Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
    
The scalar product isn't bilinear; it's sesquilinear (unless the base field is the reals, in which case the two concepts are equivalent). –  Darsh Ranjan Dec 5 '09 at 2:29
add comment

1 Answer

up vote 8 down vote accepted

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, 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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.