Orthogonalities and functional equations

In this survey we show how various notions of orthogonality appear in the theory of functional equations. After introducing some orthogonality relations, we give examples of functional equations postulated for orthogonal vectors only. We show their solutions as well as some applications. Then we discuss the problem of stability of some of them considering various aspects of the problem. In the sequel, we mention the orthogonality equation and the problem of preserving orthogonality. Last, but not least, in addition to presenting results, we state some open problems concerning these topics. Taking into account the big amount of results concerning functional equations postulated for orthogonal vectors which have appeared in the literature during the last decades, we restrict ourselves to the most classical equations.


Introduction
During the last years many papers concerning various aspects of orthogonalities in the field of functional equations and inequalities have appeared. In this paper we want to give some overview on these results as well as to collect a number of items from the literature dealing with the subject. It is worth mentioning papers by Paganoni and Rätz [139] from 1995, Rätz [156] from 2001 and Chmieliński [42,44] from 2006, 2012, respectively, where the reader can find some partial collections of the results in this domain.

Various definitions of the orthogonality relation
As long as we are working in inner product spaces, usually there is no doubt what kind of orthogonality relation we have in mind. Namely, it is the one derived from an inner product and then vectors x and y are orthogonal (x ⊥ y) if and only if x|y = 0. The situation looks completely different if we consider normed spaces or more general structures. We start the survey with listing some orthogonality relations described in normed spaces. More details and other orthogonality relations can be found in Amir [8], Alonso et al. [2,3], Alsina et al. [7], Alonso et al. [4], and in the references therein. Later on, we go on with some definitions in linear spaces, C * -algebras and groups.

Birkhoff orthogonality.
Let (X, · ) be a real normed linear space. For vectors x and y from X, we say that x is orthogonal to y (x ⊥ B y) in the sense of Birkhoff (x is Birkhoff orthogonal to y) if x + λy ≥ x for all λ ∈ R. This orthogonality was introduced by Birkhoff [26], however since in normed linear spaces Birkhoff orthogonality is in fact equivalent to normality as it was introduced by Carathéodory, some ideas of this kind can already be found in Blaschke's book [28]. There are also other names for this orthogonality in the literature: Birkhoff-James orthogonality, Blaschke-Birkhoff-James orthogonality. James [103,104] provided comprehensive studies on this relation. Many properties of this orthogonality relation are collected in Amir [8] and Alonso et al. [4].
If X is an inner product space, then ⊥ B coincides with the standard orthogonality in the inner product space. Moreover, the Birkhoff orthogonality is homogeneous (i.e., if x ⊥ B y, then αx ⊥ B βy for all α, β in R).
It is known (see Day [53], James [103]) that, if dim X ≥ 3, then ⊥ B is symmetric (i.e., x ⊥ B y if and only if y ⊥ B x) if and only if X is an inner product space. This result fails in two-dimensional spaces (see, e.g., Alonso et al. [4]). and x ⊥ ρ− y if and only if ρ − (x, y) = 0, respectively. Among the just defined three orthogonality relations only ⊥ ρ is homogeneous, and none of them is symmetric.

Roberts orthogonality.
It seems that Roberts was the first who introduced the orthogonality relation in normed linear spaces. Namely, he proposed his definition of orthogonality in 1934 (see [161]). We say that x is orthogonal to y in the sense of Roberts (x ⊥ R y) if and only if x + ty = x − ty for all t ∈ R.
It is obvious that this orthogonality implies both James and Birkhoff orthogonalities. Moreover, Roberts orthogonality is symmetric.
1.1.6. Semi-inner product orthogonality. Let (X, · ) be a normed linear space. A functional [·|·]: X × X → K ∈ {R, C} satisfying [λx + μy|z] = λ[x|z] + μ [y|z] for all x, y, z ∈ X and λ, μ ∈ K; [x|λy] = λ [x, y] for all x, y ∈ X and λ ∈ K; [x, x] = x 2 for all x ∈ X; |[x, y]| ≤ x y for all x, y ∈ X is called a semi-inner product in a normed space X (generating the given norm). Lumer [124] and Giles [91] proved that in any normed space there exists a semi-inner product. There can be infinitely many such semi-inner products. It is known, however, that in a normed space there exists exactly one semi-inner product if and only if the space is smooth (which means that the norm in X is smooth, that is, it is Gâteaux differentiable)(see, e.g., Day [54]).
For a given semi-inner product and vectors x, y ∈ X we define the semiinner product orthogonality x ⊥ s y if and only if [y|x] = 0.
In 1983, Diminnie [56] proposed the following orthogonality relation x ⊥ D y if and only if x, y = x · y . 220 J. Sikorska AEM He described its connections to Birkhoff orthogonality and proved that if dim X ≥ 3 then merely the additivity of the relation or the fact that the inequality x, y ≤ x · y holds true for all x, y ∈ X characterize X as an inner product space.
1.1.8. Orthogonality space. Apart from the different definitions of orthogonalities in normed spaces we may give some axiomatic definition of such a relation in linear spaces. The most often cited definition of an orthogonality space is the one given by Rätz [149]: Definition 1.1. Let X be a real linear space with dim X ≥ 2 and let ⊥ be a binary relation on X such that (01) x ⊥ 0 and 0 ⊥ x for all x ∈ X; (02) if x, y ∈ X \ {0} and x ⊥ y, then x and y are linearly independent; (03) if x, y ∈ X and x ⊥ y, then for all α, β ∈ R we have αx ⊥ βy; (04) for any two-dimensional subspace P of X and for every x ∈ P , λ ∈ [0, ∞), there exists y ∈ P such that x ⊥ y and x + y ⊥ λx − y.
An ordered pair (X, ⊥) is called an orthogonality space in the sense of Rätz, or shortly, orthogonality space.
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 [95], together with (01)-(03) there are (04 ) if P is a two-dimensional subspace of X, then for every x ∈ P , there exists a nonzero y ∈ P such that x ⊥ y; (05) if P is a two-dimensional subspace of X, then there exist nonzero vectors x, y ∈ P such that x ⊥ y and x + y ⊥ x − y.
An orthogonality space covers the case of an inner product space with the classical orthogonality as well as an arbitrary real normed linear space with the Birkhoff orthogonality. But it is also the case with the "trivial" orthogonality defined on a linear space by (01) and the condition that two nonzero vectors are orthogonal if and only if they are linearly independent.
However, there are known orthogonality relations on normed linear spaces which do not satisfy axioms (01)-(04), e.g., the isosceles orthogonality and the Pythagorean orthogonality.
In the following papers Rätz [150,151] and then Rätz and Szabó [157] developed the theory by considering various generalizations of the stated definitions (see also Szabó [179]).
Vol. 89 (2015) Orthogonalities and functional equations 221 1.1.9. Orthogonality defined via a difference operator. Given a real functional ϕ on an Abelian group (X, +) we may define a new orthogonality relation by the formula x ⊥ ϕ y if and only if x,y ϕ(z) = 0 for all z ∈ X, , and x,y = x • y for all x, y ∈ X. The above orthogonality relation was proposed by Ger in [87]. It generalizes the trapezoid orthogonality ⊥ T on a normed space defined by Alsina et al. in [5] by It means that if X is a real normed space and ϕ = · 2 , ϕ-orthogonality coincides with the T-orthogonality.
Ger [87] studies the properties of the relation ⊥ ϕ and obtains some new characterizations of inner product spaces, e.g., if X is a linear topological space and ϕ is a continuous functional, (X, ⊥ ϕ ) is an orthogonality space if and only if X is an inner product space (i.e., there exists an inner product ·|· :

C * -algebras. Suppose
A is a C * -algebra. Let X be an algebraic right A-module which is a complex linear space with a compatible scalar multiplication, i.e., (λx)a = x(λa) = λ(xa) for all x ∈ X, a ∈ A, λ ∈ C. Then X is called a (right) inner product A-module if there exists an Avalued inner product, i.e., a mapping ·|· : X × X → A satisfying x|x ≥ 0 (positive element of A) and x|x = 0 if and only if x = 0; x|λy + z = λ x|y + x|z ; x|ya = x|y a; y|x = x|y * , for all x, y, z ∈ X, a ∈ A, λ ∈ C (cf., e.g., Lance [121]).
The orthogonality relation in X is naturally defined by x ⊥ y if and only if x|y = 0.

Orthogonalities on groups.
In 1998, Baron and Volkmann [24] proposed the following axioms of orthogonality. Let (X, +) be a uniquely 2divisible Abelian group. Further, let ⊥ be a binary relation defined on X with the properties: (a) 0 ⊥ 0; (b) if x, y ∈ X and x ⊥ y, then −x ⊥ −y and x 2 ⊥ y 2 ; (c) every odd orthogonally additive mapping having values in an Abelian group is additive and every even orthogonally additive mapping is quadratic (see Sect. 2.1.1). In 2010, Fechner and Sikorska [69] were dealing with the stability of orthogonal additivity (see Sect. 3.1.1) proposing the following definition of orthogonality: Let (X, +) be an Abelian group and let ⊥ be a binary relation defined on X with the properties: (α) if x, y ∈ X and x ⊥ y, then x ⊥ −y, −x ⊥ y and 2x ⊥ 2y; (β) for every x ∈ X, there exists y ∈ X such that x ⊥ y and x + y ⊥ x − y.
Each orthogonality space satisfies these conditions as well as an arbitrary normed linear space with the isosceles orthogonality, but it is no longer the case with the Pythagorean orthogonality.
In what follows, we provide an example of a binary relation which seems to be far from any known orthogonality relations but satisfies (α) and (β). Example 1.1. (Fechner and Sikorska [69]) Take X = R and define ⊥⊂ R 2 in the following way: x ⊥ y if and only if x · y ∈ R \ Q or x · y = 0.
Considering usually at least two-dimensional spaces while dealing with orthogonalities allows us to avoid trivial situations, i.e., situations when x ⊥ y implies that x = 0 or y = 0. In the above example it is not the case. However, of course this example can be extended to higher dimensional inner product spaces, where the set of "orthogonal" vectors is considerably bigger than in the standard case.

Approximate orthogonalities
Let ε ∈ [0, 1). A natural way to define approximate orthogonality (or εorthogonality) of vectors x and y in an inner product space is: x ⊥ ε y if and only if | x|y | ≤ ε x y .
Quite similarly, in normed spaces we define the approximate semi orthogonality relation (ε-semi-inner product orthogonality) and approximate ρ-orthogonality (ε-ρ-orthogonality) as follows There are two notions of approximate Birkhoff orthogonality (motivations for using such relations are described in Mojškerc and Turnšek [134]). The first one comes from Dragomir [58,59]: For inner product spaces we have (Dragomir [59]; see also Chmieliński [44]) Another definition of approximate Birkhoff orthogonality (generally not equivalent to the just mentioned one) comes from Chmieliński [40]: x λy for all λ ∈ K. Mojškerc and Turnšek [134] showed that for any x, y from a normed space (real or complex) the relation , and the converse holds (with some ε depending on δ), e.g., in uniformly smooth spaces.
Moreover, we have the following properties.  [49,50]) For an arbitrary real normed linear space X and ε ∈ [0, 1) we have Theorem 1.2. (Chmieliński and Wójcik [50]) Let (X, · ) be an arbitrary real normed linear space and let ε ∈ [0, 1). Then for arbitrary x, y ∈ X and α ∈ R we have As a special case of the latter result we obtain the property: So, we have generalizations of the known conditions: (see, e.g., Amir [8]).
Two notions of approximate James orthogonality are given by the following conditions (see Chmieliński and Wójcik [48]): or equivalently, Obviously, for ε = 0 both versions of the approximate J-orthogonality coincide with the J-orthogonality. As observed by Chmieliński and Wójcik in [48], the second definition of approximate J-orthogonality is weaker than the first one, i.e., for an arbitrary ε ∈ [0, 1) the condition x ⊥ ε J y implies x ε ⊥ J y, but not conversely.
One can check that in the case when the norm comes from a real valued inner product, then So, the first (stronger) approximate J-orthogonality coincides with the standard notion of approximate orthogonality in inner product spaces. Example 1.2. Let (X, ·|· ) be an inner product space, x ∈ X \ {0} and y = λx for some λ > 0. Then | x|y | x 2 + y 2 = λ 1+λ 2 → 0 as λ → ∞, and | x|y | x y = 1 for all λ. Thus for arbitrary x = 0 and ε ∈ [0, 1) there exists λ such that x ε ⊥ J λx whereas x ⊥ ε J λx does not hold for any ε ∈ [0, 1).
It is known that the conditions ⊥ B ⊂⊥ J or ⊥ J ⊂⊥ B characterize (X, · ) as an inner product space, so it is quite natural to ask about the connections between approximate James orthogonalities and approximate Birkhoff orthogonalities.
Finally, similarly to Chmieliński and Wójcik [48], two notions of approximate Roberts orthogonality are given by Zamani and Moslehian [201]: for all t ∈ R.
Vol. 89 (2015) Orthogonalities and functional equations 225 We start this section with giving some historical background for the investigations in this field (see also Paganoni and Rätz [139], Rätz [156]).
In what follows we will consider the Cauchy functional equation postulated for orthogonal vectors only, that is the conditional equation (the domain, target space and the orthogonality relation will be specified later). Functions satisfying (2.1) are called orthogonally additive. The studies on (2.1) were begun (to the best of our knowledge) by Pinsker in 1938 who was considering (see [143]) orthogonally additive mappings defined on the space of continuous functionals from L 2 [a, b] with an orthogonality defined by x ⊥ y if and only if b a x(t)y(t)dt = 0 (so by means of the inner product). Later on, the studies proceeded in two directions concerning the domain. It was considered: (i) a set of continuous functions on some type of topological space or (ii) a set of measurable functions on a measure space. Then we say that x, y are orthogonal in the lattice theoretic sense (x ⊥ L y) if the set {t : x(t)y(t) = 0} is empty in case (i) or of measure zero in case (ii). A real valued functional f is L-additive if f (x+y) = f (x)+f (y) whenever x ⊥ L y. If f is L-additive and satisfies certain continuity or boundedness conditions, then f admits an integral representation giving a nonlinear generalization of the Riesz theorem. Such representations have been obtained for case (i) in [36,79] and for case (ii) in [61,78,126,131,132,174].
The above mentioned concepts of orthogonality are quite natural in the spaces considered and they are important for certain applications [81,123,159], however, there are several other concepts of orthogonality defined and developed by Birkhoff, Roberts, James and Day (see Sect. 1).
We restrict our attention to the approach (being in fact the continuation of Pinsker's [143]) which was begun in 1972 by Sundaresan [175]. He studied orthogonally additive functions defined on an inner product space or on a normed linear space with Birkhoff orthogonality, and he gave some partial results in the case of continuous orthogonally additive functions. His main result reads as follows.
Moreover, if X is not isometric with a Hilbert space (that is, there is no bilinear symmetric inner product in it from which the given norm can be derived in the customary way), then f is a continuous linear operator on X into Y .
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. For the main results the assumption about the completeness of the domain turned out to be superfluous. [95]) (Characterization of inner product spaces (i.p.s.)) Let (X, ⊥) be an orthogonality space (defined by (01)-(03) and (04')). If there exists f : X → R which is even, orthogonally additive, hemicontinuous 1 and not identically zero, then there is an inner product ·|· on X such that for any x, y ∈ X, x ⊥ y if and only if x|y = 0. In fact,

Theorem 2.2. (Gudder and Strawther
As a corollary, Gudder and Strawther [95] obtained an analogous form of solutions as in Theorem 2.1 for real functions defined on a normed space with the Birkhoff orthogonality under the assumption of hemicontinuity. Moreover, they proved a generalization of the Riesz representation theorem, showing that if X is an inner product space and f : X → R is orthogonally additive and satisfies |f (x)| ≤ M x for all x ∈ X, then f is a continuous linear functional and hence, if X is a Hilbert space, then f (x) = x|z for some z ∈ X.
In his book [55], Dhombres states the open problem whether the regularity assumption of a considered function may be omitted in order to derive that in a normed space of dimension not less than 2 the existence of an even nonzero orthogonally additive mapping characterizes inner product spaces.
So, the next step was to get rid of the regularity conditions and to characterize in an abstract framework-the general even and the general odd solution of (2.1) with values in an Abelian group (Lawrence [122], Rätz [149][150][151], Szabó [176], Rätz and Szabó [157]). Roughly speaking, in many important situations, the general even solution is quadratic, and the general odd solution is additive.
Vol. 89 (2015) Orthogonalities and functional equations 227 Theorem 2.3. (Rätz [149]) Let (X, ⊥) be an orthogonality space and (Y, +) be a uniquely 2-divisible Abelian group. If f : X → Y is a solution of (2.1), then it has the form f = a + q, where a is additive and q is quadratic. [147][148][149]) Let (X, · ) be an inner product space and (Y, +) be a uniquely 2-divisible Abelian group. Then f : X → Y is a solution of (2.1) if and only if there exist additive mappings a : The above result in the case of a uniquely 2-divisible Abelian group (Y, +) was found independently by Ger and Szabó (see [84]). Under the assumption that (Y, +) is 2-torsion-free it was proved by Rätz and Szabó in [157]. Baron and Rätz in [23] (for inner product spaces) and then Baron and Volkmann in [24] showed that the assumption that Y is uniquely 2-divisible as well as the 2-torsion-freeness of Y may be omitted. We present here a theorem from [24]. [24]) Let X be a linear space over a field of characteristic different from 2 (or let X be a uniquely 2-divisible group), (Y, +) be an Abelian group and let f :

Theorem 2.5. (Baron and Volkmann
(ii) every odd orthogonally additive mapping from X to Y is additive and every even orthogonally additive mapping is quadratic.
Then f is orthogonally additive if and only if with a : X → Y being additive and b : X × X → Y being biadditive, symmetric and such that b(x, y) = 0 whenever x ⊥ y. Moreover, in this case the functions a, b and q(x) = b(x, x), x ∈ X, are uniquely determined; they are given by respectively.
If X is an inner product space then various assumptions force orthogonally additive functions f to be of the form (2.2), namely: -Y is a separated topological R-vector space and f is continuous (Rätz [149,Corollary 11]; cf. Theorem 2.1 and Sundaresan [175]); -f : X → R satisfies |f (x)| ≤ m x for all x ∈ X and a fixed m ≥ 0 (Gudder and Strawther [95, Corollary 2.4], Rätz [149,Corollary 12]); -f : X → R satisfies f (x) ≥ 0 for all x ∈ X; then we obtain (2.2) with h = 0 and with nonnegative c (see Gudder and Strawther [96]; Rätz [149,Corollary 13]); -f : X → R is bounded on a second category Baire subset of X (follows from Rätz [149], Ger [82]; see also Ger and Sikorska [89]); -Y is a topological Abelian group and f : X → Y is continuous at a point (Baron and Kucia [21,Theorem 4.3]). The last result was generalized first by Brzdęk [32] (with the domain being an orthogonality space and with the assumption of continuity at the origin) and then by Wyrobek [198] who was working in an Abelian topological group in the domain with the assumption of continuity at an arbitrary point.
It is possible to characterize Hilbert spaces among real inner product spaces in terms of the boundedness behavior of R-valued orthogonally additive mappings. The result is related to the Riesz representation theorem. Theorem 2.6. (Rätz [149]) (Characterization of Hilbert spaces) For an inner product space X the following conditions are equivalent: (i) for every orthogonally additive mapping f : For several years mathematicians were trying to find the connection between the property (e)Hom ⊥ (X, Y ) = {0} (where (e)Hom ⊥ (X, Y ) stands for the set of all even orthogonally additive functions from X to Y ) and the property that X is an inner product space. Theorem 2.7. (Rätz [149], Szabó [176]) (Characterization of i.p.s.) Let (X, · ) be a real normed space, dim X ≥ 2, with Birkhoff orthogonality ⊥ B and (Y, +) be an Abelian group. (X, · ) is an inner product space if and only if not every In [149], the above theorem is proved for at least three-dimensional spaces. Moreover, in [176], it is additionally proved that if Y is an arbitrary group (so not necessarily Abelian), then every even orthogonally additive mapping is identically zero.
The following fact concerns the symmetry of relation ⊥. It is known that in at least three dimensional normed spaces X, the symmetry of ⊥ B characterizes X as an inner product space. Lawrence [122] proved that if dim X = 2 and ⊥ B is not symmetric, then every orthogonally additive mapping is additive. In fact, we can prove this result in an arbitrary orthogonality space.
Vol. 89 (2015) Orthogonalities and functional equations 229 Theorem 2.8. (Rätz [150]) For any orthogonality space (X, ⊥) and any Abelian group In 1990, Szabó proved the following (see [177]). Theorem 2.9. Let (X, ⊥) be an orthogonality space and let (Y, +) be an Abelian group. If dim X ≥ 3 and there is a nontrivial even orthogonally additive mapping f : X → Y , then X is an inner product space.
In [179], Szabó went on trying to answer the question about dimension 2. By strengthening the fourth assumption in the definition of orthogonality space he proved that the above result is also true when dim X = 2. In 2001, Rätz [156] came back to the problem and asked whether it is also true in arbitrary orthogonality spaces X with dim X = 2.
An affirmative answer was given by Yang in 2006. She showed (see [200]) is an orthogonality space and there exists a nontrivial even orthogonally additive function f : X → Y for some Abelian group (Y, +). Then X is an inner product space.
Interesting results were obtained lately by Baron [16][17][18]. He was working with orthogonally additive involutions, functions with orthogonally additive second iterate, finally with orthogonally additive bijections in real inner product spaces.

Theorem 2.11. Any orthogonally additive bijection from a real inner product space into an Abelian group is additive.
In [190], Turnšek extended this result to complex or quaternionic inner product spaces.
The following examples show that none of the assumptions: injectivity or surjectivity, may be omitted (see Baron [16]).
Example 2.1. Assume that X is a real inner product space. Let H 0 be a basis of R over Q and let H be a basis of X over Q. If H 1 and H 2 are disjoint subsets of H such that 1 ≤ card H 1 ≤ c and card H 2 = card H, and a : R → X and b : X → X are additive functions such that a(H 0 ) = H 1 , b(H) = H 2 and b is injective, then f : X → X given by (2.3) is orthogonally additive, injective and it is not additive.
Then the function f : X → X given by (2.3) is orthogonally additive, f (X) = X and f is not additive. Theorem 2.12. (Baron [17]) Suppose (X, ·|· ) is a real inner product space. Assume f : X → X and f 2 are orthogonally additive. If f is surjective, then it is additive.
It is easy to observe that neither the orthogonal additivity of f implies the orthogonal additivity of f 2 , nor the converse.
So far we have been considering orthogonalities which were homogenous. One can ask what is the form of orthogonally additive mappings defined on a normed space with isosceles orthogonality, Pythagorean orthogonality and norm derivative orthogonality.
Some partial answers were given by Szabó [180,181]. It seems that the Pythagorean orthogonality is the most difficult one for this kind of investigations. Of course in a normed space with Pythagorean orthogonality we cannot expect a similar result to Theorem 2.14 since in all such spaces, for arbitrarily fixed z 0 ∈ X \ {0} the function f (x) := x 2 z 0 , x ∈ X, is nonzero, even and orthogonally additive.
Looking for the form of solutions of the conditional equation we proved (see Alsina et al. [6,7]) that a real normed space X with dimension at least 2 with ρ-orthogonality is in fact an orthogonality space in the sense of Rätz [149], and so, we know the form of solutions. As a by-product we obtained an alternative proof for the fact that a real norm space with Birkhoff orthogonality is an example of an orthogonality space (cf. Szabó [177,179]).
In [153], Rätz showed still another analogous characterization of inner product spaces: Instead of considering orthogonally additive mappings defined on the whole space we may study such conditional forms on a more restricted domain, for example on balls (see, e.g., Sikorska [166,167]).
Another direction of studies is to generalize the classical James orthogonality by defining the orthogonality relation as follows: where ϕ is a given function with some properties (cf. Ger and Sikorska [90,Theorem 5], and Sikorska [169]). Of course, in case ϕ = · , we have the isosceles orthogonality.
Let now A be a C * -algebra and let (X, ·|· ) be a Hilbert C * -module over A.
Obviously, if a is an additive mapping on X and b is an additive mapping on A, then the mapping f defined by is orthogonally additive.
During the 15th ICFEI (May 2013, Ustroń, Poland), Ilišević presented results obtained jointly with Turnšek and Yang giving the conditions when the converse is true.
It is worth mentioning that the form a+q of orthogonally additive functions, where a is additive and q is quadratic, is not always achieved.
Rätz [152] studied orthogonally additive mappings on free Z-modules. In the case dim Z X = 2, he showed a deviation from the described situation, e.g., in the inner product space case (see also Rätz [156], Kuczma [120]).
At the end of this section we want to point out that also an additive setfunction f : M → R defined on an algebra M, i.e., the function which satisfies a conditional equation is in fact an example of an orthogonally additive mapping.

Applications.
We have already mentioned some applications of orthogonal additivity in mathematics. Namely, with its help we can give several characterizations of inner product spaces among normed spaces as well as of Hilbert spaces among Banach spaces. The equation of orthogonal additivity can also give rise to some other mathematical problems (see, e.g., Maksa et al. [125] or Matkowski [128]). For other fields of mathematics where some kind of orthogonal additivity appears see Rätz [156].
There are quite interesting interactions of orthogonal additivity outside mathematics.
Equation (2.1) has its applications in physics, in the theory of ideal gas (see Aczél and Dhombres [1], Truesdell and Muncaster [188], Arkeryd and Cercignani [10]). In the three-dimensional Euclidean space, by means of (2.1) we obtain the formula for the distribution law of velocities in an ideal gas at a fixed temperature. Since for physical reasons, it is generally assumed that the distribution function is continuous, positive and even, the so called Maxwell-Boltzmann distribution law has the form where a, A are some positive constants, and the formula a = m 2kT connects a with the mass of a molecule m, absolute temperature T and the Boltzmann constant k (cf. also the Boltzmann-Gronwall Theorem on summational invariants; [188]). In 1860, Maxwell obtained the above mentioned formula using another approach but also solving a functional equation, however, he made stronger assumptions (see [188]).
Equation (2.1) has got its applications also in actuarial mathematics in a premium calculation principle. It is shown (see Heijnen and Goovaerts [97]), that the variance principle is the only covariance-additive premium principle, i.e., satisfies the condition π(x + y) = π(x) + π(y) for all risks x, y with cov (x, y) = 0.

Jensen functional equation
It is easy to see that a function which satisfies the orthogonal form of the Jensen functional equation between an orthogonality space X and an Abelian group divisible by 2 is of where h is orthogonally additive and c is a real constant. However, if one knows that h is of the form a+q, where a is additive and q is quadratic, the immediate consequence is that the solutions of this conditional Jensen equation are unconditionally Jensen ones (see Ger [86]).
This nice fact we obtain thanks to the equality f ( valid for all x ∈ X, which is in the sequel, a consequence of the relation x ⊥ 0 for all x ∈ X. This fact implies that while studying this conditional equation we are Vol. 89 (2015) Orthogonalities and functional equations 233 closer to investigations used for studies of unconditional forms rather than their conditional analogues. A modified version of the Jensen equality was presented by Szostok [185]. A starting point for his considerations was the inequality postulated for some constant γ ∈ 0, 1 2 (see Kolwicz and P luciennik [118]), having its background in Orlicz spaces. Namely, Szostok was studying the equality where f maps a real normed linear space into the space of reals, which led in the sequel to an orthogonal Jensen equation (2.6) with the isosceles orthogonality. These investigations gave rise to the studies of a generalized Cauchy equation (see Szostok [186,187]).

Quadratic functional equation
The studies of orthogonally quadratic functional equations, that is, conditional equations of the form f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X with x ⊥ y, started in 1966 with the paper by Vajzović [193], who described the form of continuous orthogonally quadratic functionals on a Hilbert space of dimension at least 3. More exactly, he proved that if f : X → K, where (X, ·|· ) is a real or complex Hilbert space, dim X ≥ 3, and K ∈ {R, C}, satisfies the condition f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X with x|y = 0, (2.7) then there exist a continuous linear operator B and continuous quasi-linear operators C and D (C is quasi-linear if C(x + y) = C(x) + C(y) and C(λx) = λC(x) for all x, y ∈ X and λ ∈ C) such that (2.8) His result was generalized in 1986 by Drljević to A-orthogonality on a real (or complex) Hilbert space (see [63]). Namely, Drljević was considering (2.7) with x|y = 0 replaced by x|Ay = 0 (we say that x is A-orthogonal to y), where A : X → X is a continuous selfadjoint operator with dim A(X) > 3. Looking for the general continuous solution, he obtained the same form of f as in (2.8).
In the same year Fochi [70] showed that in inner product spaces of dimension not less than 3, each real valued orthogonally quadratic mapping is unconditionally quadratic. In fact, this result remains true for values in a uniquely 2-divisible Abelian group.
In [71], Fochi proved even more, namely, she showed that both for real and complex valued functions, if dim A(X) > 2, then the solutions of the conditional A-orthogonal quadratic equation have to be quadratic (unconditionally).
Szabó [178] generalized the above mentioned results to a symmetric orthogonality introduced by a sesquilinear form on a linear space and for arbitrary mappings with values in an Abelian group. Before stating the main result given by Szabó, we introduce some notation.
Let Φ be a field such that char Φ ∈ {2, 3, 5}, let X be a vector space over Φ with dim Φ X ≥ 3 and let (Y, +) be a 6-torsion-free Abelian group, i.e., multiplication by 6 in Y is injective. Furthermore, let ϕ : X 2 → Φ be a sesquilinear functional with respect to an automorphism ξ : Φ → Φ, i.e., ϕ is biadditive and ϕ(αx, βy) = αξ(β)ϕ(x, y) for all x, y ∈ X and α, β ∈ Φ. Then define the ϕ-orthogonality relation ⊥ ϕ on X by x ∈ X} which is a linear subspace of the algebraic conjugate space X * of X. Theorem 2.17. Suppose that the ϕ-orthogonality on X is symmetric, dim X * ϕ ≥ 3, and there exists a non-isotropic vector in X. If Y is 6-torsion-free, then every ϕ-orthogonally quadratic mapping from X to Y is quadratic.
The problem of determining all solutions of the orthogonally quadratic functional equation on an arbitrary orthogonality space, or in a normed space with, e.g., Birkhoff, isosceles or Pythagorean orthogonalities remains open.
Some partial results were presented by Szabó during his lectures at the 5th International Conference on Functional Equations and Inequalities and on the 33rd International Symposium on Functional Equations in 1995 (see [184] and [182], respectively). However, we do not know his proofs, since the results were not published. We cite here the main theorems presented then. Theorem 2.18. (Szabó, 5 ICFEI, 1995) Assume that (X, · ) is a real normed space equipped with the Birkhoff orthogonality and (Y, +) is an Abelian group. If dim X ≥ 5 and the norm is Gateaux differentiable, then 2f is unconditionally quadratic whenever f : X → Y is a Birkhoff orthogonally quadratic mapping. Theorem 2.19. (Szabó, 33 ISFE, 1995) If (X, · ) is a strictly convex real normed space such that dim X ≥ 4 and (Y, +) is an Abelian group, then for any isosceles orthogonally quadratic mapping f : X → Y , 2f is unconditionally quadratic. In [76], Fochi was looking for the solutions of some pexiderized forms of an orthogonally quadratic equation, namely In an orthogonality space X in the sense of Rätz (see Sect. 1.1.8) with a symmetric relation of orthogonality ⊥ she proved the following Theorem 2.20. The general solution f, g, h : X → R of the functional equation (2.9) is given by for all x ∈ X, where A : X → R is an additive function and Q : X → R is orthogonally quadratic.
The final problem, however, stays unsolved. We do not know the general form of an orthogonally quadratic function.
For the next result assume that (X, ·|· ) is an inner product space, dim X > 2.

Exponential functional equation
Assume that (X, ·|· ) is an inner product space of dimension at least 2. Some immediate consequences of Theorem 2.4 concerning the solutions f : X → R of the conditional equation can be found, e.g., in Fochi [70]. We cite here two results from Baron and Rätz [22] and Baron and Forti [19], respectively. for every x ∈ U , then either f vanishes on X, or (2.14) Before stating the next result we will recall some notion. Namely, we say that f : X → C is measurable on rays if and only if for every x ∈ X the function t → f (tx), t ∈ R, is Lebesgue or Baire measurable. Baron et al. in [20] were studying solutions f : X → C of (2.11), different from zero at every point, assuming that the function x → f (x) |f (x)| , x ∈ X, is continuous at the origin or measurable on rays. As a result they obtained among others the following. (ii) if f is measurable on rays, then either f vanishes on X or has the form (2.13), or f has the form (2.15) with some complex constant c and an R-linear function g : X → C.
Later on, still with the assumption that X is an inner product space, the studies were going in two directions.
First, instead of having (2.12) on a neighbourhood of zero it can be assumed that it is valid on a Christensen measurable set which is not a Christensen zero set (and β = 2) on a Polish space, or it is valid on a second category set with the Baire property (and β = 2), or it is valid on a set that has an algebraically interior point 2 . In each case we derive that the solution f has to have the form (2.14) with some uniquely determined additive functions a : R → R and A : X → R, a linear function g : X → R, and a real constant c (see Brzdęk [30,Corollary 5] Another result under some measurability assumptions can be found in Brzdęk [33,Corollary 3]. We may also generalize the domain. Instead of an inner product space, we may consider an orthogonality space. We cite here one of the results from Brzdęk [32]. Theorem 2.26. Let (X, ⊥) be an orthogonality space and f : X → C be a nonzero solution of (2.11), hemicontinuous at the origin. Then there exist c ∈ C, unique linear functionals a 1 , a 2 : X → R, and a symmetric bilinear functional L : X 2 → R, unique up to a multiplicative constant, such that L(x, y) = 0 whenever x ⊥ y and Instead of an orthogonality space in the domain we may also consider a normed space with the isosceles James orthogonality. Brzdęk [34] proved the following.
Theorem 2.27. Let X be a real normed linear space which is not an inner product space, dim X ≥ 3, and let (S, ·) be a commutative semigroup with a neutral element. Suppose f : X → S satisfies (2.11) and there exists a nonzero element x 0 in X such that f (x 0 ) is invertible in S. Then f is (unconditionally) exponential.
A simple consequence of the above is the fact that if we have a (commutative) field in the domain, then each orthogonally exponential mapping is (unconditionally) exponential.

D'Alembert equation.
It is interesting to compare the families of solutions of the functional equation characterizing the cosine function with its correspondent equation postulated only for orthogonal vectors (see Fochi [72]).
We restrict ourselves to inner product spaces (X, ·|· ) with dimension not less then 2.
It is easy to see that the function f : X → R, where h : R → R is an arbitrary additive function, is a solution of In [73], Fochi was dealing with some other classical trigonometric functional equations related to (2.16), namely Similarly as in previous cases, we will study the conditional equation One can see (cf. Fochi [74]) that already in inner product spaces the solutions of (2.19) differ from solutions of the orthogonal additivity equation (2.1).
Example 2.3. Let h : R → R be a given non-trivial additive function and Then f satisfies (2.19), but it is not orthogonally additive.
First, we present results describing separately the odd and the even solutions of (2.19). [74]) Let X be an inner product space with dim X ≥ 3 and let f : X → R be a solution of the functional equation (2.19).

Theorem 2.29. (Fochi
(i) If f is odd, then there exists an additive mapping h : X → R such that |f (x)| = |h(x)| for all x ∈ X. (ii) If f is even, then f is orthogonally additive, i.e., there exists an additive mapping ϕ : With some additional assumption on f we have the following result. Similar investigations concerning the orthogonal forms of functional equations are done in Fochi's paper [75], where the author considers the conditional equation

Cocycle equation.
In 1998, Sikorska [165] asked about solutions of the conditional cocycle equation For a non-conditional case of the equation we have the following Theorem. (Davison and Ebanks [52]) Let M be a cancellative Abelian monoid and let G be a divisible Abelian group. Then for every symmetric solution F : M 2 → G of the equation Hypothesis. Let (X, ·|· ) be a real inner product space, dim X ≥ 3, and let (G, +) be a divisible Abelian group (for example G = R).
Then for every symmetric solution F : X 2 → G of the conditional functional equation (2.20) there exists a function f : X → G such that The assumption dim X ≥ 3 in the Hypothesis allows us to avoid the necessity of using zero vectors. Rätz [154,155] showed that just in these trivial cases we already have some variety. Namely, if dim X ≤ 1, the function F has to be a Cauchy difference, while if dim X = 2, it is not the case.
Problem 2.6. Prove or disprove the above Hypothesis.
It is worth pointing out that an interesting and fruitful approach for arithmetic functions satisfying a conditional cocycle equation was done by Kochanek in [111, Lemma 2.2].

Arithmetic functions
As already observed by Rätz [156], there are connections between Pinsker theory, as sometimes the considered theory is called, and additive numbertheoretical (arithmetic) functions. Consider for functions f : N → R the conditional functional equation (2.21) where in this case the orthogonality sign ⊥ rp means that two elements are relatively prime, i.e., (m, n) = 1. Several mathematicians were looking for conditions which force an arithmetic additive function to be of the form c log n. The first two results of this Vol. 89 (2015) Orthogonalities and functional equations 241 type, due to Erdős [65], assert that it is the case if f satisfies one of the following conditions: Rényi in [158] gave a simplified and elegant proof of the Erdős theorem in the case (ii). Later, Kátai [108] and Máté [127] strengthened the assertion, assuming instead of (ii). Schoenberg [162] extended (2.21) to the form

Hyers-Ulam stability.
The origin of the stability problem traces back to Ulam (see [191,192]), who in 1940 asked to give conditions for the existence of a linear mapping near an approximately linear one. If f is a function from a normed linear space (X, · ) into a Banach space (Y, · ) which satisfies with some ε > 0 the inequality then Hyers [98] proved that there exists a unique additive mapping a : X → Y such that Moreover, if R t → f (tx) ∈ Y is continuous for any fixed x ∈ X, then a is linear (see also Rassias [146]). It should be mentioned that a version of Ulam's problem for real sequences appeared in the book of Pólya and Szegő [144].
We start this section with citing a result by Ger and Sikorska [89, Theorem 1 and Remark 3] concerning the stability of the Cauchy functional equation postulated for orthogonal vectors in an orthogonality space. Theorem 3.1. Let (X, ⊥) be an orthogonality space. Given ε ≥ 0 and a real Banach space (Y, · ), let f : X → Y be a mapping such that Then there exists exactly one orthogonally additive mapping g : X → Y such that Actually, the norm structure in Y may be avoided. We have (see Ger and Sikorska [89,Remark 4]) Theorem 3.2. Let (X, ⊥) be an orthogonality space and let Y be a real sequentially complete Hausdorff linear topological space. Assume that a bounded convex and symmetric with respect to zero set V ⊂ Y and a mapping f : X → Y are given such that Then there exists exactly one orthogonally additive mapping g : In fact, some stability results for A-orthogonal vectors in Hilbert spaces appeared already in the paper by Drljević and Mavar [64], but just [89] gave probably a rise to a huge number of papers considering various kinds of stability problems of various functional equations postulated for orthogonal vectors.
It is worth recalling that the orthogonally additive mapping appearing in the assertion of Theorems 3.1 and 3.2 is of the form a + q, where a is additive and q is quadratic (cf. Theorem 2.3).
Similar results can be obtained in case (X, · ) is a real normed linear space with dim X ≥ 2 and with the James orthogonality relation on X (see Sikorska [164] or [168]). In 2010, Fechner and Sikorska [69] published a generalization of the above results (see also Sikorska [173]). Also, the estimating constant was sharpened.

Theorem 3.3. Let X be an Abelian group and let ⊥ be a binary relation defined on X with the properties:
(α) if x, y ∈ X and x ⊥ y, then we have x ⊥ −y, −x ⊥ y and 2x ⊥ 2y; (β) for every x ∈ X, there exists y ∈ X such that x ⊥ y and x + y ⊥ x − y. Vol. 89 (2015) Orthogonalities and functional equations 243 Further, let (Y, · ) be a (real or complex) Banach space. Given ε ≥ 0, let f : X → Y be a mapping such that Then there exists a mapping g : X → Y such that

3)
and Moreover, the mapping g is unique on the set 2X.
In case X is uniquely 2-divisible, we get (3.4) on the whole group X; however, there are examples of non-trivial groups with 2X = {0} for which our assertion does not bring much information.
Remark 3.1. The above results can be applied both in an orthogonality space and in a normed space with James orthogonality. However, the problem remains open in the case of Pythagorean orthogonality.
Along the results for functions with the domain being the whole space we may consider orthogonal vectors only from some set. We give some results where the role of this set is played by a ball (see Sikorska [166,167]).
We start with the results in an inner product space.
Remark 3.2. It is easy to see that g = a + b • · 2 is orthogonally additive but, in general, such g is not uniquely determined.
x ≤ 1} and let f : B → R be an additive function. Obviously, for all vectors x, y ∈ B such that x + y ∈ B and x ⊥ y we have Then both g 1 := f and g 2 (x) := f (x) + c · x, x ∈ R 2 , with constant c ∈ R 2 such that c < kε, fulfil the condition (the sign "·" stands here for the standard inner product in R 2 ).
Assume now that the domain (X, · ) is a real normed linear space with Birkhoff orthogonality, dim X = N ≥ 2, the target space Y is a real sequentially complete linear topological space, and V is a nonempty, bounded subset of Y which is convex and symmetric with respect to zero.
The above result is slightly weaker than expected. Apart from the fact that the approximating function g := a + q is not uniquely determined, it does not need to be orthogonally additive, it means that the quadratic part may fail to be orthogonally additive.
Example 3.2. Take f : R 2 → R which is additive and assume that a norm · in R 2 does not come from an inner product. Consider the Birkhoff orthogonality in R 2 . Then of course, f satisfies (3.5).
Take an arbitrary ε > 0 and define g : and with a real constant c such that |c| ≤ 4Kε α , where α is a positive number such that for all x ∈ R 2 we have (x 2 1 + x 2 2 ) ≤ α x 2 (since the Euclidean norm and · are equivalent). It is easy to show that g − f is quadratic.
Moreover, for every x ∈ 1 2 B, where B is a unit ball in R 2 , we have |f (x) − g(x)| = c(x 2 1 + x 2 2 ) < Kε. However, the function g is not orthogonally additive on the half-ball. To see this, take arbitrary x = (x 1 , x 2 ) and y = (y 1 , y 2 ) from 1 2 B such that x ⊥ y and note that g(x + y) − g(x) − g(y) = 2c(x 1 y 1 + x 2 y 2 ).
The above difference cannot always be zero. Otherwise the orthogonality relation in the sense of Birkhoff would be equivalent to the orthogonality relation connected with some inner product defined on R 2 , which leads to a contradiction.
Assume now that (X, · ) is a real uniformly convex space 3 with Birkhoff orthogonality relation, dim X = N ≥ 2, and Y , V and B are the same as before. This time we get an approximation on the whole ball.
3. An analogous approach is used while studying the generalized orthogonal stability of the Jensen functional equation y) for all x, y ∈ X with x ⊥ y, or Pexider functional equation. For the latter one, see the beginning of the paper by Fechner and Sikorska [68]. For the former one we should have a function ϕ with the properties (a) for every x ∈ X the series ∞ n=1 2 1−n ϕ(2 n x, 0) is convergent or for every x ∈ X the series ∞ n=1 2 n ϕ(2 1−n x, 0) is convergent; denote such a sum by Λ(x); and (c) with conditions corresponding to the respective cases from (a) . We use here only (i) and (iii) with α = β from the properties of the orthogonality space.
Letf := f − f (0). Thenf (0) = 0 and by the assumptions we get the existence of an orthogonally additive mapping a such that 3.1.3. Some applications. By means of the results from the last section we may prove various kinds of "sandwich" theorems, where we separate orthogonally subadditive (3.7) and orthogonally superadditive (3.8) functions. We give here one example (cf. Fechner and Sikorska [68,Proposition 3]). More examples and more general forms of the theorem can be found in [68].
Theorem 3.8. Let (X, · ) be a real normed space, dim X ≥ 2, with Birkhoff orthogonality. Assume that p, q : X → R satisfy and If p(x) − q(x) ≤ c x r for all x ∈ X, where c, r are positive constants and r > 2, then there exists a unique orthogonally additive mapping f : X → R such that with some positive constant d, With some additional assumptions imposed on the functions p and q we get the approximation q ≤ f ≤ p in the above theorem (see [68,Theorem 3]).
Some other results on orthogonally superadditive functions can be found in Fechner [67].

Functional congruences.
Let X be a real linear space and let ⊥ be an orthogonality relation defined in this space. Let F and F (2) be classes of functions defined on X and on X 2 , respectively, and with values in a group (Y, +). We say that the pair (F, F (2) ) has the orthogonal double difference property if every function f : (2) for orthogonal vectors is of the form f = g + A, where g ∈ F and A is orthogonally additive.
Vol. 89 (2015) Orthogonalities and functional equations 247 In the classical stability problem the classes F and F (2) are the classes of bounded functions. But similarly we can consider other pairs (F, F (2) ), namely with integer valued functions or, more general, classes of functions with values in a given discrete subgroup.
The first result comes from papers by Baron and Rätz [22] and Baron [15].
Theorem 3.9. Let X be a real inner product space with dim X ≥ 2, (G, +) a topological Abelian group, and K a discrete subgroup of G. If f : X → G fulfils the condition

9)
and it is continuous at a point, then there exist continuous additive functions a : R → G and A : X → G such that (3.10) In fact the above result (see Baron [15]) was first proved by Baron and Rätz in [22] under the additional assumption that G is continuously divisible by 2 (the function u → 2u is a homeomorphism of G onto G), and f is continuous at the origin. Brzdęk [31] generalized the result of Baron and Rätz [22] showing that f can be supposed continuous at any point and that the assumption concerning G can be replaced by a weaker one: 2u = 0 for u ∈ G \ {0}.
The representation obtained in the above theorem does not remain valid without a regularity condition. In order to see this we may consider a function ϕ : R → R such that ϕ(s + t) − ϕ(s) − ϕ(t) ∈ Z for all s, t ∈ R, but for every additive function a : R → R there exists t ∈ R such that ϕ(t) − a(t) ∈ Z. The existence of such a function follows from Godini's paper [92,Example 2]. We use this ϕ in the following two examples (see Baron and Rätz [22]).
and one can show that there is no additive function A : and one can show that there are no additive functions a : R → R and A : A particular case where the target space is the space of reals and the discrete subgroup is the set of integers was examined first by Baron and Forti in [19].

Theorem 3.10. Let X be a real inner product space with
(3.11) If there exist a neighbourhood U of the origin and γ ∈ (0, 1/4) such that f (U ) ⊂ (−γ, γ)+Z, then there exist a real constant c and a continuous linear functional h : X → R such that Brzdęk [30] showed that an analogous result can be obtained in the cases: U is a set of the second category and with the Baire property, or U is a Christensen measurable nonzero set, or U has an algebraically interior point. Theorem 3.11. Let X be a real inner product space with dim X ≥ 2, γ ∈ R, γ > 0, D ⊂ X, and let f : X → R be a functional satisfying (3.11) such that f (D) ⊂ (−γ, γ) + Z. Suppose that one of the three following conditions is valid: (i) X is a Polish space, D is a Christensen measurable set which is not a Christensen zero set, and γ = 1 6 ; (ii) D is of the second category and with the Baire property and γ = 1 6 ; (iii) D has an algebraically interior point and γ < 1 4 . Then there exist a unique linear functional h : X → R and a unique constant c ∈ R such that (3.12) is satisfied. Moreover, if (ii) holds then h is continuous.
The following theorem describes the functions which are Christensen or Baire measurable and for which the Cauchy difference is in a discrete subgroup (Brzdęk [31]).

Theorem 3.12.
Assume that X is a real inner product space with dim X ≥ 2, (G, +) is an Abelian topological group and K a discrete subgroup of G, x+x = 0 for x ∈ G, x = 0. Let f : X → G be a function satisfying (3.9). If one of the conditions (i) X is a Polish space, G is σ-bounded 4 and f is Christensen measurable; (ii) X is a Baire space is satisfied, then there exist continuous additive functions a : R → G and A : X → G such that (3.10) holds.
In all the results we were considering so far the domain was an inner product space. Of course, it is also possible to think of a linear space with an abstract orthogonality relation, or even of a group with orthogonality in the sense of Baron and Volkmann [24] (see Sect. 1.1.11).
Vol. 89 (2015) Orthogonalities and functional equations 249 In [33], Brzdęk studied universally, Christensen or Baire measurable functions defined on a real linear topological space with axiomatic orthogonality relation by Rätz, and with values in C. In [198], Wyrobek-Kochanek proved the following result. Theorem 3.13. Assume that (G, +) is an Abelian topological group such that the mapping u → 2u, u ∈ G, is a homeomorphism and the following condition holds: every neighbourhood of zero in G contains a neighbourhood U of zero such that U ⊂ 2U and G = {2 n U : n ∈ N}. Assume that ⊥ is an orthogonality relation in G in the sense of Baron and Volkmann, (H, +) is an Abelian topological group and K is a discrete subgroup of H. Then a function f : G → H continuous at a point satisfies if, and only if, there exist a continuous additive function a : G → H and a continuous biadditive and symmetric function b : and b(x, y) = 0 for all x, y ∈ G with x ⊥ y.
(3.15) Moreover, a and b are uniquely determined.
Theorem 3.13 generalizes earlier results from the paper by Baron and Kucia [21] and also Theorem 2.9 from [32] (where Brzdęk obtained the continuity of q(x) := b(x, x), x ∈ X, only at a point).
Assume that G is a topological Abelian group, M is a σ-algebra and I is a proper σ-ideal of subsets of G which fulfil the condition Continuing the studies of Brzdęk from [33] (for functions from an orthogonality space to the complex field), Kochanek and Wyrobek [114], working now on groups with the orthogonality relation in the sense of Baron and Volkmann, faced a problem: under what assumptions does an M-measurable mapping f from (G, +) into an Abelian topological group (H, +), which is orthogonally additive modulo K, a discrete subgroup of H, admit a factorization (3.14) with a continuous additive function a : G → H and a continuous biadditive Namely, they have obtained the following results. Baire and Christensen measurable solutions of (3.13) were examined before by Brzdęk in [31] for the orthogonality given by an inner product (in inner product spaces) and in [33] for more abstract orthogonality in linear topological spaces.

Orthogonal additivity almost everywhere.
Assume that f is defined on the Euclidean space X = R n and takes values in an Abelian group (Y, +).
Kochanek and Wyrobek-Kochanek [115] were studying the functions which satisfy (2.1) almost everywhere in a sense that where Z is a negligible subset of the (2n − 1)-dimensional manifold ⊥⊂ R 2n .
They have concluded that f coincides almost everywhere with some orthogonally additive mapping.
Considerations of this type go back to a problem posed by Erdős [66], concerning the unconditional version of the Cauchy functional equation. It Vol. 89 (2015) Orthogonalities and functional equations 251 was solved by de Bruijn [29] and, independently, by Jurkat [106], and also generalized by Ger [83].
Then there exist a constant C < 45 and an additive set-function μ : M → R such that Pawlik [142] gave an example showing that C ≥ 3 2 . The above theorem was a motivation for Kochanek to study the stability problem for vector measures (understood as finitely additive set functions) (see [112]). He was investigating the properties of those Banach spaces which have the so called SVM (stability of vector measures) property; namely, we say that a Banach space X has the SVM property if there exists a constant v(X) < ∞ (depending only on X) such that given any set algebra M and any mapping ν : M → X satisfying there is a vector measure μ : M → X such that

Arithmetic functions.
A natural stability question for arithmetic additive functions may be formulated as (see Kochanek [109]): assume that for a fixed ε ≥ 0 we have the conditional inequality |f (mn) − f (m) − f (n)| ≤ ε for all m, n ∈ N with m ⊥ rp n. (3.16) Does it imply that f is approximately equal to some arithmetic additive function, that is a function satisfying (2.21)? Of course the condition that m and n are relatively prime, appearing in (3.16), causes that the direct method using Cauchy sequences cannot be used. Considering results of Erdős, Kátai, Máté (see [65,108,127]), Kochanek [109] proved the following. Then there exists c ∈ R such that |f (n) − c log n| ≤ ε for all n ∈ N.
In order to express the next result we make some notations. Let P be the set of all prime numbers and for each n ∈ N let P n = {p ∈ P : p|n}. Theorem 3.18. (Kochanek [111]) There is an absolute constant C ≤ 89 2 having the property: if a function f : N → R satisfies (3.16) and |f (m) − f (n)| ≤ 2ε for all m, n ∈ N with P m = P n , where ε ≥ 0 is a fixed constant, then there exists a strongly additive function 5 g : N → R such that |f (n) − g(n)| ≤ Cε for all n ∈ N.
The above theorem gives a stability result for strongly additive functions, but the basic problem remains open.
Problem 3.2. (Kochanek [111]) Assume that f : N → R satisfies (3.16) with some ε ≥ 0. Does there exist an additive arithmetic function g : N → R such that |f (n) − g(n)| ≤ Lε for all n ∈ N, where L is an absolute constant? Some other stability results for additive arithmetic functions one can find in the papers [109][110][111] by Kochanek.

Quadratic functional equation
In this section we consider the stability problem for the quadratic equation for functions from a space X with an orthogonality relation into a real Banach space. So, our starting point is the conditional functional inequality f (x + y) + f (x − y) − 2f (x) − 2f (y) ≤ ε for all x, y ∈ X with x ⊥ y (3.17) or, in more general form, f (x + y) + f (x − y) − 2f (x) − 2f (y) ≤ ϕ(x, y) for all x, y ∈ X with x ⊥ y, (3.18) for some function ϕ defined on X 2 .
To the best of our knowledge the first result on stability of an orthogonally quadratic functional equation was given by Drljević [62]. His orthogonality relation was defined on a complex Hilbert space (X, ·|· ) by means of a selfadjoint operator A : X → X as follows x ⊥ y if and only if Ax|y = 0.
He assumed that the functions involved are continuous and his result states what follows.
Furthermore, there exists a real number ε > 0 such that For the case A = id, so with the classical definition of orthogonality defined on an inner product space, and without the continuity assumption of a function mapping now into a Banach space, we have the following (see Sikorska [168,Theorem 5.1]). Theorem 3.20. Let (X, · ) be a real normed linear space in which the norm comes from an inner product, dim X ≥ 3, and let (Y, · ) be a real Banach space. If a function f : X → Y satisfies f (x + y) + f (x − y) − 2f (x) − 2f (y) ≤ ε( x p + y p ) (3.19) for all x, y ∈ X with x ⊥ y, with some ε ≥ 0 and p ∈ R \ {2}, then there exists a unique quadratic mapping q : X → Y such that where X p = X if p ≥ 0 (with 0 0 := 1), and X p = X \ {0} if p < 0.
The next theorem is based on a result by Moslehian [135] and proves the stability of a pexiderized form of the orthogonally quadratic equation (2.9) for functions defined on an orthogonality space. It does not, however, generalize the previous results since it is assumed that f is odd. f (x + y) + f (x − y) − 2g(x) − 2h(y) ≤ ε for all x, y ∈ X with x ⊥ y, In fact, also the converse can be proved: if for real normed spaces X and Y , the James orthogonality preserving property is stable, then (X, Y ) ∈ A (see Chmieliński [44], Wójcik [196]). So, the two properties are equivalent.
A suitable example of spaces for which the stability of the above property cannot be proved is given by Chmieliński [44] (see also Protasov [145]).
From Theorem 5.18, it follows that the property of orthogonality preservation is also stable if we consider the assumption with respect to the relation ⊥ ε J .
In the case of Birkhoff orthogonality an answer to the stability problem was given by Mojškerc and Turnšek [134,Theorem 4.1].
Theorem 5.19. Assume that (X, Y ) ∈ A and let f : X → Y be a linear mapping satisfying (5.8). Then there exists a linear mapping g : X → Y preserving the Birkhoff orthogonality and such that with some function δ (depending only on X and Y ) satisfying lim ε→0 + δ(ε) = 0.
Problem 5.5. Is the converse true, that is, for (real) normed spaces X and Y , if the Birkhoff orthogonality preserving property is stable (with approximation given by the relation ⊥ ε B ), does it necessarily follow that (X, Y ) ∈ A?
In the case of the approximation given by the relation ε ⊥ B , if X and Y are normed spaces such that the stability of the orthogonality preserving property holds, then (X, Y ) ∈ A (Mojškerc and Turnšek [134,Proposition 4.2]).
In case Y is uniformly smooth, the two kinds of stability properties and the property (X, Y ) ∈ A are equivalent (Mojškerc and Turnšek [134,Theorem 4.3]).
If X and Y are finite dimensional normed spaces and f : X → Y approximately preserves orthogonality in the sense of ε ⊥ B , then it is close to a multiple of a linear isometry, i.e., it satisfies (5.9) (Mojškerc and Turnšek [134,Proposition 4.4]).
In [196], Wójcik gave some other conditions imposed on the spaces X and Y which imply that the Birkhoff orthogonality preserving property is stable (with approximation given by the relation ε ⊥ B ) as well as he showed an example of spaces for which the Birkhoff orthogonality preserving property is not stable.
Problem 5.6. Describe the set of all pairs (X, Y ), for which the Birkhoff orthogonality preserving property (in the sense of ε ⊥ B ) is stable.
Similarly as above, the stability problem of the property of preserving the ρ (ρ + , ρ − )-orthogonality is connected with the property of approximate orthogonality preservation as well as with the stability of isometries for given spaces.
Since, by Theorem 5.5, the properties of preserving ρ-, ρ + -and ρ −orthogonality are equivalent as well as the corresponding properties of approximate preservations, the above theorem can be stated in the same form also for ρ + -and ρ − -orthogonality.