Let X be a Banach space and ν a countably additive X-valued measure defined on a σ-algebra. We discuss some generation properties of the Banach space L¹(ν) and its connection with uniform Eberlein compacta. In this way, we provide a new proof that L¹(ν) is weakly compactly generated and embeds isomorphically into a Hilbert generated Banach space. The Davis-Figiel-Johnson-Pełczyński factorization of the integration operator $I_{\nu }:L¹\left(\nu \right)\to X$ is also analyzed. As a result, we prove that if $I_{\nu }$ is both completely continuous...

Let K be a compact Hausdorff space, $C_p\left(K\right)$ the space of continuous functions on K endowed with the pointwise convergence topology, D ⊂ K a dense subset and $t_p\left(D\right)$ the topology in C(K) of pointwise convergence on D. It is proved that when $C_p\left(K\right)$ is Lindelöf the $t_p\left(D\right)$-compact subsets of C(K) are fragmented by the supremum norm of C(K). As a consequence we obtain some Namioka type results and apply them to prove that if K is separable and $C_p\left(K\right)$ is Lindelöf, then K is metrizable if, and only if, there is a countable and dense...