On orthogonally exponential functionals

Janusz Brzdȩk

Vol. 181 (1997), No. 2, 247–267

DOI: 10.2140/pjm.1997.181.247

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:

1. There are unique linear functionals a₁,a₂ : X → R with

2. there are a ⊥-equivalent inner product ⟨⋅,⋅⟩ in X, c ∈ C, and unique linear functionals a₁,a₂ : X → R such that

   where ∥x∥ = ⟨x,x⟩ for x ∈ X.

We also prove some auxiliary results concerning functions f mapping a real linear (orthogonality) space X into a commutative group (G,+) and satisfying one of the following two conditions:

	f(x + y) + f(x − y) − 2f(x) − 2f(y) ∈ K for x,y ∈ X,
	f(x + y) − f(x) − f(y) ∈ K whenever x ⊥ y,

where K is a subgroup of G.

Milestones

Received: 10 January 1996

Published: 1 December 1997