# Freeman, Daniel B. (2009-08). Upper Estimates for Banach Spaces. Doctoral Dissertation. Thesis

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### abstract

• 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 $\omega^{\alpha\omega}$.
• 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 $Cgeq1$ 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 $omega^{alphaomega}$.

• August 2009