, Volume 48, Issue 2-3, pp 129-138

Representation ofL p-norms and isometric embedding inL p-spaces

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Abstract

For fixed 1≦p<∞ theL p-semi-norms onR n are identified with positive linear functionals on the closed linear subspace ofC(R n ) spanned by the functions |<ξ, ·>| p , ξ∈R n . For every positive linear functional σ, on that space, the function Φσ:R n R given by Φσ is anL p-semi-norm and the mapping σ→Φσ is 1-1 and onto. The closed linear span of |<ξ, ·>| p , ξ∈R n is the space of all even continuous functions that are homogeneous of degreep, ifp is not an even integer and is the space of all homogeneous polynomials of degreep whenp 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 anyL p unlessp=2.

Supported by the National Science Foundation MCS-79-06634 at U.C. Berkeley.