') then U*^U V In fct, we hve for x^o: U x (x) ^ U(x) nd U t (x) ^ U(x); consequently U x^ U nd U x 7^ U, whence U x > sup (U, U) = U*, q. e. d. Remrk 4. The opertion U* my be lso defined s follows: for x^o U*(x)= sup \U(x')\ = sup U(x'). (10) \x'\^x In fct, first of ll we hve evidently \x f \^x sup U(x')= sup [sup( /(*'), U( **))] = \x'\^x \x'\^x sup (sup ( /(*')),-( /(*'))]= sup (\U(x'){)

16 Liner opertions in semi-ordered spces 225 nd thus sup (U(x')) = sup \x'\<^x \x'\& {\U(x')\)=U(x). We shll prove tht U* (x) = 0 (x) (for x^o). If x' is such tht x* < x y then we hve by (9) f/m <^(U' )<(/ 4 w, whence U(x)^U*(x). (11) On the other hnd if х г -\-х 2 = х, х г^0, x 2^0, then (becuse ^-л 2 <^- -^ = ^ U (x,) U (x 2 ) = U (x, x 2 ) < / (*) whence (by (8 bis)) follows tht U*(x)^U(x). (12) From inequlities (11) nd (12) follows tht U* (x) = U(x), q. e. d. Remrk 5. With this definition of U* the following definition of regulr opertion (equivlent to the originl definition) is connected: An dditive opertion U is regulr, if for every x ^ 0 we hve U(x)= sup!/(*')<+oo. \x'\& In fct if the opertion U is regulr (in the old sense), then U (x) = U* (x) is finite; nd conversely, if U(x) is finite then the opertion U* defined for x ^ 0 s follows: U*(x) = sup \U(x') U(x x')] O^x'^x is finite positive dditive opertion surpssing U. Consequently (/ is regulr opertion, q. e. d. 3. The convergence in the spce H T of regulr opertions. We pss now to the reltion U n -+U(o) in H r. We shll prove tht in the cse when the sequence {U n } is monotonous, this reltion is equivlent to the following: for every x X lim U n (x) = U(x). П-ЮО In fct, suppose, e. g., tht the opertions U n form decresing sequence nd tht there exists finite limit \imu n = mfu n = U. * Then, if x^so, the sequence U n (x) decreses nd is bounded (becuse U ^U(x) for ny n). Therefore there exists the limit (x)^ lim U n (x) = U(x)^U(x). The opertion U(x) (defined for x^o) is dditive nd bounded by regulr opertion (viz. и г ); consequently it is regulr opertion. Besides, we hve U ^U n for ny #, nd consequently

17 '224 L. Kntorovitch Since, on the other hnd, U < U t it follows tht U=U (if x Is not ^ 0, U{x) is, of course, defined by (6)). We hve now for ny x (o) lim U n (x)}= lim U n ((x) + (x)_) = Hm U ((x)+) lir U n ((x)j = U((x) + ) - U((x)_) = U(x), q. e. d. Suppose gin tht { / } is decresing sequence nd tht the sequence {U n (x)\ possesses for every x finite limit (o) lir П-+0О U n {x)=u{x). Then U(x) is n dditive opertion nd, being bounded by the regulr opertion U v is itself regulr. Consequently the sequence U n is bounded nd (being monotonous) possesses finite limit, which, s we hve lredy seen, coincides with U. In the generl cse when U n is n rbitrry sequence of regulr opertions converging towrds /, the equlity lim U n (x) = U(x) (for every x X) remins true, but not its converse, i. e. there my exist such opertions U n nd U tht for ny x lim U n (x) = U(x), while the sequence U n is divergent. In order to prove the first of these ssertions we must only show tht, setting U =lim U n nd U= lim U n, we hve for ny x^o (supposing U nd U to be finite) becuse then in the cse U= U=U U (x) ^ Ш U n (x); U (x) < lim U n (x), (13) я-* 00 n^oo we shll hve for x^so: iim U n (x) = lim U n (x) = U (x), whence we shll be ble to conclude s bove tht for ny x: UmU n (x) = U(x). It evidently suffices to prove the first of the inequlities (13); we hve U = lim U n = lim sup U n я-х» rc>nn nd if we set U m = sup U n, then U m^u n for ny n^m, or, whtever be x ^ 0, we hve С/ да (х) ^ / л (х) nd ccordingly U m (x)^supu n (x). ri>im On the other hnd the opertions U m form decresing U=limU m, whence, s we hve lredy proved, U(x) = lim U m (x) ^ lim sup U n (x) = lim 6/ я (х), sequence with q. e. d. 4. The csewhen Y K$. The spce К my be regulr spce (of the type KQ~ ог even ^e) w^nout H t being necessrily of the sme type. But the cse is different with the type K$. Viz., we hve the following theorem:

18 Liner opertions in semi-ordered spces 225 Theorem 7. // Г / then H^K^ (we suppose tht XZKJ. Proof. Let E= {U^gz be non-void set of regulr opertions possessing the following property: (). If Х й -> 0 nd S S (n= 1, 2,...), then \ n U^ -> 0 (o) in H t. We must prove tht then there exists positive opertion U 0 such tht [7 e <t/ 0 for ny H. First of ll we my suppose tht ll U^ ^ 0, becuse otherwise we my substitute U^\ for every U$ this substitution does not ffect the property (): in fct if 1 п Ц ы -> 0 (о), then [ l n U in \ = \ l n \. ЦJ 0 (o) nd hence XJ ЦJ 0 (o). Further we my suppose tht if n opertions U v..., U n E, then their sup(^7 1,..., U n ) lso belongs to E; in fct if we dd to E the supremt of ll finite groups of elements of E, the property () will not be ffected by this extension of E. Indeed, let m /, = sup(^>,..., ^i>); C/C/> (/=1, 2,... ; j= 1,..., k.) nd X^. *-0; then the sequence ki times k 2 times / л ч ^ * -s Aj, Aj,..., Aj, A 2, A 2,..., A 2,... lso tends to 0 nd therefore the sequence \Щ\... \Щ*, \Щ),... ). 2 ЦЧ... (o)-converges to 0 in Я г, whence follows tht the sequence \U V X 2 / 2, (o). Let now л: be positive element of X Whtever be X., >0 nd U. E, we hve l.u.-^ 0 (o) nd consequently, ccording to 3 (see p. 25, f)), X^. (л:) -* *- 0 (o). Hence, the spce Г being of the type K (see Introduction, p. 5), the set {U(x)}$3 (x is constnt) is bounded. We set U(x) = suput(x). The opertion U is thus defined for every лг^о, it is finite nd positive (i. e. ^0) for ny such x; we shll prove now tht it is dditive. Let x== = x x -\- x 2. where x x ^ 0 nd x 2 ^ 0; then for ny whence u% (*) = ^ (*i) + Ц (* 2 ) < и (*i) + # (* 2 ). sup Ц(лг) = Jw^^W-f^^). (14) On the other hnd, the opertion Ц = sup (U^, U^) E nd we hve Ц, (*i) + Ц, (* 2 ) < Ц, (*i) + u ft,, (* 8 ) = Ц, ( *) < > W, whence U{x x ) + U(x 2 )^U{x). (15> The inequlities (14) nd (15) show tht U(x l ) + U(x 2 )=U(x 1 +x 2 ). 2 Математический сборник, т, 7 (49), N. 2.

19 226 L. Kntorovitch Thus the opertion U (being positive nd dditive) is regulr nd, besides, U^U^ for ny. This proves tht the set Щ} is bounded, q. e. d. 5. The clsses H r nd H 0. In order Lo estblish (in certin cses) the identity of the clsses H r nd HQ we shll prove first lemm, immedite corrolries of which will include the bove mentioned identity nd some other results. Suppose tht in spce X of the type /Q there is given definition of -convergence to 0 * 0(J). Then we cn define the clss H of dditive opertions trnsforming X into Y nd Q-continuous t the point 0 (of course we must suppose tht Y K~). One such definition of -convergence merits our specil ttention. Definition. We shll sy tht x n -^~ 0 (а а ), if there exists sequence {\ n } of positive integers such tht \ n ^ oc, while the set {^J-^J} is bounded. Lemm. Let Y(zK, X(iK v Then the clss of regulr opertions coincides with the clss of (^-continuous opertions, i. e. H r = H Gl. Proof. A. Я г ся. Let U H X. If x n ^0 (^, then (by the definition of (Gj)-convergence) we cn find such positive integers l n - oo nd such n element x 0 tht Wx n \^x 0 * ог я=1, 2,... On the other hnd, U being regulr, there exists the opertion U*=\U\ nd we hve (by (9)) \U(x n )\ = ±\U(l n x n )\^±U*(l n \x n \)^±U*(x 0 ), whence (cf. H. R., p. 136, Theorem 21, e)) h n U(x n )-+0(o), i. e. / //, q. e. d. B. ffi x H v Let / # ; we must prove tht U H V To this end it is sufficient to show, tht ll the sets of the form {U(x')}\ X' \^x 0 (where x 0 ^ 0 is n element of X) re bounded (see Remrk 5 to Theorem 6). Let {l n \ be sequence of positive integers tending to oo, nd {x' n } sequence of elements of X h n h n such tht \x' n \^x 0 (for я~1, 2,...); then \ Г Х 'п\^х^ L e ' whence (since U # ) /(*')-> O(O). < < ) Now if e n > 0 nd if we denote by l n the integrl prt of j, then Х л ->оо, whence ^-U(x' n )-+ 0 (o), but evidently \s n U(x' n )\^\^ U(x' n )\. We hve thus proved tht whtever be sequence {U(x' n )} of elements of the set nd {U(x')}\x'\^x 0 sequence {e n } of rel numbers converging to zero, we hve lwys WO 0(o).

20 Liner opertions in semi-ordered spces 227 Since Theorem 25 of H. R. holds in K, it follows tht the set {U(x)}, x, ] ^x is (o)-bounded, q. e. d. The Lemm is thus completely proved. The following two theorems re its immedite consequences. Theorem 8. If X K7 nd Y^Ks, then for these spces the clss of regulr opertions coincides with the clss of (o)-continuous opertions, H x = Hl (16) This theorem is prticulr cse of the preceding, lemm becuse in spce of the type ATs~ the (o)-convergence coincides with the (з г )-convergence (H. R., p. 142, Theorem 27). Remrk. The bove theorem my be lso stted differently (see Remrk s to Theorem 6), viz.: In order tht U Hl it is necessry nd sufficient tht the imge of every (o)-bounded set in X be (o)-bounded in Y (s bove X KQ, Y Ks). In this form Theorem 8 is nlogous to Theorem 5. Theorem 9 *. Is X=C, Мог M* nd Y^Ks then Hl = H r (17) This theorem is lso prticulr cse of the foregoing lemm, becuse the (b)-convergence in the spces C, M, M* (the lst two spces we consider s simultneously normed nd semi-ordered, i. e. s M nd M (resp. M* nd M* ) simultneously) coincides with the (,)-convergence in these spces. Remrk 1. Let now Y K Q nd suppose tht in X there is defined (j)- convergence stisfying the following three conditions: (1) x n -+0(j implies x n >0(). (2) If x n -+0() nd x' n ^ x n \ (n=l, 2,...) then л^->0(а). (3) If () x n > 0 (); (b) /z 0 = 0, n v /z 2,... is sequence of incresing integers К<я 1 </г 2 <...); (с) y m = x k for n k _ t <m*^n k {k=\, 2,...); then y m >0 (). Then [/^Я 0 implies U* = \ U\ tf (U* exists owing to condition (1) which implies tht # ся = Я, becuse К g/f e cz/ ). In fct let x n >0(j); suppose first tht x n^0 nd let us consider the opertion U*(x)= sup U(x') (see (10)). I x' I s^ x The spce Y being regulr, we cn find for every /z, finite number of elements x{ n \..., дк я ) such tht nd Consider now the sequence 14»> <* я (/=1,...,A ) (18) sup (U(xf)),..., U(xW)) - ^ (x n ) -+ 0 (o). (19) * In the cse of functionl this theorem my be considered s lredy known. In the cse when X = M> e. g., it is stted in Bnch ([*], p. 217). 2*

21 228 L. Kntorovitch From (18) nd conditions (2) nd (3) ppers tht this sequence -converges to 0 (becuse x n > 0 ()) nd consequently the sequence (o)-converges to 0, whence U (*(i>),...,u (40), U (xf)),..., U(x$),... sup ( /(*(»)),... ^/(^ягоо ( ) nd combining this result with (19) we find tht /*(* ) 0(o), q. e. d. In the generl cse (when not ll x n ^ 0) from x n * 0 () follows (by condition (2)) лг я *- 0 () whence (\x n \ being positive elements) U* ( x n \) >- 0 (o), ^* ( * ) < I/* ( x n ) nd therefore /* (х я ) -* 0 (o), q. e. d. In prticulr the (b)-convergence nd the (k)-convergence in spce of the type B x nd the (o)-nd the (t)-convergence in spce of the type K b stisfy ll the conditions (1) (3), whence follows tht if U H y tf, Щ or H, then U* = \U\ belongs to the sme clss (if Fg/tQ. Remrk 2. If X is the spce of the type K b (but not Кб) then positive opertion y=u(x) my be not n H ; for U H Q it is necessry nd sufficient tht U(x n ) *0 for ny monotonous sequence х п -+0, х г^х 2^.. In fct if x n >- 0 is n rbitrry sequence nd we set x n = sup{x n,x n+ly...); * я =1пЦ* я> х я+1,...), we hve x n -^0, x n -^ 0 nd therefore /(лг я ) *0, U (x n ) ^ 0. But we hve x n^x n^x n nd /(# я ) ^ /(л; я ) ^» U(x n ), therefore t/(х я ) 0. It follows in prticulr from this ssertion tht if U is positive nd U H\, then U H Q. In the cse when Y is the spce of rel numbers the bove ssertion holds for regulr opertions, i. e. if f(x) is regulr functionl nd f(x n ) + 0 for ny sequence x n -^ 0, such tht x x ^ л: 2 ^=..., then / is (o)-continuous. It is sufficient to prove tht if the lst condition is stisfied for / it is lso stisfied for / + (nd /_), becuse pplying to / + nd /_ the foregoing ssertion we come to our theorem. Let us suppose, in contrry, tht there exists sequence \x n } such tht x n -* 0, л?! ^'л; 2 ^... nd lim / + (x n ) =e >0. We tke e k such tht 2ns n -\- /z-> oo n e - ~ 3 Ve^<C у for ny я. We cn suppose, tking if it is necessry subsequence, l tht /+(* )<е + е я nd /(* я )<е я (л=1, 2,...) We hve / + ( *: )= sup /(л:') nd we cn find therefore x' n such tht 0^x' n^x n nd /(4) >Д (л- ) (я= 1, 2,...) We shll prove t first tht /*(K ^!)<3s A + 2 /l (/ =!/!=/*+/_; «>&).

22 Liner opertions in semi-ordered spces 229 In fct f-(\x' n -x' k \)^f_(x' n -}-x' /t )=f + (x' n + x' k )-f(x' n + x' k )^f + (x n + x k ) /K + *;)^e + e ft, / + (K-4 l)</ + K-<)+/ + (**-^)<2/ + (x k ) -f(x' n )-f(x' k )< nd the inequlity is verified. Let now We hve then but <e ft + (8 + s ft ) (e 8 я )=е я + 2е й, x* n = mi(x[,x' 2,...,x' n ). x* 5= 4 5s...; x* n >- 0, n-a >/K)-S Г(К-^1)>г-г я -2(2г л +Зг А )5 : _ (2да +32 & )>4 1 Therefore for the monotonous sequence \x^\ we hve lim /(-*;*) > y in contrn -> oo diction with our supposition. We note tht the ssertion just proved is pplicble lso in the cse when X is semi-ordered group (H. R., Axioms I, II, III nd V). 6. (o)-con ti n uous opertions in the spces of the type B 2. By Theorem 8 if X nd Y re both spces of the type B 2 the clss of regulr opertions trnsforming X into Y coincides with Hi (see H. R., p. 154, Theorem 40, nd Remrk on p. 150). Definition. The ( )-norm of regulr opertion U is the ( )-norm of the opertion U* == U\: \W\t=Wt (20) From this definition follows tht \\U\l= sup \\U*(x)\\ = sup sup /(*'). H * ^ l \\x\\^l \x>\^\x\ Theorem 10. If X nd Y re spces of the type B 2 then in order tht n dditive opertion U should belong to the clss H =H* it is necessry nd sufficient tht the number P= sup 1 U(x x ),...,U(x n )\\ ll*i,... *nj ssl should be finite (we recll tht by x ly..., x n \\ we denote ] sup ( х г,..., кп и ; Besides, if P is finite, then P=\\Ul*. (21) Proof. First of ll it is lmost evident tht P= sup [ sup U( Xl ),..., U(x n ) ]. (22) 11*11^1 ijpil^ljfi * A similr condition for (k)-continuous opertions ws obtined by B. V u 1 i с h [*], though without the identity (21).

23 230 L. Kntorovitch We pss now to the proof of Theorem 10. A. Necessity. Let U H nd U* = \U\; then \\U( Xl ),... f /(jf I ) = sup( t/(^1) >..., U(x n )\)\\^ < y(sup( ^1 l... J*J)) < ] /*ltk,... у х п \\=\\и\\цх 1У whence follows immeditely tht...,*j f /><те (23) nd therefore P is finite. B. Sufficiency. Suppose tht P is finite nd let x X nd x.^x (/=1,..., л), IxJ^I^I. Let r be ny rtionl number such tht r^\x\. 1 1 Then x. I A 1 l r -X 1 nd consequently (see (22)) -л: nd r ±MU{ r Xl ),...,U(x )\\ = \\u(z),...,и(ь)\\^р, whence, r ^ x being rbitrry, we conclude tht \\U(x x ) 9..., /(* ) <P *. Consequently, ccording to Theorem 40, d) of H. R. (p. 154) the set {U(x')}\ ^ ssjc is (o)-bounded. Therefore the opertion U is regulr, i.e. U H i = H Q (Theorem 8, Remrk, p. 227). Now since Y K Q we cn choose (supposing x ^ 0) (H. R., p. 139, Theorem 23, )) sequence x v x 2,... such tht x. \ ^x (i = 1, 2,...) nd U*(x) = sup U(x') =sup U(x n ) \=hm[snp(\u(x 1 )\ 1..., /(* Л ) )]. I x' ^л; я я-» oo Hence U* (x) =UmJ sup ( U (x,) \,..., \U(x n )\)\\= ' provided tht [ A; ^ 1, i. e. = lim f/(x 1 ),..., /(* ) </> Я-> OO 11^*115 = I tf &<Л (24) This inequlity together with the inequlity (23) gives (21), q. e. d. Remrk 1. Note the following importnt inequlity \\U(x 1 ),...,U(x n )\\^\\Ug-\\x v..,,x l l\, which is the immedite consequence of the equlity P= /. Remrk 2. A similr theorem exists in the more generl cse when X nd Y re spces of type /?*, i. e. spces of the type R B, stisfying the condition p(x n ) > oo, if x n»oo monotonously (cf. H. R., p. 150, Theorem 35). 7. A theorem on the coincidence of clsses. In conclusion of this section we shll prove the following Theorem 11. If X B 2 nd Y B 2 nd if in the spce X the following condition is stisfied: ] x x -\-x % = [I x x [] -f- [ х г \[ provided tht x x >0, x 2^>0 (25)

24 Liner opertions in semi-ordered spces 231 ^-conti then the clss of (t)-continuous opertion coincides with the clss of nuous opertions, I. e. nd for every U H o. Proof. Let U H\. Set for x^o Но о = н\ = н\=н* \\u\l=\w\l U(x) = sup _ ( U( Xl ) U(x n ) ) (26) Xj^0(/=1,...,«) We shll prove tht 0(x) is finite. In fct, denote by E the set of ll elements (of the spce Y) of the form U (x x ) \ -{-...-]- U {x n ), where x. ^ 0 nd x x =...-\-x n =x. Then for ny у E \\y\\ = \\\ ui*i) I I u(* n ) I IK II ^ W II + + II "(*«) II < <II^IE(II^II+---+1I^II) = 1I / IMI^II- < 27 > Further, if / E nd у б ' ", then У= /К) /«) ; У= /«) /«)!, where x'.^z 0 (/ = 1,..., n) x" ^ 0 (/= 1,..., m) nd x\ x' n = *" -f- -[-...- "AT" = x. From the Lemm proved on p. 220 esily follows (induction by m) tht we my represent x in the form x=== 2 */,*» x i,k^* 0 > x ';=2i x i,k ( l=l > -.«); m The element ^*=2*л* (*=i,,«).v= 2 1<Д*/,*)1 &^1,, m evidently belongs to E. On the other hnd m n m n m nd in the sme wy y^y f. We hve thus proved tht if У nd y", then ' contins n element j/ ^ sup (y\ y"). It follows tht if J/J, ->.,y n re ll elements of ", then " contins n element y^sup(y 1,...,j/ ). Hence II sup(y v...,л)11<11^11<1 г/1е-11^1 (by (27)), whtever be y v...,y n E. Applying now Theorem 40, d) of H. R. (p. 154), we conclude tht E is bounded set, i. e. tht U (x) = sup E is finite nd tht l tf(*)u<l tf &- *l[- (28)

25 232 L. Kntorovitch From the definition of 0 (x) (i. e. (26)) nd (8 bis) follows tht if U(x) is finite, then U* (x) is lso finite nd U* (*)< U(x) (x^ 0). Accordingly (by (28)), if x^o, then ] U* (x) < иц- \\x\\. If x is rbitrry, then by (9) II U*(x) = \U* (x)\ ^ U*(\x\) < / b. x ] = ] / g *. We hve proved thus tht: 1) if U H\, then U* exists, i. e. U H j in other words tht #{ся ; but since lwys Я ся, it follows tht H\ = H o \ 2) У* (x) < UI x, i. e. tht t/* g= / *< /. But since, on the other hnd, / < /], it follows tht ] i/ g=h /. The theorem is proved in ll its prts. Remrk 1. Note tht the condition of Theorem 11 is stisfied if X=L r V, A or / (nd F 2 ). Remrk 2. In similr mnner we could prove tht if X B V Y B 2 nd in the spce X the condition (25) is fulfilled, then H\ = H\. 4. Opertions of the clsses # nd # (strongly continuous opertions) 1. A necessry nd sufficient condition for n opertion to belong to the clss //. Theorem 12 *. If X B~, Y K, then in order tht n dditive opertion U should belong to the clss H, it is necessry nd sufficient tht there should exist n element y 0 Y such tht (whtever be x X) \U[x)\^y 0 \\x\\. (29) J Proof. A. Sufficiency. If the condition (29) is stisfied, nd If x n -+0 (b), - e - II x nii - t h e n \ и ( х hence п)\ > ( ); q. e. d. B. Necessity. Let / Я nd consider the set \U(x)} \\ x \\^i- This set is (o)-bounded. In fct, if {l n } is sequence of rel numbers converging to 0 nd {x n } is sequence of elements of X with А: Л ^ 1, then Х л лс л = Х л «лг д 0^ nd consequently (since / // ), l n U(x n ) = U(l n x n )-+0(o). But this proves tht the set {U(x)}\\ x \\^x is (o)-bounded (since Y K nd therefore, by the definition of the spces /, Theorem 25 of H.R. holds in Y). We hve proved tht It is obvious tht for ny x X q. e. d. y Q = sup /(*) <+oo. 11*11*21 we hve tf(*) <JVlM> * The opertions of the clss # hve been introduced in previous pper by the uthor (see Kntorovitch [ 3 ]).