Direct sums and abstract Kadets--Klee properties

Let $\mathcal{X} = \{ X_{\gamma} \}_{\gamma \in \Gamma}$ be a family of Banach spaces and let $\mathcal{E}$ be a Banach sequence space defined on $\Gamma$. 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 $(\bigoplus_{\gamma \in \Gamma} X_{\gamma})_{\mathcal{E}}$. 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 $X_{\gamma}$'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, $X_{\gamma} = X$ for all $\gamma \in \Gamma$ and some Banach space $X$, we return to the well-explored ground of K{\"o}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{\"o}the--Bochner sequence spaces.