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The subspace problem for weighted inductive limits of spaces of holomorphic functions. (English) Zbl 0841.46014
Let \(G\) be an open subset of \(\mathbb{C}^n\) and \((v_k)^\infty_{k= 1}\) a decreasing sequence of weight functions on \(G\). One can define the Banach space \(Cv_k(G)\) (resp. \(Hv_k(G)\)) of continuous (resp. holomorphic) functions on \(G\) such that \(v_k(z)|f(z)|\) is bounded and the inductive limit \({\mathcal V}C(G)\) (resp. \({\mathcal V}H(G)\)) of these spaces. The space \(Hv_k(G)\) is a topological subspace of \(Cv_k(z) G\) but the authors give an example of \(G\) in \(\mathbb{C}^2\) and \(v_k\) for which the space \({\mathcal V}H(G)\) is not a topological subspace of \({\mathcal V}C(G)\).
This answers by the negative a question of K. D. Bierstedt, R. Meise and W. H. Summers [Trans. Am. Math. Soc. 272, 107-160 (1982; Zbl 0599.46026)].
Reviewer: P.Mazet (Paris)

46E10 Topological linear spaces of continuous, differentiable or analytic functions
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
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