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The property ( β ) of Orlicz-Bochner sequence spaces

Paweł Kolwicz (2001)

Commentationes Mathematicae Universitatis Carolinae

A characterization of property ( β ) of an arbitrary Banach space is given. Next it is proved that the Orlicz-Bochner sequence space l Φ ( X ) has the property ( β ) if and only if both spaces l Φ and X have it also. In particular the Lebesgue-Bochner sequence space l p ( X ) has the property ( β ) iff X has the property ( β ) . As a corollary we also obtain a theorem proved directly in [5] which states that in Orlicz sequence spaces equipped with the Luxemburg norm the property ( β ) , nearly uniform convexity, the drop property and...

Unconditionally p-null sequences and unconditionally p-compact operators

Ju Myung Kim (2014)

Studia Mathematica

We investigate sequences and operators via the unconditionally p-summable sequences. We characterize the unconditionally p-null sequences in terms of a certain tensor product and then prove that, for every 1 ≤ p < ∞, a subset of a Banach space is relatively unconditionally p-compact if and only if it is contained in the closed convex hull of an unconditionally p-null sequence.

Upper and lower estimates in Banach sequence spaces

Raquel Gonzalo (1995)

Commentationes Mathematicae Universitatis Carolinae

Here we study the existence of lower and upper ℓ p -estimates of sequences in some Banach sequence spaces. We also compute the sharp ℓ p estimates in their basis. Finally, we give some applications to weak sequential continuity of polynomials.

Weak orthogonality and weak property ( β ) in some Banach sequence spaces

Yunan Cui, Henryk Hudzik, Ryszard Płuciennik (1999)

Czechoslovak Mathematical Journal

It is proved that a Köthe sequence space is weakly orthogonal if and only if it is order continuous. Criteria for weak property ( β ) in Orlicz sequence spaces in the case of the Luxemburg norm as well as the Orlicz norm are given.

Weak sequential completeness of sequence spaces.

Charles Swartz (1992)

Collectanea Mathematica

Köthe and Toeplitz introduced the theory of sequence spaces and established many of the basic properties of sequence spaces by using methods of classical analysis. Later many of these same properties of sequence spaces were reestablished by using soft proofs of functional analysis. In this note we would like to point out that an improved version of a classical lemma of Schur due to Hahn can be used to give very short proofs of two of the weak sequential completeness results of Köthe and Toeplitz....

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.

λ-Properties of Orlicz sequence spaces

Shutao Chen, Huiying Sun (1994)

Annales Polonici Mathematici

It is proved that every Orlicz sequence space has the λ-property. Criteria for the uniform λ-property in Orlicz sequence spaces, with Luxemburg norm and Orlicz norm, are given.

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