Juan Carlos Díaz

The density condition in subspaces and quotients of Frechet spaces

The aim of this note is to investigate the topological structure (in particular the density condition) of subspaces and separated quotients of Fréchet spaces. Our main result is the following one: LetE be a Fréchet space which is neither Montel nor isomorphic to a closed subspace ofX × ε, withX a Banach space, also assume thatE can be written asF⊕G withF andG infinite dimensional closed subspaces ofE not isomorphic to ε, thenE contains a closed subspace with basis and not satisfying the density condition. We also prove that every Köthe echelon space of orderp, 1<p<∞, which is not quasinormable has a separated quotient with basis which does not satisfy the density condition.

On duals of weakly acyclic ()-spaces

For countable inductive limits of Frechet spaces ((LF)-spaces) the property of being weakly acyclic in the sense of Palamodov (or. equivalently. having condition (Mo0) in the terminology of Retakh) is useful to avoid some important pathologies and in relation to the problem of well-located subspaces. In this note we consider if weak acyclicity is enough for a (LF)-space E := ind E, to ensure that its strong dual is canonically homeomorphic to the projective limit of the strong duals of the spaces E,. First we give an elementary proof of a known result by Vogt and obtain that the answer to this question is positive if the steps E, are distinguished or weakly sequentially complete. Then we construct a weakly acyclic (LF)-space for which the answer is negative.

Two problems of Valdivia on distinguished Frechet spaces

In this paper we construct the following spaces: (a) a Fréchet spaceE with basis which is non-distinguished and has no subspace isomorphic to ℓ1 (b) a non-distinguished Fréchet spaceF such that every separable subspace ofF is distinguished (even more, every separable subspace ofF has a separable dual). These examples answer in the negative two questions posed by Valdivia.

Existence of Unconditional Bases in Spaces of Polynomials and Holomorphic Functions

Our main result shows that every Montel Kothe echelon or coechelon space E of order 1 < p ≤ ∞ is nuclear if and only if for every (some) m ≥ 2 the space ((mE), τ0) of m-homegeneus polynomials on E endowed with the compact-open topology τ0 has an unconditional basis if and only if the space (ℋ(E), τδ) of holomorphic functions on E endowed with the bornological topology τδ associated to τ0 has an unconditional basis (for coechelon spaces τδ equals τ0). The main idea is to extend the concept of the Gordon-Lewis property from Banach to Frechet and (DF) spaces. In this way we obtain techniques which are used to characterize the existence of unconditional basis in spaces of m-th (symmetric) tensor products and, as a consequence, in spaces of polynomials and holomorphic functions.

On Non-Distinguished Matrix Domains

In this note matrices A and B are presented such that the matrix domains (ℓ∞)A and (ℓ1)B are not distinguished. We also give an example of a matrix domain cA whose bidual is not distinguished.