The algebraic equalities and their topological consequences in weighted spaces. (English) Zbl 1319.46021
The paper deals with weighted Banach, Fréchet or (LB)-spaces of holomorphic functions (in fact, the authors consider a more general class of objects for which holomorphic functions are one of the examples). The authors use the following notation: \(v\) – a weight on a locally compact, \(\sigma\)-compact Hausdorff space \(X\), \(A(X)\) – a semi-Montel subspace of \(C(X)\),
\[
A_v(X):=\{f\in A(X):\,\,\|f\|_v:=\sup_{x\in X}\frac{|f(x)|}{v(x)}<\infty\},
\]
\[ A_{v0}(X):=\{f\in A(X):\,\,\frac{f(x)}{v(x)} \text{ vanishes at infinity on } X\}. \] For a decreasing (increasing) sequence \((v_n)_n\) of weights, they denote \(AV(X):=\text{proj}_nA_{v_n}(X)\) (\(\mathcal{V}A(X):=\text{ind}_n A_{v_n}(X)\)). The spaces \(AV_0(X)\) and \(\mathcal{V}A_0(X)\) are defined similarly.
The following results are proved. The algebraic equality \(A_v(X)=A_{v0}(X)\) [\(AV(X)=AV_0(X)\), \(\mathcal{V}A(X)=\mathcal{V}A_0(X)\)] implies that the respective spaces are finite dimensional [Montel, (DFS)] – Theorem 3.3. Under an additional condition (called (CD)) they also prove the converse – Theorem 3.8.
The above results strengthen those of [K. D. Bierstedt and J. Bonet, North-Holland Math. Stud. 170, 113–133 (1992; Zbl 0804.46007)]; [K. D. Bierstedt et al., Mich. Math. J. 40, No. 2, 271–297 (1993; Zbl 0803.46023)]; [K. D. Bierstedt and W. H. Summers, J. Aust. Math. Soc., Ser. A 54, No. 1, 70–79 (1993; Zbl 0801.46021)].
\[ A_{v0}(X):=\{f\in A(X):\,\,\frac{f(x)}{v(x)} \text{ vanishes at infinity on } X\}. \] For a decreasing (increasing) sequence \((v_n)_n\) of weights, they denote \(AV(X):=\text{proj}_nA_{v_n}(X)\) (\(\mathcal{V}A(X):=\text{ind}_n A_{v_n}(X)\)). The spaces \(AV_0(X)\) and \(\mathcal{V}A_0(X)\) are defined similarly.
The following results are proved. The algebraic equality \(A_v(X)=A_{v0}(X)\) [\(AV(X)=AV_0(X)\), \(\mathcal{V}A(X)=\mathcal{V}A_0(X)\)] implies that the respective spaces are finite dimensional [Montel, (DFS)] – Theorem 3.3. Under an additional condition (called (CD)) they also prove the converse – Theorem 3.8.
The above results strengthen those of [K. D. Bierstedt and J. Bonet, North-Holland Math. Stud. 170, 113–133 (1992; Zbl 0804.46007)]; [K. D. Bierstedt et al., Mich. Math. J. 40, No. 2, 271–297 (1993; Zbl 0803.46023)]; [K. D. Bierstedt and W. H. Summers, J. Aust. Math. Soc., Ser. A 54, No. 1, 70–79 (1993; Zbl 0801.46021)].
Reviewer: Krzysztof Piszczek (Poznan)
MSC:
| 46E10 | Topological linear spaces of continuous, differentiable or analytic functions |
| 46A13 | Spaces defined by inductive or projective limits (LB, LF, etc.) |
References:
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