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Is there such a thing as a "trilinear inner product"? The definition of an inner product is:

Let $H$ be a vector space over $\mathbb{K}\in \{\mathbb{R,C}\}$. An inner product is a map $\langle \cdot|\cdot\rangle: H^2 \to \mathbb{K}$ such that for all $x,y,z \in H$ and $\lambda \in \mathbb{K}$ the following properties hold:

  • Bilinearity: $\langle x+\lambda y | z\rangle = \langle x|z\rangle + \lambda \langle y|z\rangle $

  • Complex conjugacy: $\overline{\langle y | x \rangle} = \langle x | y \rangle$

  • Positive definiteness: $||x||^2:=\langle x | x \rangle$ > 0 if $x \neq 0$

Can this be modified to have a trilinear map $\langle \cdot |\cdot| \cdot \rangle : H^3 \to \mathbb{K}$? Would it for example be possible to make $L^3$ into a "trilinear inner product space" like $L^2$ is a "bilinear inner product space"? What is so special about the number $2$ in this context? Of course $2$ is the only number that is conjugate to itself in the sense that $\frac{1}{2}+\frac{1}{2}$, so there would be no nice identification of this trilinear inner product space with it's dual.

I guess that there is no useful notion because the complex conjugacy can't be modified to get a trilinear inner product: $\mathbb{C}$ is a field extension of degree $2$ of $\mathbb{R}$, but there is no field extension of degree $3$ of the reals this inner product could be defined over. What if the "trilinear space" is solely defined over $\mathbb{R}$?

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  • $\begingroup$ Given a one dimensional vector space over the reals, try to come up with a trilinear map with the properties you listed. $\endgroup$
    – Somos
    Apr 17, 2019 at 22:48
  • $\begingroup$ @Somos Yeah defining it as $\langle x | y | \z \rangle := xyz$ the problem is with the positive definiteness. But what about "$2n$-linear inner products"? $\endgroup$ Apr 18, 2019 at 9:44

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There is a notion of higher order inner products: https://terrytao.wordpress.com/tag/inner-product-spaces/

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  • $\begingroup$ This is exactly what I was looking for, thank you! $\endgroup$ Apr 18, 2019 at 11:03

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