Corollary 4
Tag 05SN, permanent. Cite it as [Bourbaki, Tag 05SN].
Algebra, chapter II, § 7. Vector spaces
Let E be a vector space and M, N two subspaces of E of finite codimension; if M', N' are the orthogonals of M and N in E, the orthogonal of M ∩ N in E* is M' + N'.
As M (resp. N) is the orthogonal of M' (resp. N') in E (Theorem 7), M ∩ N is the orthogonal of M' + N' in E and hence M' + N' is the orthogonal of M ∩ N in E* (Theorem 7 (iii)).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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