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arXiv · 1604.04408

Unconditional and bimonotone structures in high density Banach spaces

Jarno Talponen
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Abstract

It is shown that every normalized weakly null sequence of length $κ_λ$ in a Banach space has a subsequence of length $λ$ which is an unconditional basic sequence; here $κ_λ$ is a large cardinal depending on a given infinite cardinal $λ$. 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 $κ$ 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.

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Jarno Talponen. 2016-07-07. Unconditional and bimonotone structures in high density Banach spaces. https://arxiv.org/abs/1604.04408

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