The absolute-value estimate for symmetric multilinear forms (Q912951)

A shorter proof is given for an algebraic function \(T(v_ 1,v_ 2,...,v_ n)\) which is symmetric on the vectors \(v_ 1,v_ 2,...,v_ n\) in some r dimensional space satisfying the inequality \(| T(v_ 1,v_ 2,...,v_ n)| \leq \| v_ 1\|...\| v_ n\| M\) where \(M=\max_{\| v\| =1}| T(v,v,...,v)|\). It is shown that for nonzero \(v_ i\) equality can occur only in very special cases when the \(v_ i's\) are not all parallel. Some detailed precise conditions are derived for the equality to hold.