Maximal vector topologies

Suppose that κ is an infinite cardinal, V κ = ⊕ α < κ R α is a vector space of dimension κ over R , τ κ is the box topology on V κ , μ κ and ν κ are the maximal and maximal locally convex vector topologies on V κ respectively. We prove that τ ω = μ ω = ν ω but, for every κ > ω , μ κ ⊃ ν κ ⊃ τ κ . For every κ, the topological vector spaces ( V κ , μ κ ) , ( V κ , ν κ ) , ( V κ , τ κ ) are complete but not sequential.