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Upper Estimates for Banach SpacesFreeman, Daniel B. 2009 August 1900 (has links)
We 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 $\omega^{\alpha\omega}$.
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