Proposition 15

Tag 05T0, permanent. Cite it as [Bourbaki, Tag 05T0].

If $(E_\lambda)_{\lambda \in L}$ is a family of right vector spaces over a field K and $(F_\mu)_{\mu \in M}$ a family of left vector spaces over K, the canonical mapping

\left( \prod_{\lambda \in L} E_\lambda \right) \otimes_K \left( \prod_{\mu \in M} F_\mu \right) \to \prod_{(\lambda, \mu) \in L \times M} (E_\lambda \otimes_K F_\mu)

(§ 3, no. 7, formula (22)) is injective.

We write $F = \prod_{\mu \in M} F_\mu$ ; the mapping (26) is the composition of the canonical mappings

\left( \prod_{\lambda \in L} E_\lambda \right) \otimes_K F \to \prod_{\lambda \in L} (E_\lambda \otimes_K F)

and

\prod_{\lambda \in L} (E_\lambda \otimes_K F) \to \prod_{\lambda \in L} \left( \prod_{\mu \in M} (E_\lambda \otimes_K F_\mu) \right);

F

and the

E_\lambda

are vector spaces over

K

, this reduces to § 3, no. 7, Corollary 3 to Proposition 7.

When the conditions of Proposition 15 are fulfilled, the tensor product

\left( \prod_{\lambda \in L} E_\lambda \right) \otimes_K \left( \prod_{\mu \in M} F_\mu \right)

is often identified with its canonical image in

\prod_{\lambda, \mu} (E_\lambda \otimes_K F_\mu).

With this convention:

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

Proposition 15

Cited by

Cites