Abstract
We discuss some properties of the Banach‐valued sequence space ℓp[X](1 ≤ p < ∞), the space of weakly p‐summable sequences on a Banach space X. For example, we characterize the reflexivity of ℓp[X], convergent sequences on ℓp[X], and compact subsets of ℓp[X].
Highlights
It is known that the general theory of scalar-valued sequence spaces (SVSS) plays an important role in the theory of topological vector spaces
The theory of generalized sequence spaces, or vector-valued sequence spaces (VVSS) which has emerged as an outgrowth of the development of SVSS plays an important role in the theory of locally convex spaces, especially in the investigation of nuclear spaces through λ-summing operators
Wu and Bu [26] have shown a representation of the Köthe dual of the space p[X] and a characterization that p[X] is a Grothendieck space
Summary
It is known that the general theory of scalar-valued sequence spaces (SVSS) plays an important role in the theory of topological vector spaces (cf. [11, 21]). By using this result, we characterize the reflexivity of the space p[X] in terms of the reflexivity and (q)-property of a Banach space X. The space p[X] (1 < p < ∞) is a GAK-space if and only if X has the (q)-property
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