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...

We study in terms of corresponding Köthe matrices when every continuous linear operator between two Köthe spaces is bounded, the consequences of the existence of unbounded continuous linear operators, and related topics.

Characterizations of pairs (E,F) of complete (LF)?spaces such that every continuous linear map from E to F maps a 0?neighbourhood of E into a bounded subset of F are given. The case of sequence (LF)?spaces is also considered. These results are similar to the ones due to D. Vogt in the case E and F are Fréchet spaces. The research continues work of J. Bonet, A. Galbis, S. Önal, T. Terzioglu and D. Vogt.

In this paper we generalize some results concerning bounded variation functions on sequence spaces.