#### Vol. 14, No. 5, 2021

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Uniform subsequential estimates on weakly null sequences

### Milena Brixey, Ryan M. Causey and Patrick Frankart

Vol. 14 (2021), No. 5, 743–774
##### Abstract

We provide a generalization of two results of Knaust and Odell. We prove that if $X$ is a Banach space and ${\left({g}_{n}\right)}_{n=1}^{\infty }$ is a right dominant Schauder basis such that every normalized, weakly null sequence in $X$ admits a subsequence dominated by a subsequence of ${\left({g}_{n}\right)}_{n=1}^{\infty }$, then there exists a constant $C$ such that every normalized, weakly null sequence in $X$ admits a subsequence $C$-dominated by a subsequence of ${\left({g}_{n}\right)}_{n=1}^{\infty }$. We also prove that if every spreading model generated by a normalized, weakly null sequence in $X$ is dominated by some spreading model generated by a subsequence of ${\left({g}_{n}\right)}_{n=1}^{\infty }$, then there exists $C$ such that every spreading model generated by a normalized, weakly null sequence in $X$ is $C$-dominated by every spreading model generated by a subsequence of ${\left({g}_{n}\right)}_{n=1}^{\infty }$. We also prove a single, ordinal-quantified result which unifies and interpolates between these two results.

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##### Keywords
Banach spaces, Ramsey theory
##### Mathematical Subject Classification
Primary: 46B03, 46B15