Direct sums and abstract Kadets--Klee properties

TL;DR

The paper develops a general framework to study abstract Kadets--Klee properties $H(\mathfrak{T})$ for direct sums $(\bigoplus_{\gamma\in\Gamma} X_\gamma)_{\mathcal{E}}$, replacing weak convergence with convergence in a linear Hausdorff topology $\mathfrak{T}$. It introduces the notion of $\oplus$-compatible topologies and proves two pivotal results: an Inheritance Theorem ensuring $H(\mathfrak{T})$ is hereditary under a Schur/SM dichotomy condition, and a Lifting Theorem proving stability under a dual set of assumptions, with a compatibility result linking these conditions. The framework applies to Köthe--Bochner spaces and yields both known and refined results for classical Kadets--Klee properties with respect to weak and local convergence in measure topologies, while also pointing toward extensions to direct integrals and uniform Kadets--Klee properties. The work thus provides a unifying approach to transfer Kadets--Klee-type behavior from components to direct sums and offers a blueprint for analyzing Kadets--Klee properties in broader topological contexts. The findings have implications for the geometric theory of Banach spaces and the analysis of Köthe–Bochner constructions, with potential applications in related topologies and spaces.

Abstract

Let $\mathcal{X} = \{ X_γ \}_{γ\in Γ}$ be a family of Banach spaces and let $\mathcal{E}$ be a Banach sequence space defined on $Γ$. The main aim of this work is to investigate the abstract Kadets--Klee properties, that is, the Kadets--Klee type properties in which the weak convergence of sequences is replaced by the convergence with respect to some linear Hausdorff topology, for the direct sum construction $(\bigoplus_{γ\in Γ} X_γ)_{\mathcal{E}}$. As we will show, and this seems to be quite atypical behavior when compared to some other geometric properties, to lift the Kadets--Klee properties from the components to whole direct sum it is not enough to assume that all involved spaces have the appropriate Kadets--Klee property. Actually, to complete the picture one must add a dichotomy in the form of the Schur type properties for $X_γ$'s supplemented by the variant of strict monotonicity for $\mathcal{E}$. Back down to earth, this general machinery naturally provides a blue print for other topologies like, for example, the weak topology or the topology of local convergence in measure, that are perhaps more commonly associated with this type of considerations. Furthermore, by limiting ourselves to direct sums in which the family $\mathcal{X}$ is constant, that is, $X_γ = X$ for all $γ\in Γ$ and some Banach space $X$, we return to the well-explored ground of K{ö}the--Bochner sequence spaces $\mathcal{E}(X)$. Doing all this, we will reproduce, but sometimes also improve, essentially all existing results about the classical Kadets--Klee properties in K{ö}the--Bochner sequence spaces.

Figures (1)

• Figure 1: Three natural topologies on a Banach sequence space

Theorems & Definitions (94)

• Theorem A: Inheritance Problem; see Theorem
• Theorem B: Lifting Problem; see Theorem
• Theorem C: Compatibility result; see Theorem
• Definition 2.c.1
• Lemma 2.c.2
• proof
• Definition 2.d.1: Direct sum
• Remark 2.d.2
• proof
• Remark 2.d.3