Susanne Dierolf

On Distinguished Fréchet Spaces

Abstract We report on recent results about distinguished Frechet spaces concerning the behaviour of biduals, stability properties and characterizations in the context of Kothe echelon spaces, of Frechet spaces of Moscatelli type and projective tensor products. Some new results on an extension of a classical example of Amemiya are also included (see section 5).

Bornological spaces of null sequences

It is proved that c0(E) is an LB-space, whenever E is either a Montel LB-space or a coechelon space k∞(v) of type ∞ (a partial solution of a problem of Schmets and Bierstedt). In the second case we show that c0(k∞(v)) is even the Mackey completion of c0 ⊗e k∞(v). Introduction. J. Schmets has successfully studied various linear-topological properties of spaces of vector-valued continuous functions [Sch2]. Unfortunately, the problem of bornologicity of such spaces has been left open (comp. [Sch1]). Up to now a solution has been obtained only in very special cases (see [Bo1], [Mu], [Sch2, I.7.3 and IV.4.7], [DG], [BoS1] and [BoS2]), in particular, it is still unclear when c0(E) is bornological. Bierstedt and Schmets asked if c0(E), E a DFM-space, is bornological (or, equivalently, an LB-space). We solve this problem completely. Then we show that the problem of bornologicity of c0(E) for general LB-spaces E (also asked by Schmets and Bierstedt [Sch1], see also [Sch2, Ch. IV]) reduces in some sense to the same problem for Montel E. By this reduction we solve the problem for arbitrary coechelon spaces E = k∞(v). We use our “factorization” approach developed in [DiDo1] and [DiDo2]. The problem of boronologicity of C(K,E), K compact, at least in case when E = indn∈N En is an LBor LF-space, is closely related to the question of interchanging of e-products and inductive limits (comp. [BoS2]). Indeed, by a result of Mujica ([Mu], [Sch2, I.7.2]), it is known that C(K,E) = C(K)e indn∈N En contains a bornological space indn∈N(C(K)eEn) as a topological subspace. Thus, if (∗) C(K)e ind n∈N En = ind n∈N (C(K)eEn) holds algebraically, then it holds topologically and C(K,E) is a fortiori bornological (see the result of Mujica [Sch2, I.7.3] or Marquina and Schmets [MS] and a more general result of Defant and Govaerts [Sch2, IV.4.7], [DG, Th. 13, Cor. 14]). If the space E is compactly regular (i.e., each compact subset of E is compact in some step En), then (∗) holds. Roughly speaking, for LB-spaces the case of 1991 Mathematics Subject Classification. Primary 46A11, 46A13, 46E40.

A note on biduals of strict (LF)-spaces

Grothendieck asked in 1954 in [1] the following questions. (1) Is the bidual of a strict inductive limit of a sequence of locally convex spaces the inductive limit of the biduals? (2) Is the bidual of a strict (LF)-space again an (LF)-space? (3) Is the bidual of a strict (LF)-space complete? M. Valdivia gave a (negative) answer to the first question in 1979 in [5]. Since his counterexample is not an (LF)-space, problem (2) remained open. The aim of this note is to present a negative solution to questions (2) and (3). The answer to question (2) is negative even if every step of the (LF)-space is distinguished, in which case the strong bidual is complete by a result of Grothendieck. Moreover, we show that the strong dual of a strict (LF)-space need not be countably barrelled.

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.

On S-Barrelled Spaces

Locally convex spaces E for which every linear map of E into an arbitrary Frechet space with sequentially closed graph is continuous are studied.

Examples on projective spectra of (LB)-spaces

SummaryIn this note we present examples of projective spectraɛ=(En)n∈ℕ of (LB)-spaces satisfying proj1 ε≠0 such that the inductive spectrum (En′)n∈ℕ of the duals is strict. Moreover, we characterize proj1ε=0 for projective spectra of Moscatelli type.

On Completions of LB-spaces of Moscatelli Type (Dedicated to the Memory of Walter Roelcke, doctoral promotor of the first author)

We first present a class of LF-spaces, extending the class of LF-spaces of Moscatelli type, for which regularity implies completeness. Then we utilize the obtained results to describe the completions of LB-spaces of Moscatelli type. In particular, we prove that the completions of LB-spaces of that type are again LB-spaces.ResumenPresentamos primero una clase de espacios LF, que extiende los espacios LF de tipo Moscatelli, para la cual la regularidad del límite inductivo implica la completitud. A continuación utilizamos los resultados obtenidos para describir la completación de los espacios LB de tipo Moscatelli usual. En particular, demostramos que la completación de un espacio LB de ese tipo es también un espacio LB.

Strong duals of projective limits of (LB)-spaces

We investigate the problem when the strong dual of a projective limit of (LB)-spaces coincides with the inductive limit of the strong duals. It is well-known that the answer is affirmative for spectra of Banach spaces if the projective limit is a quasinormable Fréchet space. In that case, the spectrum satisfies a certain condition which is called “strong P-type”. We provide an example which shows that strong P-type in general does not imply that the strong dual of the projective limit is the inductive limit of the strong duals, but on the other hand we show that this is indeed true if one deals with projective spectra of retractive (LB)-spaces. Finally, we apply our results to a question of Grothendieck about biduals of (LF)-spaces.

Inductive limits of vector-valued sequence spaces

Let L be a normal Banach sequence space such that every element in L is the limit of its sections and let E = ind En be a separated inductive limit of the locally convex spaces. Then ind L(En) is a topological subspace of L(E).

A Note on Strict LF-Spaces

This note is an abstract of the talk delivered by the second author at this Workshop on Frechet Spaces. Publication of a more detailed version containing all the proofs is intended.