Representation ofL p-norms and isometric embedding inL p-spaces
Copyright policy )Representation ofL p-norms and isometric embedding inL p-spaces
For fixed \(1\leq p|^ p\), \(\xi \in {\mathbb{R}}^ n\). For every positive linear functional \(\sigma\), on that space, the function \(\phi_{\sigma}:{\mathbb{R}}^ n\to {\mathbb{R}}\) given given by \(\phi_{\sigma}(\xi)=\sigma (| |^ p)^{1/p}\) is an \(L_ p\)-semi-norm and the mapping \(\sigma \to \phi_{\sigma}\) is 1-1 and onto. The closed linear span of \(| |^ p\), \(\xi \in {\mathbb{R}}^ n\) 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 \(p=2\).
\(L_ p\)-semi-norms, space of all homogeneous polynomials, positive linear functionals, Norms (inequalities, more than one norm, etc.) of linear operators, norm inequalities that characterizes linear isometric embeddability, Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
\(L_ p\)-semi-norms, space of all homogeneous polynomials, positive linear functionals, Norms (inequalities, more than one norm, etc.) of linear operators, norm inequalities that characterizes linear isometric embeddability, Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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