Corollary

Tag 05SF, permanent. Cite it as [Bourbaki, Tag 05SF].

For every exact sequence

0 \longrightarrow E' \xrightarrow{u} E \xrightarrow{v} E'' \longrightarrow 0

of vector spaces over the same field K and linear mappings the sequence

0 \longrightarrow E''* \xrightarrow{t_v} E^* \xrightarrow{t_u} {E'}^* \longrightarrow 0

is exact and splits.

It follows in particular that for every vector subspace M of E the canonical homomorphism $E^*/M' \to M^*$ , where $M'$ is the subspace of $E^*$ orthogonal to M (§ 2, no. 4), is bijective.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Theorem 6, § 7
Remark 2, § 20
Proposition 3, § 20
§ 21. Linear Representations of Finite Groups

Cites

§ 2, no. 4