We introduce max-plus analogues of basic Euclidian geometry notions: scalar product is replaced by a scalar division, and the associated distance is essentially Hilbert’s projective distance. We introduce an orthogonal projection and prove a Hahn-Banach type theorem: a point can be separated from a semimodule by a hyperplane orthogonal to the direction of projection. We use these results to sep...

We show that for a given C*-algebra A and for any pair of Hilbert A-modules {{M, 〈., .〉},N ⊆ M} every bounded A-linear mapping r : N → A can be continued to a bounded A-linear mapping r : M → A so that (i) ‖r‖ = ‖r‖, (ii) r restricted to N equals r and (iii) the extended mappings of N ′ form a Banach A-submodule of M wherein the extensions {r n : n ∈ N} of the standardly embedded mappings {rn =...

Every continuous linear functional defined on a vector subspace of a real normed space can be extended to the whole space so as to remain linear and continuous, and with the same norm(2). The extension of continuous linear transformations between two real normed spaces has been studied by several authors and for a long time it has been recognized that this problem has a close connection with th...

This paper provides new versions of the Farkas lemma characterizing those inequalities of the form f(x) ≥ 0 which are consequences of a composite convex inequality (S ◦ g)(x) ≤ 0 on a closed convex subset of a given locally convex topological vector space X, where f is a proper lower semicontinuous convex function defined on X, S is an extended sublinear function, and g is a vector-valued S-con...

Abstract Many authors consider that the main pillars of Functional Analysis are Hahn–Banach Theorem, Uniform Boundedness Principle and Open Mapping Principle. The first one is derived from Zorn’s Lemma, while latter two usually obtained Baire’s Category Theorem. In this paper we show these three should be either just or at least eight, since Principle, another five theorems equivalent, as in a ...

We discuss a new version of the Hahn-Banach theorem, with applications to linear and nonlinear functional analysis, convex analysis, and the theory of monotone multifunc-tions. We show how our result can be used to prove a " localized " version of the Fenchel-Moreau formula — even when the classical Fenchel-Moreau formula is valid, the proof of it given here avoids the problem of the " vertical...