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Weighted $$(LB)$$-spaces of holomorphic functions: $$\nu H(G)=\nu_{0}H(G)$$ and completeness of $$\nu_{0}H(G)$$. (English) Zbl 1119.46028
In their interesting paper, the authors show that algebraic equalities between weighted spaces of holomorphic functions may have strong topological implications. Let $$\mathcal{V}:= \{v_n\mid n\in \mathbb{N}\}$$ be an inductive system of weights on an open set $$G\subset \mathbb{C}^N$$. Then $$\mathcal{V}$$ defines inductive limits $$\mathcal{V} H(G)$$ ($$\mathcal{V}_0H(G)$$) of holomorphic functions by means of $$O$$-growth conditions ($$o$$-growth conditions, respectively). Let $$H \overline{V}(G)$$ ($$H \overline{V}_0(G)$$) be the projective hulls of $$\mathcal{V} H(G)$$ ($$\mathcal{V}_0H(G)$$, respectively). Assume that, for each $$n\in \mathbb{N}$$, every discrete sequence in $$G$$ contains a subsequence which is interpolating in $$Hv_n(G)$$. Then the algebraic equality $$\mathcal{V} H(G)= \mathcal{V}_0H(G)$$ implies that $$\mathcal{V} H(G)$$ and $$\mathcal{V}_0H(G)$$ are $$(DFS)$$-spaces. Moreover, the algebraic equality $$H \overline{V}(G)=H \overline{V}_0(G)$$ implies that $$H \overline{V}(G)$$ is semi-Montel.
The interpolating assumption is discussed in detail, e.g., it is always satisfied if $$G\subset \mathbb{C}$$ is connected and no connected component of the complement of $$G$$ in the Riemann sphere consists of one point or if $$G\subset \mathbb{C}^N$$ is absolutely convex and bounded. If the biduality conditions are satisfied, i.e., if for each $$n\in \mathbb{N}$$ the closed unit ball $$B_n$$ of $$Hv_n(G)$$ is contained in the compact open closure of the unit ball $$C_n$$ of $$H(v_n)_0(G)$$, then the algebraic equality $$H \overline{V}(G)=H \overline{V}_0(G)$$ is equivalent to $$H \overline{V}_0(G)$$ being semireflexive. Finally, let $$\mathbb{D}\subset \mathbb{C}$$ be the unit disc and assume that $$\mathcal{V}_0 H(\mathbb{D})= H \overline{V}_0(\mathbb{D})$$ algebraically and topologically. Then the inductive limit $$\mathcal{V}_0 H(\mathbb{D})$$ is boundedly retractive.

##### MSC:
 46E10 Topological linear spaces of continuous, differentiable or analytic functions 30H05 Spaces of bounded analytic functions of one complex variable 46A13 Spaces defined by inductive or projective limits (LB, LF, etc.) 46M40 Inductive and projective limits in functional analysis
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