---- > [!theorem] Theorem. ([[when tensors are canonically realized as multilinear forms]]) > Let $M_{1},\dots,M_{\ell}$ be [[free module|free]] [[ideal generated by a subset|finitely generated]] $R$-[[module|modules]] for $R$ a [[commutative ring]], and $M_{1}^{*},\dots,M_{\ell}^{*}$ their [[dual vector space|dual modules]]. With $\otimes$ denoting the [[tensor product of modules]] and $L(M_{1},\dots,M_{\ell};R)$ the space of [[multilinear map|multilinear forms]] $M_{1} \times \dots \times M_{\ell} \to R$, there is a canonical [[isomorphism]] of $R$-[[module|modules]] $M_{1}^{*} \otimes \dots \otimes M_{\ell}^{*} \cong L(M_{1},\dots,M_{\ell}; R)$ under which the elements of the [[tensor product of modules]] correspond to [[algebra of multilinear forms|tensor products]] of 1-[[dual vector space|linear forms]]. > > Moreover, since we are assuming finitely generated + [[projective module|projective]], we can [[double dual of a finite-dimensional vector space is naturally isomorphic to that space|also canonically identify]] each $M_{i} \cong M_{i}^{* *}$ to obtain $M_{1} \otimes \dots \otimes M_{\ell} \cong L(M_{1}^{*},\dots, M_{\ell}^{*}).$ > > ^theorem - [ ] cast in terms of tensor-hom adjunction > [!proof]- Proof. ([[when tensors are canonically realized as multilinear forms]]) > Define the obvious map $\begin{align} \Phi: M_{1}^{*} \times \dots \times M_{\ell}^{*} &\to L(M_{1},\dots,M_{\ell}; R) \text{ by}\\ \Phi(\omega^{1}, \dots, \omega^{\ell})(v_{1},\dots, v_{\ell})& := \omega^{1}(v_{1}) \cdots \omega^{\ell}(v_{\ell}). \end{align}$ It is not hard to see that $\Phi$ itself is a [[multilinear map]]. The [[tensor product of modules|universal property of tensor products]] then induces a unique [[linear map|linear map]] $\overline{\Phi}:M_{1}^{*} \otimes \dots \otimes M_{\ell}^{*} \to L(M_{1},\dots,M_{\ell}; R)$ satisfying $\overline{\Phi}(\omega^{1} \otimes \dots \otimes \omega^{\ell})(v_{1},\dots,v_{\ell}) = \Phi(\omega^{1},\dots,\omega^{\ell})(v_{1},\dots,v_{\ell})=\omega^{1}(v_{1})\cdots\omega^{\ell}(v_{\ell}).$ Then be convinced that the product and bases are preserved. ---- #### ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```