Representation of L_p-norms and isometric embedding in L_p-spaces

For fixed 1≦p<∞ the L_p-semi-norms on Rⁿ are identified with positive linear functionals on the closed linear subspace of C(Rⁿ) spanned by the functions |<ξ, ·>|^p, ξ∈Rⁿ. For every positive linear functional σ, on that space, the function Φ_σ: Rⁿ→R given by Φ_σ is an L_p-semi-norm and the mapping σ→Φ_σ is 1-1 and onto. The closed linear span of |<ξ, ·>|^p, ξ∈Rⁿ 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.