Subspaces of $L_p$, $p>2$, determined by partitions and weights

Volume 159 / 2003

Dale E. Alspach, Simei Tong Studia Mathematica 159 (2003), 207-227 MSC: Primary 46B20; Secondary 46E30. DOI: 10.4064/sm159-2-4


Many of the known complemented subspaces of $L_p$ have realizations as sequence spaces. In this paper a systematic approach to defining these spaces which uses partitions and weights is introduced. This approach gives a unified description of many well known complemented subspaces of $L_p$. It is proved that the class of spaces with such norms is stable under $(p,2)$ sums. By introducing the notion of an envelope norm, we obtain a necessary condition for a Banach sequence space with norm given by partitions and weights to be isomorphic to a subspace of $L_p$. Using this we define a space $Y_n$ with norm given by partitions and weights with distance to any subspace of $L_p$ growing with $n$. This allows us to construct an example of a Banach space with norm given by partitions and weights which is not isomorphic to a subspace of $L_p$.


  • Dale E. AlspachDepartment of Mathematics
    Oklahoma State University
    Stillwater, OK 74078, U.S.A.
  • Simei TongDepartment of Mathematics
    University of Wisconsin-Eau Claire
    Eau Claire, WI 54702, U.S.A.

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