Subspaces of $L_p$, $p>2$, determined by partitions and weights
Volume 159 / 2003
Abstract
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 $L_p$.
