Upper Estimates for Banach Spaces

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dc.contributorSchlumprecht, Thomas
dc.creatorFreeman, Daniel B.
dc.date.accessioned2010-10-12T22:31:32Z
dc.date.accessioned2010-10-14T16:02:36Z
dc.date.accessioned2017-04-07T19:57:36Z
dc.date.available2010-10-12T22:31:32Z
dc.date.available2010-10-14T16:02:36Z
dc.date.available2017-04-07T19:57:36Z
dc.date.created2009-08
dc.date.issued2010-10-12
dc.description.abstractWe study the relationship of dominance for sequences and trees in Banach spaces. In the context of sequences, we prove that domination of weakly null sequences is a uniform property. More precisely, if $(v_i)$ is a normalized basic sequence and $X$ is a Banach space such that every normalized weakly null sequence in $X$ has a subsequence that is dominated by $(v_i)$, then there exists a uniform constant $C\geq1$ such that every normalized weakly null sequence in $X$ has a subsequence that is $C$-dominated by $(v_i)$. We prove as well that if $V=(v_i)_{i=1}^\infty$ satisfies some general conditions, then a Banach space $X$ with separable dual has subsequential $V$ upper tree estimates if and only if it embeds into a Banach space with a shrinking FDD which satisfies subsequential $V$ upper block estimates. We apply this theorem to Tsirelson spaces to prove that for all countable ordinals $\alpha$ there exists a Banach space $X$ with Szlenk index at most $\omega^{\alpha \omega +1}$ which is universal for all Banach spaces with Szlenk index at most $\omega^{\alpha\omega}$.
dc.identifier.urihttp://hdl.handle.net/1969.1/ETD-TAMU-2009-08-7100
dc.language.isoen_US
dc.subjectupper estimates
dc.subjectuniform estimates
dc.subjectweakly null sequences
dc.subjectSzlenk index
dc.subjectuniversal space
dc.subjectembedding into FDDs
dc.subjectEfros-Borel structure
dc.subjectanalytic classes
dc.titleUpper Estimates for Banach Spaces
dc.typeBook
dc.typeThesis

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