> [!definition] > > Let $R$ be a commutative [[Ring|ring]]. Let $\seqf{E_j}$ and $F, G$ be $R$-[[Module|modules]], and denote > $ > L^n(E_1, \cdots, E_n; F) > $ > as the module of $n$-[[Multilinear Map|multilinear]] maps from the product $\prod_{j}E_j$ to $F$. Let $f \in L^n(\seqf{E_n}; F)$ and $g \in L^n(\seqf{E_n}; F)$. A morphism from $f$ to $g$ is a linear map $h: F \to G$ such that the following diagram commutes: > > ```tikz > \usepackage{tikz-cd} > \begin{document} > \begin{tikzcd} & F \\ {\prod_j E_j} \\ & G \arrow["h", from=1-2, to=3-2] \arrow["f", from=2-1, to=1-2] \arrow["g"', from=2-1, to=3-2] \end{tikzcd} > \end{document} > ``` > > This makes the multilinear maps from $\seqf{E_j}$ a [[Category|category]]. A [[Universal Object|universally repelling]] object in this category > ```tikz > \usepackage{tikz-cd} > \begin{document} > \begin{tikzcd} {\prod_jE_j} & F \\ {\bigotimes_j E_j} \arrow["f"{description}, from=1-1, to=1-2] \arrow["\varphi"{description}, from=1-1, to=2-1] \arrow["{f_*}"{description}, from=2-1, to=1-2] \end{tikzcd} > \end{document} > ``` > > is a **tensor product** of $\seqf{E_j}$. By the universal property, it is unique up to unique isomorphisms. > [!theorem] > > Let $\seqf{E_j}$ be a family of $R$-modules, then their tensor product exists. > > The module constructed in the proof, $M/N$ and the map $\varphi: \prod_j E_j \to M/N$ is known as **the** tensor product, from which we identify $M/N = \bigotimes_{j} E_j$. For any sequence $\seqf{x_j}$, its image > $ > \varphi(x_1, \cdots, x_n) = x_1 \otimes \cdots \otimes x_n > $ > which follows multilinear addition and multiplication. > > *Proof*. Let $M$ be the [[Free Module|free module]] of formal linear combinations generated by elements of $\prod_j E_j$. Let $N$ be the submodule generated by elements of the following form: > - $(x_1, \cdots, x_j + x_j', \cdots, x_n) - (x_1, \cdots, x_j, \cdots, x_n) - (x_1, \cdots, x_j', \cdots, x_n)$ (multilinear addition). > - $(x_1, \cdots, \alpha x_j, \cdots , x_n) - \alpha(x_1, \cdots, x_n)$ (multilinear multiplication). > > Let $\iota: \prod_j E_j \to M$ be the canonical injection, and $\pi: M \to M/N$ be the canonical quotient map. Define > $ > \varphi: \prod_j E_j \to M/N \quad \varphi = \pi \circ \iota > $ > From the definition of $N$, we know that $\varphi$ is multilinear. Let $f \in L(\seqf{E_j}; F)$ be a multilinear map, then $f$ admits a unique extension $\overline{f}$ to $M$. > > ```tikz > \usepackage{tikz-cd} > \begin{document} > \begin{tikzcd} {\prod_j E_j} && M \\ & {M/N} \\ & F \arrow["\iota"{description}, from=1-1, to=1-3] \arrow["\varphi"{description}, from=1-1, to=2-2] \arrow["f"{description}, from=1-1, to=3-2] \arrow["\pi"{description}, from=1-3, to=2-2] \arrow["{\overline{f}}"{description}, from=1-3, to=3-2] \arrow["{f_*}"{description}, from=2-2, to=3-2] \end{tikzcd} > \end{document} > ``` > > Since $f|_N = 0$, $\ol{f}$ can be factored through the [[First Isomorphism Theorem]] to obtain $f_*$. The module $M/N$ is the desired product. > [!theorem] > > Let $\seqf{E_j}$ be a family of $R$-modules, then there exists a canonical injective linear map > $ > \bigotimes_{j = 1}^nE_j^* \hookrightarrow \braks{\bigotimes_{j = 1}^n E_j}^* > $ > If each $E_j$ is a finite-dimensional [[Vector Space|vector space]], then this map is an isomorphism. > > *Proof*. Define > $ > \Phi: \prod_j E_j^* \to L(E_1, \cdots, E_n; R) \quad \Phi(f_1, \cdots, f_n)(x) = \prod_{j}f_j(x_j) > $ > then $\Phi$ is a multilinear injective map, which is a linear functional on the tensors.