Hereditary approximation property

• 成果类型: Article
• 署名作者: Johnson, W. B.; Szankowski, A.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2012.176.3.10
• 发表日期: 2012
• 页码: 1987-2001
• 关键词: banach-space; subspaces; quotients; bases

摘要

If X is a Banach space such that the isomorphism constant to l(2)(n) from n-dimensional subspaces grows sufficiently slowly as n -> infinity, then X has the approximation property. A consequence of this is that there is a Banach space X with a symmetric basis but not isomorphic to l(2) so that all subspaces of X have the approximation property. This answers a problem raised in 1980 [7]. An application of the main result is that there is a separable Banach space X that is not isomorphic to a Hilbert space, yet every subspace of X is isomorphic to a complemented subspace of X. This contrasts with the classical result of Lindenstrauss and Tzafriri [14] that a Banach space in which every closed subspace is complemented must be isomorphic to a Hilbert space.