Unconditional and bimonotone structures in high density Banach spaces

Abstract

It is shown that every normalized weakly null sequence of length ${\kappa }_{\lambda }$ in a Banach space has a subsequence of length $\lambda$ which is an unconditional basic sequence; here ${\kappa }_{\lambda }$ is a large cardinal depending on a given infinite cardinal $\lambda$. Transfinite topological games on Banach spaces are analyzed which determine the existence of a long unconditional basic sequence. Then 'asymptotic disentanglement' condition in a transfinite setting is studied which ensures a winning strategy for the unconditional basic sequence builder in the above game. The following problem is investigated: When does a Markushevich basic sequence with length uncountable regular cardinal $\kappa$ admit a subsequence of the same length which is a bimonotone basic sequence? Stabilizations of projectional resolutions of the identity (PRI) are performed under a density contravariance principle to gain some additional strong regularity properties, such as bimonotonicity.

References in corpus (1)

• Banach spaces with projectional skeletons