Many of the known complemented subspaces of 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 . 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 . Using this we define a space Yₙ with norm given by partitions and weights with distance to any subspace of 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 .
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm159-2-4,
author = {Dale E. Alspach and Simei Tong},
title = {Subspaces of $L\_{p}$, p > 2, determined by partitions and weights},
journal = {Studia Mathematica},
volume = {157},
year = {2003},
pages = {207-227},
zbl = {1057.46014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm159-2-4}
}
Dale E. Alspach; Simei Tong. Subspaces of $L_{p}$, p > 2, determined by partitions and weights. Studia Mathematica, Tome 157 (2003) pp. 207-227. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm159-2-4/