The Hyers–Ulam and Hahn–Banach Theorems and Some Elementary Operations on Relations Motivated by Their Set-Valued Generalizations

In the first part of this paper, we provide several historical facts on the famous Hyers–Ulam stability theorems, Hahn–Banach extension theorems, and their set-valued generalizations with numerous references. These generalizations will clearly show that the essence of the above mentioned theorems is nothing but the statement of the existence of a certain homogeneous, additive, or linear selection function of a particular relation. In the second part of this paper, motivated by the above generalizations, we briefly review the most basic additivity and homogeneity properties of relations and investigate, in greater detail, some elementary operations on relations. More concretely, for any relation F on one group X to another Y, we define two relations −F and $\check{F}$ on X to Y such that $\check{F}(x)=F(-x)$ and (−F)(x)=−F(x) for all x∈X. Moreover, we also define $\hat{F}=-\check{F}$ and $F^{\vartriangle}= F\cap\hat{F}$ . Furthermore, if in particular Y is a vector space over ℚ, then for any k∈ℤ, with k≠0, we also define a relation F k on X to Y such that F k (x)=k −1 F(kx) for all x∈X. Moreover, we also define $F^{\star}=\bigcap_{ n=1}^{\infty} F_{n}$ and $F^{\ast}=F^{\vartriangle\star}$ . The above operations and the intersection convolutions of relations, which can only be sketched here, will certainly allow of instructive treatments of some hoped-for common relational generalizations of the Hyers–Ulam and Hahn–Banach theorems.