A Hilbert space is a complete vector space equipped with scalar product, i.e. a symmetric positive definite bilinear form.


What if we replace 'bilinear' by 'n-linear'? One might wonder, whether the $l^3$-norm might be induced by a trilinear form in a similar fashion like the $l^2$-norm by a bilinear form is.


Is there any interesting theory on this?



Hilbert spaces are induced by a bilinear form. How about n-linear forms?
As long as the form is positive definite and the unit ball is convex, you get a perfectly good Banach space using any symmetric $n$-linear form on a real vector space $V$. The degree $n$ is necessarily even. It is equivalent to defining the norm as the $n$th root of a homogeneous degree $n$ polynomial. $\ell^p$ is an example for any even integer $p$. There are many other examples. I found a paper, [Banach spaces with polynomial norms](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-82/issue-1/Banach-spaces-with-polynomial-norms/pjm/1102785074.full), by Bruce Reznick, that studies these norms. He obtains various results; the most appealing one to me at a glance is that these Banach spaces are all reflexive.


Off-hand I can't think of any simple way to recover positive definiteness starting with odd polynomials. The cube of the norm on $\ell^3$ is a polynomial in the absolute values of the coordinates rather than the coordinates themselves.


Addendum: To address Darsh's comment, what you would look at in the complex case is self-conjugate polynomials of degree $(n,n)$. Equivalently, as with all complex Banach norms, the realification is a real Banach norm which is invariant under complex scalar rotation.


