Hilbert spaces are induced by a bilinear form. How about n-linear forms? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T02:42:07Z http://mathoverflow.net/feeds/question/7828 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/7828/hilbert-spaces-are-induced-by-a-bilinear-form-how-about-n-linear-forms Hilbert spaces are induced by a bilinear form. How about n-linear forms? Martin 2009-12-05T02:09:06Z 2009-12-05T03:40:32Z <p>A Hilbert space is a complete vector space equipped with scalar product, i.e. a symmetric positive definite bilinear form.</p> <p>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.</p> <p>Is there any interesting theory on this?</p> http://mathoverflow.net/questions/7828/hilbert-spaces-are-induced-by-a-bilinear-form-how-about-n-linear-forms/7832#7832 Answer by Greg Kuperberg for Hilbert spaces are induced by a bilinear form. How about n-linear forms? Greg Kuperberg 2009-12-05T02:37:21Z 2009-12-05T02:52:58Z <p>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, <a href="http://projecteuclid.org/DPubS?verb=Display&amp;version=1.0&amp;service=UI&amp;handle=euclid.pjm/1102785074&amp;page=record" rel="nofollow">Banach spaces with polynomial norms</a>, 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.</p> <p>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.</p> <p>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.</p>