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

Dale E. Alspach¹ ; Simei Tong²

¹ Department of Mathematics Oklahoma State University Stillwater, OK 74078, U.S.A.
² Department of Mathematics University of Wisconsin-Eau Claire Eau Claire, WI 54702, U.S.A.

Studia Mathematica, Tome 159 (2003) no. 2, pp. 207-227

Résumé

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 $L_p$.

Détail

DOI : 10.4064/sm159-2-4

Keywords: many known complemented subspaces have realizations sequence spaces paper systematic approach defining these spaces which uses partitions weights introduced approach gives unified description many known complemented subspaces proved class spaces norms stable under sums introducing notion envelope norm obtain necessary condition banach sequence space norm given partitions weights isomorphic subspace using define space norm given partitions weights distance subspace growing allows construct example banach space norm given partitions weights which isomorphic subspace