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Subspaces of $L_{p}$, p > 2, determined by partitions and weights

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Alspach D. , Tong S.

tom 159

nr 2

207-227

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 Yₙ 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 $L_{p}$.