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Many of the known complemented subspaces of Lp 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 Lp. 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 Lp. Using this we define a space Yn with norm given by partitions and weights with distance to any subspace of Lp 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 Lp.

Subspaces of Lp, p>2, determined by partitions and weights

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