We analyse several examples of separable Banach spaces, some of them new, and relate them to several dichotomies obtained in [11],by classifying them according to which side of the dichotomies they fall.
It is proved that for any Banach space X property (β) defined by Rolewicz in [22] implies that both X and X* have the Banach-Saks property. Moreover, in Musielak-Orlicz sequence spaces, criteria for the Banach-Saks property, the near uniform convexity, the uniform Kadec-Klee property and property (H) are given.
We provide a partial answer to the question of Vladimir Kadets whether given an ℱ-basis of a Banach space X, with respect to some filter ℱ ⊂ 𝒫(ℕ), the coordinate functionals are continuous. The answer is positive if the character of ℱ is less than 𝔭. In this case every ℱ-basis is an M-basis with brackets which are determined by an element of ℱ.