Definition 3

Let E be an A-module and E* its dual; an element x ∈ E and an element x* ∈ E* are called orthogonal if ⟨x, x*⟩ = 0.

A subset M of E and a subset M' of E* are called orthogonal sets if, for all x ∈ M, x* ∈ M', x and x* are orthogonal. In particular, x* ∈ E* (resp. x ∈ E) is called orthogonal to M (resp. M') if it is orthogonal to every element of M (resp. M'). If x* and y* are orthogonal to M, so is x* + y* and x*α for all α ∈ A by virtue of (10) and (12) (no. 3), which justifies the following definition: