Suitable norms for simultaneous approximation (Q1307378)

The most common way to study simultaneous approximation of two points \(x,y\) of a normed linear space \(E\) from a subset \(M\) of \(E\), is to consider a suitable norm in \(E\times E\) (for instance, \(\| (u,v)\| =\sup(\| u\| ,\| v\|)\), \(\| (u,v)\| =\| u\| +\| v\|)\) and to reduce this problem to the ordinary approximation of the single point \((x,y)\in E\times E\) from the diagonal set of \(M\times M\). In this paper norms on \(E\times E\) that are suitable for simultaneous approximation will be classified, through simple propositions and examples.