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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)

##### MSC:
 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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