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 $\omega^{\alpha\omega}$.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-2009-08-7100 |
| Date | 2009 August 1900 |
| Creators | Freeman, Daniel B. |
| Contributors | Schlumprecht, Thomas |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | application/pdf |
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