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On asymptotically symmetric Banach spaces

100%

Junge M., Kutzarova D., Odell E.

Studia Mathematica

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2006

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tom 173

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nr 3

203-231

EN

A Banach space X is asymptotically symmetric (a.s.) if for some C < ∞, for all m ∈ ℕ, for all bounded sequences $(x_{j}^{i})_{j=1}^{∞} ⊆ X$, 1 ≤ i ≤ m, for all permutations σ of {1,...,m} and all ultrafilters 𝒰₁,...,𝒰ₘ on ℕ, $lim_{n₁,𝒰₁} ... lim_{nₘ,𝒰ₘ} ||∑_{i=1}^{m} x_{n_i}^{i}|| ≤ C lim_{n_{σ(1)},𝒰_{σ(1)}} ... lim_{n_{σ(m)},𝒰_{σ(m)}} ||∑_{i=1}^{m} x_{n_{i}}^{i}||$. We investigate a.s. Banach spaces and several natural variations. X is weakly a.s. (w.a.s.) if the defining condition holds when restricted to weakly convergent sequences $(x_{j}^{i})_{j=1}^{∞}$. Moreover, X is w.n.a.s. if we restrict the condition further to normalized weakly null sequences. If X is a.s. then all spreading models of X are uniformly symmetric. We show that the converse fails. We also show that w.a.s. and w.n.a.s. are not equivalent properties and that Schlumprecht's space S fails to be w.n.a.s. We show that if X is separable and has the property that every normalized weakly null sequence in X has a subsequence equivalent to the unit vector basis of c₀ then X is w.a.s. We obtain an analogous result if c₀ is replaced by ℓ₁ and also show it is false if c₀ is replaced by $ℓ_{p}$, 1 < p < ∞. We prove that if 1 ≤ p < ∞ and $||∑_{i=1}^{n} x_{i}|| ∼ n^{1/p}$ for all $(x_{i})_{i=1}^{n} ∈ {X}ₙ$, the nth asymptotic structure of X, then X contains an asymptotic $ℓ_{p}$, hence w.a.s. subspace.

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Banach spaces of bounded Szlenk index

100%

Odell E., Schlumprecht T., Zsák A.

Studia Mathematica

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2007

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tom 183

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nr 1

63-97

EN

For a countable ordinal α we denote by $𝓒_{α}$ the class of separable, reflexive Banach spaces whose Szlenk index and the Szlenk index of their dual are bounded by α. We show that each $𝓒_{α}$ admits a separable, reflexive universal space. We also show that spaces in the class $𝓒_{ω^{α·ω}}$ embed into spaces of the same class with a basis. As a consequence we deduce that each $𝓒_{α}$ is analytic in the Effros-Borel structure of subspaces of C[0,1].

3

Banach spaces of bounded Szlenk index II

100%

Freeman D., Odell E., Schlumprecht T., Zsák A.

Fundamenta Mathematicae

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2009

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tom 205

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nr 2

161-177

EN

For every α < ω₁ we establish the existence of a separable Banach space whose Szlenk index is $ω^{αω+1}$ and which is universal for all separable Banach spaces whose Szlenk index does not exceed $ω^{αω}$. In order to prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with Tsirelson type upper estimates.

4

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The unit ball of every infinite-dimensional normed linear space contains a (1+ɛ)-separated sequence

99%

Elton J., Odell E.

Colloquium Mathematicum

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1981

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tom 44

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nr 1

105-109

5

On the structure of separable $L_{p}$ spaces (1 < p < ∞)

75%

Alspach D., Enflo P., Odell E.

Studia Mathematica

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1977

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tom 60

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nr 1

79-90

Ograniczanie wyników

3 Studia Mathematica

1 Colloquium Mathematicum

1 Fundamenta Mathematicae

5 Odell E.

2 Schlumprecht T.

2 Zsák A.

1 Alspach D.

1 Elton J.

1 Enflo P.

1 Freeman D.

1 Junge M.

1 Kutzarova D.

1 2009

1 2007

1 2006

1 1981

1 1977

On asymptotically symmetric Banach spaces

Banach spaces of bounded Szlenk index

Banach spaces of bounded Szlenk index II

The unit ball of every infinite-dimensional normed linear space contains a (1+ɛ)-separated sequence

On the structure of separable $L_{p}$ spaces (1 < p < ∞)