Definition 10

Tag 0DJC, permanent. Cite it as [Bourbaki, Tag 0DJC].

Algebra, chapter IX, § 1. Formes sesquilinéaires

The hypotheses and notations being as previously, one says that the homomorphism $u^*$ from $F'_1$ into F satisfying (30) is the left adjoint of $u$ , and that $F'_1$ is the defining submodule of $u^*$ .

The right adjoint of a homomorphism $\varphi$ from F into $F'$ is defined analogously by the formula

\Phi'(x', \varphi(y)) = \Phi(\varphi^*(x'), y) \quad (x' \in E'_1, y \in F),

where $E'_1$ denotes the submodule of $E'$ defined analogously to $F'_1$ .

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

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