Addition theorems and representations of topological semigroups

Let a representation T of semigroup G on linear space X be given. We call x ∈ X a finite vector if its orbit T (G) is contained in a finite-dimensional subspace. In this paper some statements on finite vectors will be proved and applied to functional equations f (x + y + z) = a1(x)b1(y, z)+ a2(y)b2(x, z)+ a3(z)b3(x, y) + m k=1 αk(x)βk(y)γk(z) and f (x1 +· · ·+xn) = γ Δ∈γ a γ Δ i∈Δ xi , where γ runs through all possible partitions of the set {1, . . . , n} = Δ1 ∪ · · · ∪ Δr , r > 1, Δi = ∅, and a γ Δ-continuous functions on G.  2005 Elsevier Inc. All rights reserved.