Remark

Tag 05Q3, permanent. Cite it as [Bourbaki, Tag 05Q3].

Algebra, chapter II, § 4. Relations between tensor products and homomorphism modules

If E, F, G are any three C-modules, it is immediately verified that the diagram

\begin{array}{ccc} E^* \otimes F^* \otimes G^* & \xrightarrow{\mu \otimes 1} & (E \otimes F)^* \otimes G^* \\ 1 \otimes \mu \downarrow & & \downarrow \mu \\ E^* \otimes (F \otimes G)^* & \xrightarrow{\mu} & (E \otimes F \otimes G)^* \end{array}

is commutative, by virtue of formula (24).

We note also that, without any hypothesis on the C-modules E, F, there are canonical isomorphisms

(28)

(E \otimes F)^* \to \text{Hom}(E, F^*)

(29)

(E \otimes F)^* \to \text{Hom}(F, E^*)

which are just the isomorphism (6) and (5) of no. 1 for G = C, A = B = C.

Thus a canonical one-to-one correspondence has been defined between the bilinear forms on E × F, the homomorphisms of E into F and the homomorphisms of F into E: if u (resp. v) is a homomorphism of E into F* (resp. of F into E*), the corresponding bilinear form is given by

(x, y) \mapsto \langle y, u(x) \rangle \quad (\text{resp. } (x, y) \mapsto \langle x, v(y) \rangle).

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

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