MAZUR‐ORLICZ Type Theorems with some Applications

Abstract

Eingegangen am 1'7.10.1975) 0. Introduction In 1953 MAZUR and ORLICZ published the following remarkable theorem ([ll], p. 147). Let E be a vector .upace, T a nonempiy set, p a sublinear functional on E , d a functional on T , and y a mapping of T into E . Then the following statements are equivalent : (i) There exists a linear functional v on E such that v ( x ) s p ( x ) for each ~E E, v ( y ( t ) ) z d ( t ) for each t c T . 1 (ii) Whenever m is a positive integer and Ai, . . . , Ams 0, t,, . . . , tmE T , then It is known that this theorem admits remarkable generalizations ahd various interesting applications (see e.g. ALTMAN [i] and KONIG [7], [S]). Concerning the proofs of the MAZUR-ORLICZ theorem, we expecially refer to the short and elegant proof of SIMONS [14].I n Section 1 of the present note the authors prove a very general extension theorem for linear functionals (Theorem 2), which is at the same time a generalization of the MAZUR-ORLICZ theorem.In the following sections the results of Section 1 are used t o present a very simple proof of a general momeht theorem of CARLING, and to deduce separation theorems for functionals as well as decomposition theorems for linear functionals. '1. MAZUR-ORLICZ type theoremsI n all that follows we only consider vector spaces over the field R of the real numbers. As usual, we call a nonempty subset P of a vector space a cone if P + P s C P and t P & P for all t z0. For any subset F of a vector space, let K ( F ) dehote Gie cone geherated by F and let co (E") denote the convex hull of F .

Bootstrapping the Mazur--Orlicz--König theorem

“…This paper is about extensions of the Mazur-Orlicz theorem, which first appeared in [5]: Let E be a vector space, S : E → R be sublinear and C be a nonempty convex subset of E. Then there exists a linear map L : E → R such that L ≤ S on E and inf C L = inf C S. Early improvements and applications of this result were given, in chronological order, by Sikorski [6], Pták [4], König [1], Landsberg-Schirotzek [3] and König [2].…”

The moment problem in the spaceC 0(S)

“…We have to prove that this can be restated as follows. see [2] and [7] for related results. This means that ~}P=0 is a Besselian basis in C(S).…”

“…Sometimes such conditions are obtained using the Hahn and Banach theorem assuming that T is denumerable and Pi are linearly independent, see [6]. If T is not denumerable or p~ are linearly dependent then some solvability criterions can be found in [9], [2] and [7].…”

“…We wish to thank the referee for calling our attention of [2], [7] and criticism on the first draft of this paper.…”

Mazur–Orlicz equality

“…Different proofs and generalizations of this theorem abound in the literature (see, for example, [1], [4], [5], [8], [9], [11], [12]) and it is now usually referred to as the Mazur-Orlicz theorem.…”