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Holomorphic functions of uniformly bounded type on nuclear Fréchet spaces. (English) Zbl 0657.46003

The authors investigate necessary as well as sufficient conditions for nuclear Fréchet spaces E to satisfy the relation \(H(E)=H_{ub}(E)\). Here, H(E) is the space of all holomorphic functions on E while \(H_{ub}(E)\) is the space of all holomorphic functions of uniformity bounded type. As a main result they obtain that all nuclear Fréchet spaces E with the linear property (\({\tilde \Omega}\)) satisfy \(H(E)=H_{ub}(E)\), and that all nuclear Fréchet spaces E with \(H(E)=H_{ub}\) possess the linear property \((LB^{\infty}).\) The two conditions (\({\tilde \Omega}\)) and \((LB^{\infty})\) for Fréchet spaces E have been introduced in [D. Vogt, J. Reine Angew. Math. 345, 182- 200 (1983; Zbl 0514.46003)], where it has been shown that (\({\tilde \Omega}\)) is strictly stronger than \((LB^{\infty})\). It remains open in the article under review whether the condition \(H(E)=H_{ub}(E)\) for nuclear Fréchet spaces defines a new class of nuclear Frechet spaces of whether this holomorphic property coincides with (\({\tilde \Omega}\)) or \((LB^{\infty}).\)
Moreover, with an appropriate definition of \(H_{ub}(U)\) for open subsets U of E it is shown that a nuclear Fréchet space E has property (\({\tilde \Omega}\)) [cf. Vogt, loc. cit.] if \(H(P)=H_{ub}(P)\) for all polycylindrical open subsets P of E.
Reviewer: M.Schottenloher

MSC:

46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
46A04 Locally convex Fréchet spaces and (DF)-spaces
46E10 Topological linear spaces of continuous, differentiable or analytic functions
46G20 Infinite-dimensional holomorphy
46A45 Sequence spaces (including Köthe sequence spaces)

Citations:

Zbl 0514.46003
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