The general linear equation on open connected sets

Abstract

Fix non-zero reals α 1 , . . . , α n with n ≥ 2 and let K be a non-empty open connected set in a topological vector space such that i≤n α i K ⊆ K (which holds, in particular, if K is an open convex cone and α 1 , . . . , α n > 0). Let also Y be a vector space over F := Q(α 1 , . . . , α n ). We show, among others, that a function f : K → Y satisfies the general linear equationThe main tool of the proof is a general version of a result Radó and Baker on the existence and uniqueness of extension of the solution on the classical Pexider equation.