Zur allgemeinen Theorie der halbgeordneten Räume

Abstract

The paper ''On the general theory of semi-ordered spaces'' (''Zur allgemeinen Theorie der halbgeordneten Raume'') was written by L.V. Kantorovich and G.R. Lorentz sometime in 1937-1939, and this is the first time it appears in print. The following is a short history of this manuscript. In his letter to I.P. Natanson written on October 11, 1937, G.G. Lorentz mentioned a talk on joint work with L.V. Kantorovich that he gave at a Session on Functional Analysis in Moscow earlier that year. The records of the Academy of Sciences of USSR indicate that a Session on Functional Analysis took place in Moscow during September 27-29, 1937, and that G.R. Lorentz gave a talk ''Topological theory of semi-ordered spaces'' there, and that L.V. Kantorovich was speaking on ''Theory of linear operations in semi-ordered spaces''. The manuscript ''On the general theory of semi-ordered spaces'' was found in the archives of L.V. Kantorovich. According to Vsevolod Leonidovich Kantorovich, L.V. Kantorovich's son, it was submitted to Trudy Tomskogo Gosudarstvennogo Universiteta imeni V. V. Kuibysheva (Proceedings of Tomsk State University). The typed version of the manuscript has a handwritten note by N. Romanov dated by August 31, 1939 stating that the manuscript is accepted for publication. The manuscript was never published (probably because of the World War II) and around 1945 was returned to L.V. Kantorovich. It has been decided to publish this manuscript in its original language (German), and translate the extended abstract accompanying this manuscript from Russian to English. The manuscript appears here in its original form with only minor editorial corrections. Publication of this historical document would not have been possible without the assistance and effort of many people. In particular, the significant help of C. de Boor, Ya.I. Fet, V.L. Kantorovich, V.N. Konovalov, and S.S. Kutateladze is acknowledged and greatly appreciated. Extended abstract The current manuscript is devoted to the investigation of general semi-ordered spaces that are not necessarily linear. Hence, it may be considered a generalization of the work of L.V. Kantorovich [Linear semi-ordered spaces, Mat. Sbornik, 2 (1) 1937, 121-168]. We say that a set Y={y} is a semi-ordered space if its elements are partially ordered using a relation '' "~y"n"""k"""""""i. This type of convergence, *-convergence, turns out to be identical with the topological convergence that we arrive at if we turn Y into a topological space using the convergence defined initially. Relationships among various limits which we can define using the above approaches as well as some properties of these limits are studied in & 1 and & 2. In & 3, we study semi-ordered spaces equipped with a nonnegative metric function @r(y"1,y"2) defined for all pairs y"1, y"2 such that y"1=y monotonically, then @r(y"n,y)->0 (or @r(y,y"n)->0). 5^@?.If y"n monotonically tends to infinity, then the condition lim"n","m"->"[email protected](y"n,y"m)=0 should not hold. Let @r(y"1,y"2,...,y"n)[email protected](inf(y"1,...,y"n),sup(y"1,...,y"n)). Then y"n->y turns out to be equivalent to @r(y,y"n,...,y"n"+"p)->0 when n->~, and y"n->y(*) is equivalent to @r(y,y"n)->0. In addition, Cauchy's convergence principle holds. Moreover, if Y is distributive, i.e., inf(y,sup(y"1,y"2))=sup(inf(y,y"1),inf(y,y"2)), then it is also strongly distributive: inf(y,supny"n)=supn(inf(y,y"n)). In & 4, we study similar spaces under weaker assumptions. Particular examples of such spaces are the Hausdorff space of closed sets (see Hausdorff ''Set theory'', p. 165) and the space of semicontinuous functions. & 5 is devoted to applications of the general theorems to the theory of semicontinuous functions y=f(x) that map a metric space {x}=X into a semi-ordered space {y}=Y. Under some additional assumptions (Y is regular, distributive, and between any two elements y"1 and y"2 such that y"1