Definition 4
Tag 05NC, permanent. Cite it as [Bourbaki, Tag 05NC].
Algebra, chapter II, § 2. Modules of linear mappings. Duality
Given a subset M of E (resp. a subset M' of E*), the set of x* ∈ E* (resp. the set of x ∈ E) which are orthogonal to M (resp. M') is called the submodule totally orthogonal to M (resp. M') (or simply the submodule orthogonal to M (resp. M') if no confusion can arise).
By definition of a linear form, the submodule of E* orthogonal to E is reduced to 0; the submodule of E* orthogonal to {0} is identical with E*.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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