We examine the so-called three-space-stability for some classes of linear topological and locally convex spaces for which this problem has not been investigated.

We discuss various results on the existence of ‘true’ preimages under continuous open maps between $F$-spaces, $F$-lattices and some other spaces. The aim of the paper is to provide accessible proofs of this sort of results for functional-analysts.

We construct the example of the title.

We prove that if 0 < p < 1 then a normalized unconditional basis of a complemented subspace of $c_0\left(l_p\right)$ must be equivalent to a permutation of a subset of the canonical unit vector basis of $c_0\left(l_p\right)$. In particular, $c_0\left(l_p\right)$ has unique unconditional basis up to permutation. Bourgain, Casazza, Lindenstrauss, and Tzafriri have previously proved the same result for $c_0\left(l₁\right)$.

We prove that the quasi-Banach spaces ${\ell }_p\left(c₀\right)$ and ${\ell }_p\left(\ell ₂\right)$ (0 < p < 1) have a unique unconditional basis up to permutation. Bourgain, Casazza, Lindenstrauss and Tzafriri have previously proved that the same is true for the respective Banach envelopes $\ell ₁\left(c₀\right)$ and ℓ₁(ℓ₂). They used duality techniques which are not available in the non-locally convex case.

The aim of the present paper is to study the class of tvs which we define by ommiting the word increasing in the definition of *-suprabarrelled spaces. We prove that the product of Baire tvs is *-UBL and hence the class of *-UBL spaces is stricty larger than the class of Baire spaces.

The area of research of this paper goes back to a 1930 result of H. Auerbach showing that a scalar series is (absolutely) convergent if all its zero-density subseries converge. A series ${\sum }_n{x}_n$ in a topological vector space X is called ℒ-convergent if each of its lacunary subseries ${\sum }_k{x}_{n_k}$ (i.e. those with $n_{k+1}-n_k\to \infty$) converges. The space X is said to have the Lacunary Convergence Property, or LCP, if every ℒ-convergent series in X is convergent; in fact, it is then subseries convergent. The Zero-Density Convergence...

We generalize the theory of Wiener amalgam spaces on locally compact groups to quasi-Banach spaces. As a main result we provide convolution relations for such spaces. Also we weaken the technical assumption that the global component is invariant under right translations, which is new even for the classical Banach space case. To illustrate our theory we discuss in detail an example on the ax+b group.

In this paper, we generalize the concept of $\varphi$-multipliers on Banach algebras to a class of topological algebras. Then the characterizations of $\varphi$-multipliers are investigated in these algebras.