Let $B$ be a symmetric bilinear form over a Euclidean space $E$. Say that $|B(v,v)|\le c\|v\|^2$ for every $v\in E$, for some $c\ge0$. Then
$$4B(v,w)=B(v+w)+B(v-w)$$
yields $2|B(v,w)|\le c(\|v\|^2+\|w\|^2)$. Replacing $v$ by $tv$, $w$ by $t^{-1}w$ and minimizing over $t$, we get $|B(v,w)|\le c\|v\|\cdot\|w\|$. Notice that we can't use Cauchy-Schwarz if $B$ is indefinite.


I am interested in symmetric $n$-linear forms $B$ over $E$.



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> If $B(v,\ldots,v)|\le c\|v\|^n$ for every $v\in E$, is it true that $|B(v\_1,\ldots,v\_n)|\le c\|v\_1\|\cdots\|v\_n\|$ ?
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At least, there exists a constant $\gamma\_n$ such that $B(v,\ldots,v)|\le c\|v\|^n$ implies $|B(v\_1,\ldots,v\_n)|\le \gamma\_nc\|v\_1\|\cdots\|v\_n\|$. This is because the map which, to a homogeneous polynomial of degree $n$ over $E$, associates its polar form (a symmetric $n$-linear form) is well-defined and linear. What is the best constant $\gamma\_n$ ?



Norm of $n$-linear symmetric forms
After a bit of searching, I found that $\gamma\_n=1$ for all $n$. 


This is known as "van der Corput-Schaake inequality" (1935), discovered before by Szegö (1928), and mentioned to have been known to Banach in the Scottish Book. 


The proof proceeds by showing that if $P$ is homogeneous of degree $n$ on a euclidean space, one has, for $B$ the unit ball, $$\sup\_{B\times B} |\nabla P(x)\cdot y|\leq n \sup\_B |P|,$$ reducing first to the 2-dimensional case and reasoning about trigonometric polynomials.


This implies by differentiating successively that $$\sup\_{x\_i\in B}|D^nP(x\_1,\dots,x\_n)| \leq n! \sup\_B |P|,$$
hence the result since $D^nP/n!$ is the polarization of $P$.


I must admit that I don't understand the proof yet.


Here is a link to van der Corput-Schaake's article


<http://archive.numdam.org/article/CM_1935__2__321_0.pdf>


EDIT: I found more recent references with simpler proofs, for instance in Chapter 4 of DeVore and Lorentz "Constructive approximation" (Springer 1993), cf the Szegö inequality, and in an [expository text](http://www.maths.tcd.ie/pub/ims/bull36/bull36_19-33.pdf) by Lawrence A. Harris. Hope this helps.


