---- > [!definition] Definition. ([[algebra of multilinear forms]]) > The **algebra of multilinear forms** on a [[vector space]] $V$ is the [[graded algebra|graded]] [[algebra]] $\text{Mult}^{\bullet}(V)= \bigoplus_{\ell \geq 0}\text{Mult}^{\ell}(V),$ where $\text{Mult}^{\ell}(V)$ is the space of [[multilinear map|multilinear]] $\ell$-forms on $V$, with multiplication determined by the **tensor product of multilinear forms**: if $f \in \text{Mult}^{p}(V), q \in \text{Mult}^{q}(V)$, then $f \otimes q \in \text{Mult}^{p+q}(V)$ is defined by $\begin{align} \\ (f \otimes g )( v_{1},\dots, v_{p+q}) = f(v_{1},\dots, v_{p})g(v_{p+1},\dots, v_{p+q}). \end{align}$ There is no confusion with the notion of 'tensor product' found in [[tensor algebra]] because [[when tensors are canonically realized as multilinear forms|this result]] implies that there is a canonical [[isomorphism]] $\mathbb{T}^{\ell}(V^{*})=(V^{*})^{\ell} \cong \text{Mult}^{\ell}(V)$, hence $\mathbb{T}^{\bullet}(V^{*}) \cong \text{Mult}^{\bullet}(V).$ > So it is indeed appropriate to call a multilinear form a tensor. - [ ] technically need to show that the product transferred over via the isomorphism is indeed the product defined here > [!justification]- > We need to show that $f \oplus g$ is indeed a $(k+\ell)$-[[multilinear map]]. We need to show it is [[linear in the ith variable]] for $i \in [k+\ell]$; WLOG we show it is linear in the first variable. So take $\begin{align}(f \otimes g)(c\v v_{1},\dots,\v v_{k+\ell}) = & f(c \v v_{1},\dots, \v v_{k})g(\v v_{k+1},\dots, \v v_{k+\ell}) \\ = & cf(\v v_{1},\dots, \v v_{k})g(\v v_{k+1},\dots,\v v_{k+\ell}) \\ = & c(f \oplus g). \end{align}$ And $\begin{align}(f \otimes g)(\v v_{1} + \v x_{1}, \dots \v v_{k+\ell}) = & f(\v v_{1} + \v x_{1},\dots, \v v_{k})g(\v v_{k+1}, \dots, \v v)k+\ell)\\ = & f((\v v_{1}, \dots, \v v_{k})+f(\v v_{k+1}, \dots, \v v_{k+\ell}))g(\v v_{k+1}, \dots, \v v_{k+\ell}) \\ = & f(\v v_{1}, \dots, \v v_{k})g(\v v_{k+1}, \dots, \v v_{k+\ell}) + f(\v x_{1}, \dots, \v v_{k})g(\v v_{k+1}, \dots, \v v_{k+\ell}) \\ = & (f \otimes g)(\v v_{1},\dots, \v v_{k+\ell}) + (f \oplus g)(\v x_{1}, \dots, \v v_{k+\ell}). \end{align}$ We've shown that $(f \otimes g)$ is [[linear in the ith variable]] for all $i \in [k + \ell]$, as required. ---- #### > [!proof]- (this has all been subsumed elsewhere) > Let $f,g,h$ be [[multilinear map]]s on a [[vector space]] $V$. >###### 1 [[associative|Associativity]]: $f \otimes (g \otimes h)= (f \otimes g) \otimes h$ . >**Proof**. This is immediate from associativity of multiplication. > >###### 2 $(cf) \otimes g = f \otimes (cg)= c(f \otimes g).$ **Proof.** Again immediate. > >###### 3 If $f,g$ have the same order, then $(f + g) \otimes h = f \otimes h + g \otimes h \and h \otimes (f+g) = h \otimes f + h \otimes g.$ **Proof.** [[TODO]] ^proof ---- #### 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 > ```