Let {S1, S2, Sn} be a collection of subsets of a linear space X with S =

Question:

Let {S1, S2, Sn} be a collection of subsets of a linear space X with S = S1 + S2 + ... + Sn. Let f be a linear functional on X. Then x* = x1* + x*2 + ... + x*n maximizes f over S if and only if x*i maximizes f over Si for every Si. That is,
f (x*) ≥ f (x) for every x ∊ S ⇔ f (x*i) ≥ f (xi) for every xi ∊ Si for every i