Representation of \(L_ p\)-norms and isometric embedding in \(L_ p\)- spaces (Q1080116)

For fixed \(1\leq p<\infty\) the \(L_ p\)-semi-norms on \({\mathbb{R}}^ n\) are identified with positive linear functionals on the closed linear subspace of \(C({\mathbb{R}}^ n)\) spanned by the functions \(| <\xi,\cdot >|^ p\), \(\xi \in {\mathbb{R}}^ n\). For every positive linear functional \(\sigma\), on that space, the function \(\phi_{\sigma}:{\mathbb{R}}^ n\to {\mathbb{R}}\) given given by \(\phi_{\sigma}(\xi)=\sigma (| <\xi,\cdot >|^ p)^{1/p}\) is an \(L_ p\)-semi-norm and the mapping \(\sigma \to \phi_{\sigma}\) is 1-1 and onto. The closed linear span of \(| <\xi,\cdot >|^ p\), \(\xi \in {\mathbb{R}}^ n\) is the space of all even continuous functions that are homogeneous of degree p, if p is not an even integer and is the space of all homogeneous polynomials of degree p when p is an even integer. This representation is used to prove that there is no finite list of norm inequalities that characterizes linear isometric embeddability, in any \(L_ p\) unless \(p=2\).