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.

Let ${\left(E_i\right)}_{i\in I}$ be a family of normed spaces and $\lambda$ a space of scalar generalized sequences. The $\lambda$-sum of the family ${\left(E_i\right)}_{i\in I}$ of spaces is $$\lambda \left\{{\left(E_i\right)}_{i\in I}\right\}:=\{{\left(x_i\right)}_{i\in I},x_i\in E_i,\text{and}(\parallel x_i{\parallel )}_{i\in I}\in \lambda \}.$$ Starting from the topology on $\lambda$ and the norm topology on each $E_i,$ a natural topology on $\lambda \left\{{\left(E_i\right)}_{i\in I}\right\}$ can be defined. We give conditions for $\lambda \left\{{\left(E_i\right)}_{i\in I}\right\}$ to be quasi-barrelled, barrelled or locally complete.

Let F be a complemented subspace of a nuclear Fréchet space E. If E and F both have (absolute) bases $\left(e_n\right)$ resp. $\left(f_n\right)$, then Bessaga conjectured (see [2] and for a more general form, also [8]) that there exists an isomorphism of F into E mapping $f_n$ to $t_n{e}_{\pi \left(k_n\right)}$ where $\left(t_n\right)$ is a scalar sequence, π is a permutation of ℕ and $\left(k_n\right)$ is a subsequence of ℕ. We prove that the conjecture holds if E is unstable, i.e. for some base of decreasing zero-neighborhoods $\left(U_n\right)$ consisting of absolutely convex sets one has ∃s ∀p ∃q ∀r $li{m}_n\left(d_{n+1}(U_q,U_p)\right)/\left(d_n(U_r,U_s)\right)=0$ where...