Orthogonality and nonlinear functionals

Abstract

Let X be a vector space of functions. In practice, Zis taken to be (i) the set of continuous functions on some type of topological space or (ii) a set of measurable functions on a measure space. We say that x, y e X are orthogonal in the lattice theoretic sense (x ±_ L y) if {t: [13], in this note we consider orthogonalities which are standard in inner product and normed spaces. To some extent our results are less general than those in the above papers since the standard orthogonality is weaker than lattice theoretic orthogonality. On the other hand, some of our theorems apply to more general vector spaces than the above and furthermore we have obtained results for a more general class of functionals which we call orthogonally monotone functionals. Finally, we use our results to solve a nonlinear functional equation and give an application for the solution.An orthogonality vector space is a real vector space X with dim X^.2 on which there is defined a relation x_]_y such that (01) 0_L*, xj_0 for all xeX y (02) if x_[_y and x, y^O then x, y are linearly independent, (03) if x±y then ouc±py for all a 9 peR 9 (04) if B is a two-dimensional subspace of X, then for every Ojéx e B, there exists Ojéy e B such that x±_y and x+y J_x-y.It is easily seen that any real vector space of dimension ^2 is an orthogonality vector space if we define 0J_x, *_i_0 for all x and if x, y5^0 then xA_y iff x, y are linearly independent. It is also clear that an inner product space is an orthogonality vector space under the standard orthogonality AMS (MOS) subject classifications (1970). Primary 46B99, 46C05, 47H15; Secondary 46F05, 81A12.

Orthogonalities and functional equations

“…This definition is more restrictive than the ones given before by Gudder and Strawther (see [94,95]), however, none of the examples provided by them is omitted while considering the definition by Rätz. In [94], the authors define ⊥ by (01)-(03) and add (04 ) for every two-dimensional subspace P of X and for every nonzero x ∈ P , there exists a nonzero y ∈ P such that x ⊥ y and x + y ⊥ x − y.…”

“…In [94], the authors define ⊥ by (01)-(03) and add (04 ) for every two-dimensional subspace P of X and for every nonzero x ∈ P , there exists a nonzero y ∈ P such that x ⊥ y and x + y ⊥ x − y.…”

“…His main result reads as follows. Studies were continued by Gudder and Strawther [94][95][96] and Dhombres [55]. The manuscript [94] from 1974 contains a collection of properties and results without proofs, but we can find there the first axiomatization of the orthogonality relation containing the former situations as special cases.…”

“…Studies were continued by Gudder and Strawther [94][95][96] and Dhombres [55]. The manuscript [94] from 1974 contains a collection of properties and results without proofs, but we can find there the first axiomatization of the orthogonality relation containing the former situations as special cases. In the next paper [95], bringing a slightly different axiomatic definition, the authors already gave some explanations and showed the form of real solutions of (2.1) under some boundedness conditions.…”

Set-valued Orthogonal Additivity

“…We start with the following definition of the orthogonality space (see Gudder & Strawther [10], Rätz [22]). …”