A remark on constructions of certain normed and nonsingular bilinear maps
For what triples (a,b,c) does there exist an \({\mathbb{R}}\)-bilinear map \(f: {\mathbb{R}}^ a\times {\mathbb{R}}^ b\to {\mathbb{R}}^ c\) with \(\| f(x,y)\| =\| x\| \cdot \| y\|\)? Here \(\| z\| =z^ 2_ 1+...+z^ 2_ d\) for \(z\in {\mathbb{R}}^ d\). The paper under review announces a graph-theoretical approach to the classical problem above for the special case \(b=c=2^ m\). The same methods are also used for the following related question: For what triples (a,b,c) does there exist an \({\mathbb{R}}\)-bilinear map \(f: {\mathbb{R}}^ a\times {\mathbb{R}}^ b\to {\mathbb{R}}^ c\) such that \(f(x,y)=0\) implies \(x=0\) or \(y=0?\)
scientific article; zbMATH DE number 3875326