J. Math. Anal. Appl. 374 (2011) 282–289 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa A transfer principle for inequalities in vector lattices Anatoly G. Kusraev South Mathematical Institute, Vladikavkaz Science Center of the RAS, 362040, Vladikavkaz, Russia article info abstract Article history: Received 2 December 2009 Available online 26 August 2010 Submitted by M. Mathieu Keywords: Cauchy–Bunyakovski˘ı inequality Jensen inequality Vector lattice Positive bilinear operator Orthosymmetry Homogeneous functional calculus A transfer principle from inequalities with inner products to inequalities containing positive semidefinite symmetric bilinear operators with values in a vector lattices is proved. Some applications are also given. © 2010 Published by Elsevier Inc. 1. Introduction The following generalization of the classical Cauchy–Bunyakovski˘ı inequality was proved by Huijsmans and de Pagter [18]: if X is a real vector space and ·,· : X × X → E is a positive semidefinite symmetric bilinear operator with values in a semiprime f -algebra E, then x, y◦x, y  x, x◦ y, y (x, y ∈ X ), where ◦ is the f -algebra multiplication. It was shown by Bernau and Huijsmans [3] that the semiprimeness assumption can be omitted and in [11] the result was established for any almost f -algebra E by Buskes and van Rooij. Finally, it was announced in the author’s paper [21] and proved in [10, Theorem 3.8] that almost f -algebra multiplication ◦ can be replaced by an arbitrary positive orthosymmetric bilinear operator from E × E to F , where E and F are Archimedean vector lattices. In a private discussion Professor S.M. Sitnik hypothesized that there must exist some general principle allowing to pro- duce automatically new inequalities for bilinear operators, provided that the corresponding ones hold true for bilinear forms. The aim of this paper is to present a transfer principle which enables us to transform inequalities with semi-inner products to inequalities containing positive semidefinite symmetric bilinear operators with values in a vector lattice. The proof rely upon The Kre˘ıns–Kakutani Representation Theorem, homogeneous functional calculus in uniformly complete vector lattices, and a representation result for positive orthosymmetric bilinear operators. A review of different generalizations and refinements of the classical Cauchy–Bunyakovski˘ı inequality one can find in [13–15,29,30,34]. For the theory of vector lattices and positive operators we refer to the books [1,22,28,32,33]. Some aspects of positive bilinear operators in vector lattices are presented in a survey paper [8], see also [10] and [23]. Throughout the paper a vector lattice means an Archimedean vector lattice over the reals R. The positive cone of a vector lattice E is always denoted by E + . We call a vector lattice uniformly complete, if it is complete with respect to relatively uniform convergence. We use the symbol := if the equality is taken as a definition; N stands for the set of natural numbers. E-mail address: kusraev@smath.ru. 0022-247X/$ – see front matter © 2010 Published by Elsevier Inc. doi:10.1016/j.jmaa.2010.08.046