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Displaying 141 –
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A characterization of property of an arbitrary Banach space is given. Next it is proved that the Orlicz-Bochner sequence space has the property if and only if both spaces and have it also. In particular the Lebesgue-Bochner sequence space has the property iff has the property . 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 , nearly uniform convexity, the drop property and...
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.
Here we study the existence of lower and upper -estimates of sequences in some Banach sequence spaces. We also compute the sharp estimates in their basis. Finally, we give some applications to weak sequential continuity of polynomials.
It is proved that a Köthe sequence space is weakly orthogonal if and only if it is order continuous. Criteria for weak property () in Orlicz sequence spaces in the case of the Luxemburg norm as well as the Orlicz norm are given.
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....
We prove that if 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 , 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 . This extends a result of Knaust and Odell, who proved this for the cases in which is the standard basis for or c₀.
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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