On the Cauchy difference

Abstract

Assume thatE andF are real topological vector spaces (which are assumed to be Hausdorff) and letK be a countable and discrete subgroup ofF. Supposef: E → F satisfies $$f(x + y) - f(x) - f(y) \in K$$ for allx, y ∈ E. Conditions are established under whichf has the formA + k, whereA: E →F is a continuous linear operator andk takes values inK only. For example, ifF is assumed to be locally convex and iff andE satisfy suitable conditions (such asE being a Baire space andf a Baire function) thenf has the formA + k, whereA : E →F is a continuous linear operator and k(E)\(k(E) \subseteq K\)