On Universal separable reflexive stable spaces
In this note we study the topological structure of weighted James spaces J(h). In particular we prove that J(h) is isomorphic to J if and only if the weight h is bounded. We also provide a description of J(h) if the weight is a non-decreasing sequence.
Two of James’ three quasi-reflexive spaces, as well as the James Tree, have the uniform -Opial property.
We study order convexity and concavity of quasi-Banach Lorentz spaces , where 0 < p < ∞ and w is a locally integrable positive weight function. We show first that contains an order isomorphic copy of . We then present complete criteria for lattice convexity and concavity as well as for upper and lower estimates for . We conclude with a characterization of the type and cotype of in the case when is a normable space.
We study the c₀-content of a seminormalized basic sequence (χₙ) in a Banach space, by the use of ordinal indices (taking values up to ω₁) that determine dichotomies at every ordinal stage, based on the Ramsey-type principle for every countable ordinal, obtained earlier by the author. We introduce two such indices, the c₀-index and the semibounded completeness index , and we examine their relationship. The countable ordinal values that these indices can take are always of the form . These results...
We revisit Orlicz's proof of the square summability of the norms of the terms of an unconditionally convergent series in L¹. The result is then used to motivate abstract generalizations and concrete improvements.