Example 3

Tag 06DG, permanent. Cite it as [Bourbaki, Tag 06DG].

Algebra, IV, p. 55, § 5. Symmetric tensors and polynomial mappings

Let $(M_j)_{j \in J}$ be a finite family of free A-modules and $u : \prod_{j \in J} M_j \to N$ a multilinear mapping ; then u is polynomial of degree $\mathrm{Card}(J)$ .

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

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