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Vector series whose lacunary subseries converge

Lech Drewnowski, Iwo Labuda (2000)

Studia Mathematica

The area of research of this paper goes back to a 1930 result of H. Auerbach showing that a scalar series is (absolutely) convergent if all its zero-density subseries converge. A series ∑ n x n in a topological vector space X is called ℒ-convergent if each of its lacunary subseries ∑ k x n k (i.e. those with n k + 1 - n k → ∞ ) converges. The space X is said to have the Lacunary Convergence Property, or LCP, if every ℒ-convergent series in X is convergent; in fact, it is then subseries convergent. The Zero-Density Convergence...

Weak compactness and Orlicz spaces

Pascal Lefèvre, Daniel Li, Hervé Queffélec, Luis Rodríguez-Piazza (2008)

Colloquium Mathematicae

We give new proofs that some Banach spaces have Pełczyński's property (V).

Weak compactness and σ-Asplund generated Banach spaces

M. Fabian, V. Montesinos, V. Zizler (2007)

Studia Mathematica

σ-Asplund generated Banach spaces are used to give new characterizations of subspaces of weakly compactly generated spaces and to prove some results on Radon-Nikodým compacta. We show, typically, that in the framework of weakly Lindelöf determined Banach spaces, subspaces of weakly compactly generated spaces are the same as σ-Asplund generated spaces. For this purpose, we study relationships between quantitative versions of Asplund property, dentability, differentiability, and of weak compactness...

Weakly Compact Generating and Shrinking Markusevic Bases

Fabian, M., Hájek, P., Montesinos, V., Zizler, V. (2006)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 46B30, 46B03.It is shown that most of the well known classes of nonseparable Banach spaces related to the weakly compact generating can be characterized by elementary properties of the closure of the coefficient space of Markusevic bases for such spaces. In some cases, such property is then shared by all Markusevic bases in the space.

Weakly null sequences with upper estimates

Daniel Freeman (2008)

Studia Mathematica

We prove that if ( v i ) is a seminormalized 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 ≥ 1 such that every normalized weakly null sequence in X has a subsequence that is C-dominated by ( v i ) . This extends a result of Knaust and Odell, who proved this for the cases in which ( v i ) is the standard basis for ℓ p or c₀.

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