Question

Let V =(infinite direct sum)eik where k is a field. For any n, let Sn be the set of endomorphisms λ ∈ E = End(Vk) such that λ stabilizes (sum over i=1,n)eik and λ(ei) = 0 for i ≥ n + 1. Show that S =(union over n)Sn ⊆ E is a dense set of linear transformations.