Counterexamples concerning topologization of spaces of strongly almost convergent sequences

Abstract

Let λ be a sequence space, f a modulus function, and F=(f_k) a sequence of moduli. We characterize the F-normability of the sequence space λ(F) for λ⊂l_∞. In the special case if λ is the space sac₀ of strongly almost convergent to zero sequences, we give two counterexamples concerning the topologization of various extensions of sac₀(F) and sac₀(f) considered by Nanda and others. We also correct a similar inaccuracy in a previous paper of the author.