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Banach spaces which admit a norm with the uniform Kadec-Klee property

100%
J. Dilworth S., Girardi M., Kutzarova D.
Studia Mathematica
|
1994-1995
|
tom 112
|
nr 3
267-277
EN
Several results are established about Banach spaces Ӿ which can be renormed to have the uniform Kadec-Klee property. It is proved that all such spaces have the complete continuity property. We show that the renorming property can be lifted from Ӿ to the Lebesgue-Bochner space $L_2(Ӿ)$ if and only if Ӿ is super-reflexive. A basis characterization of the renorming property for dual Banach spaces is given.
2
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Minimality properties of Tsirelson type spaces

100%
Kutzarova D., Leung D., Manoussakis A., Tang W.-K.
Studia Mathematica
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2008
|
tom 187
|
nr 3
233-263
EN
We study minimality properties of partly modified mixed Tsirelson spaces. A Banach space with a normalized basis $(e_{k})$ is said to be subsequentially minimal if for every normalized block basis $(x_{k})$ of $(e_{k})$, there is a further block basis $(y_{k})$ of $(x_{k})$ such that $(y_{k})$ is equivalent to a subsequence of $(e_{k})$. Sufficient conditions are given for a partly modified mixed Tsirelson space to be subsequentially minimal, and connections with Bourgain's ℓ¹-index are established. It is also shown that a large class of mixed Tsirelson spaces fails to be subsequentially minimal in a strong sense.
3
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On the existence of almost greedy bases in Banach spaces

100%
Dilworth S., Kalton N., Kutzarova D.
Studia Mathematica
|
2003
|
tom 159
|
nr 1
67-101
EN
We consider several greedy conditions for bases in Banach spaces that arise naturally in the study of the Thresholding Greedy Algorithm (TGA). In particular, we continue the study of almost greedy bases begun in [3]. We show that almost greedy bases are essentially optimal for n-term approximation when the TGA is modified to include a Chebyshev approximation. We prove that if a Banach space X has a basis and contains a complemented subspace with a symmetric basis and finite cotype then X has an almost greedy basis. We show that c₀ is the only $ℒ_{∞}$ space to have a quasi-greedy basis. The Banach spaces which contain almost greedy basic sequences are characterized.
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