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Property (BB) and holomorphic functions on Fréchet-Montel spaces

Published online by Cambridge University Press:  24 October 2008

Andreas Defant  and
Manuel Maestre
Andreas Defant
Affiliation:
Fachbereich Mathematik, Carl von Ossieztky Universität, Ammerländer Heerstraβe 114–118, D-2900 Oldenburg, Germany
Manuel Maestre
Affiliation:
Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100 Burjasot (Valencia), Spain
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Two of the most important topologies on the space ℋ(E) of all holomorphic functions f:E → ℂ on a complex locally convex space E are the compact-open topology τ0 and the Nachbin-ported topology τε. We recall that a seminorm p on ℋ(E) is said to be τω-continuous if there is a compact K such that for every open set V with K contained in V there is a constant c > 0 satisfying

Clearly, the following natural question arises: when do the topologies τ0 and τω coincide? In the setting of Fréchet spaces equality of τ0 and τω forces E to be a Montel space; Mujica [21] proved τ0 = τω for Fréchet-Schwartz spaces and Ansemil-Ponte [1] showed that for Fréchet-Montel spaces this happens if and only if the space of all continuous n-homogeneous polynomials with the compact-open topology, (Pn(E), τ0), is barrelled. By duality, it turns out that the question τ0 = τω is intimately related to ‘Grothendieck's problème des topologies’ which asks whether or not for two Fréchet spaces E1 and E2 each bounded set B of the (completed) projective tensor product is contained in the closed absolutely convex hull of the set B1 ⊗ B2, where Bk is a bounded subset of Ek for k = 1, 2. If this is the case, then the pair (E1, E2) is said to have property (BB). Observe that every compact set B in can always be lifted by compact subsets Bk of Ek (see e.g. [20], 15·6·3). Hence, for Fréchet-Montel spaces E1 and E2, property (BB) of (El,E2) means that is Fréchet-Montel and vice versa. Taskinen[24] found the first counterexample to Grothendieck's problem. In [25] he constructed a Fréchet-Montel space E0 for which (E0,E0) does not have property (BB), and Ansemil-Taskinen [2] showed that τ0 ≠ τω on ℋ(E0).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 115 , Issue 2 , March 1994 , pp. 305 - 313
Copyright
Copyright © Cambridge Philosophical Society 1994

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