Definition 2

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

Algebra, chapter IX, § 1. Formes sesquilinéaires

Given a bilinear mapping $\Phi$ from $E \times F$ into $G$ , the mapping $d_\Phi$ from $F$ into $\mathscr{L}_A(E, G)$ (resp. the mapping $s_\Phi$ from $E$ into $\mathscr{L}_B(F, G)$ ) characterized by (5) is called the linear mapping associated on the right (resp. on the left) with $\Phi$ .

Conversely, the data of a B-linear mapping $d$ from $F$ into $\mathscr{L}_A(E, G)$ (resp. of an A-linear mapping $s$ from $E$ into $\mathscr{L}_B(F, G)$ ) determines uniquely, by the formula

\Phi(x, y) = d(y)(x) \quad \text{(resp. } \Phi(x, y) = s(x)(y))

a bilinear mapping $\Phi$ from $E \times F$ into $G$ , of which $d$ (resp. $s$ ) is the linear mapping associated on the right (resp. on the left).

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

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Cites

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