Complexity of weakly null sequences

1992/02/28 by Dale E. Alspach, Spiros A. Argyros, Alspach, Dale E. +1 · 19 citations
Mathematics · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #Advanced Topology and Set Theory

paper · pdf · doi:10.48550/arxiv.math/9202204

Abstract

We introduce an ordinal index which measures the complexity of a weakly null sequence, and show that a construction due to J. Schreier can be iterated to produce for each alpha < omega₁, a weakly null sequence (x^alpha_n)_n in C(omega^omega^alpha)) with complexity alpha. As in the Schreier example each of these is a sequence of indicator functions which is a suppression-1 unconditional basic sequence. These sequences are used to construct Tsirelson-like spaces of large index. We also show that this new ordinal index is related to the Lavrentiev index of a Baire-1 function and use the index to sharpen some results of Alspach and Odell on averaging weakly null sequences.

Cited by

• Proximity to ℓ_p and c₀ in Banach spaces
• Distorting Mixed Tsirelson Spaces
• On the structure of the spreading models of a Banach space
• Proximity to ℓ₁ and Distortion in Asymptotic ℓ₁ Spaces
• Descriptive set theoretic methods applied to strictly singular and strictly cosingular operators
• On Unconditionality and Higher-Order Schreier Unconditionality
• Unconditional structures of weakly null sequences
• The ℓ ¹-index of Tsirelson type spaces
• Operators on (C[0,1]) preserving copies of asymptotic (ℓ₁) spaces
• Ramsey dichotomies with ordinal index
• Variants of the James Tree space
• ℓ ¹-spreading models in mixed Tsirelson space
• Classes of operators determined by ordinal indices
• New examples of c₀-saturated Banach spaces II
• Weakly null sequences in the Banach space C(K)
• Types of tightness in spaces with unconditional basis
• On the complemented subspaces of the Schreier spaces
• The strength of Ramsey Theorem for coloring relatively large sets
• New examples of c₀-saturated Banach spaces

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• On (p,r)-null sequences and their relatives
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• Unconditional structures of weakly null sequences
• The Szlenk index and local l₁-indices