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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.

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References

  1. A. D. Alexandrov,A theorem on convex polyhedra, Trudy Math. Inst. Steklov., Sect. Math.4 (1933), 87 (Russian).

    Google Scholar 

  2. E. D. Bolker,A class of convex bodies, Trans. Am. Math. Soc.145 (1969), 323–345.

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Fréchet,Sur la définition axiomatique d'une classe d'espaces vectoriel distances applicables vectoriellement sur l'espace de Hilbert, Ann. of Math.36 (1935), 705–718.

    Article  MathSciNet  Google Scholar 

  4. P. Jordan and J. von Neumann,On inner products in linear metric spaces, Ann. of Math.36 (1935), 719–723.

    Article  MathSciNet  Google Scholar 

  5. J. L. Krivine,Plongment des espaces normés dans les L p pour p>2, C.R. Acad. Sci. Paris261 (1965), 4307–4310.

    MATH  MathSciNet  Google Scholar 

  6. P. Lévy,Théorle de l'addition des variables aléatoires, Gauthier-Villars, Paris, 1937.

    Google Scholar 

  7. B. Reznick,Banach spaces which satisfy linear identities, Pac. J. Math.74 (1978), 221–233.

    MATH  MathSciNet  Google Scholar 

  8. B. Reznick,Banach spaces with polynomial norms, Pac. J. Math.82 (1979), 223–235.

    MATH  MathSciNet  Google Scholar 

  9. N. W. Rickert,The range of a measure, Bull. Am. Math. Soc.73 (1967), 560–563.

    MATH  MathSciNet  Google Scholar 

  10. N. W. Rickert,Measure whose range is a ball, Pac. J. Math.23 (1967), 361–371.

    MATH  MathSciNet  Google Scholar 

  11. F. E. Sullivan,Norm characterizations of real L pspaces, Bull. Am. Math. Soc.74 (1968), 153–154.

    MATH  MathSciNet  Google Scholar 

  12. W. Weil,Zonoide und verwandte Klassen konvexer Körper, Monatsh. Math.94 (1982), 73–84.

    Article  MATH  MathSciNet  Google Scholar 

  13. H. S. Witsenhausen,A support characterization of zonotopes, Mathematika25 (1978), 13–16.

    Article  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

    Abraham Neyman

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

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Neyman, A. Representation ofL p-norms and isometric embedding inL p-spaces. Israel J. Math. 48, 129–138 (1984). https://doi.org/10.1007/BF02761158

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  • DOI: https://doi.org/10.1007/BF02761158

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