The linear subspaces of a unital algebra over a field form a Kleene algebra. Given linear subspaces "V" and "W", define "V" + "W" to be the sum of the two subspaces, and 0 to be the trivial subspace {0}. Define "V" · "W" = span {v · w | v ∈ V, w ∈ W}, the linear span of the product of vectors from "V" and "W" respectively. Define 1 = span {I}, the span of the unit of the algebra. The closure of "V" is the direct sum of all powers of "V".