On orthogonally exponential functionals

Abstract

Let (X, ⊥) be an orthogonality space and g : X → C, g(X) = {0}, be an orthogonally exponential functional, hemicontinuous at the origin. We show that then one of the follwing two conditions is valid: (i) There are unique linear functionals a 1 , a 2 : X → R with g(x) = exp(a 1 (x) + ia 2 (x)) for x ∈ X;

Measurable orthogonally additive functions modulo a discrete subgroup

“…By Lemma 1, the sections b(·, u) are continuous for u ∈ G, whence F n,k are closed for n, k ∈ N. Consequently, in case (iii) we have (3) F n,k ∈ M for n, k ∈ N.…”

“…Now we are prepared to proceed to our main result. The technical assumptions appearing below have been already considered (see [7], [3], [6] and [14]). In the last section we present a counterexample showing that condition (G2) is essential.…”

Orthogonalities and functional equations

“…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.…”

“…). 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.…”

Orthogonally additive functions modulo a discrete subgroup

“…The first three lemmas and Lemma 4(i) are very similar to some results from [2], [6] and [5], but for the reader's convenience we formulate them explicitly; however, we omit their proofs. Note that Lemma 1(ii) [ …”