Characterization of nuclear Fréchet spaces in which every bounded set is polar. (English) Zbl 0556.46003
A not identically -\(\infty\) upper semicontinuous function \(v:U\subset E\to [-\infty,+\infty)\), U an open subset of the locally convex space E, is said to be plurisubharmonic if \(v(a)\leq \frac{1}{2\pi}\int^{2\pi}_{0}v(a+be^{i\theta})d\theta\) for all \(a\in U\) and all \(b\in E\) such that \(\{a+\lambda b\); \(| \lambda | \leq 1\}\subset U\). A subset A of U is polar in U if there exists a plurisubharmonic function v on U such that \(A\subset \{x\in U;\quad v(x)=-\infty \}.\) Polar sets are one of the natural exceptional sets of complex analysis.
In his investigation of control sets which provided bounds on the growth of holomorphic and plurisubharmonic functions in locally convex spaces P. Lelong [North Holland Math. Studies, 71, 255-272 (1982; Zbl 0511.46046)] asked for a characterization of the Fréchet spaces which contained compact non-polar sets. Lelong himself gave a number of examples. In the present paper the authors give a linear characterization of the Fréchet nuclear spaces with the approximation property which contain compact non-polar subsets.
In his investigation of control sets which provided bounds on the growth of holomorphic and plurisubharmonic functions in locally convex spaces P. Lelong [North Holland Math. Studies, 71, 255-272 (1982; Zbl 0511.46046)] asked for a characterization of the Fréchet spaces which contained compact non-polar sets. Lelong himself gave a number of examples. In the present paper the authors give a linear characterization of the Fréchet nuclear spaces with the approximation property which contain compact non-polar subsets.
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 |
| 46G20 | Infinite-dimensional holomorphy |
| 31C10 | Pluriharmonic and plurisubharmonic functions |
Keywords:
upper semicontinuous function; plurisubharmonic; Polar sets; control sets; Fréchet spaces; Fréchet nuclear spaces with the approximation propertyCitations:
Zbl 0511.46046References:
| [1] | BIERSTEDT (K.-D.) , MEISE (R. G.) and SUMMERS (W. H.) . - Köthe sets and Köthe sequence spaces , Functional Analysis, Holomorphy and Approximation Theory, J. A. BARROSO (Ed.), North Holland Mathematics Studies, Vol. 71, 1982 , pp. 27-91. MR 84f:46011 | Zbl 0504.46007 · Zbl 0504.46007 |
| [2] | BOLAND (P. J.) and DINEEN (S.) . - Holomorphic functions on fully nuclear spaces , Bull. Soc. Math. Fr., Vol. 106, 1978 , pp. 311-335. Numdam | MR 81b:46060 | Zbl 0402.46017 · Zbl 0402.46017 |
| [3] | BÖRGENS (M.) , MEISE (R.) and VOGT (D.) . - Entire functions on nuclear sequence spaces , J. reine angew. Math., Vol. 322, 1981 , pp. 196-220. Article | MR 83i:46049 | Zbl 0441.46006 · Zbl 0441.46006 · doi:10.1515/crll.1981.322.196 |
| [4] | COEURÉ (G.) . - O-completion of normed linear spaces , in Proc. Coll. Anal. 1972 , Herrmann, Paris, 1975 , pp. 91-93. MR 53 #6273 | Zbl 0335.46030 · Zbl 0335.46030 |
| [5] | DINEEN (S.) . - Complex analysis in locally convex spaces , North Holland Mathematics Studies, Vol. 57, 1981 . MR 84b:46050 | Zbl 0484.46044 · Zbl 0484.46044 |
| [6] | DINEEN (S.) , MEISE (R.) and VOGT (D.) . - Caractérisation des espaces de Fréchet nucléaires dans lesquels tous les bornés sont pluripolaires , C.R. Acad. Sc. Paris, Vol. 295, 1982 , pp. 385-388. MR 84b:46004 | Zbl 0509.46001 · Zbl 0509.46001 |
| [7] | DINEEN (S.) , MEISE (R.) and VOGT (D.) . - Polar subsets of locally convex spaces , to appear in Aspects of mathematics and its applications, J. A. BARROSO (Ed.), North Holland. Zbl 0602.46049 · Zbl 0602.46049 |
| [8] | DINEEN (S.) and NOVERRAZ (P.) . - Gaussian measures and polar sets in locally convex spaces , Ark. Mat., Vol. 17, 1979 , pp. 217-223. MR 83b:32014 | Zbl 0432.28014 · Zbl 0432.28014 · doi:10.1007/BF02385469 |
| [9] | DUBINSKY (E.) . - Nuclear Fréchet spaces without the bounded approximation property , Studia Math., Vol. 71, 1981 , pp. 85-105. Article | MR 83e:46007 | Zbl 0482.46003 · Zbl 0482.46003 |
| [10] | DUBINSKY (E.) and VOGT (D.) . - Fréchet spaces with quotients failing the bounded approximation property , preprint. · Zbl 0633.46003 |
| [11] | GROTHENDIECK (A.) . - Sur les espaces (F) et (DF) , Summa Brasil Math., Vol. 3, 1954 , pp. 57-122. MR 17,765b | Zbl 0058.09803 · Zbl 0058.09803 |
| [12] | LELONG (P.) . - Ensembles de contrôle de croissance pour l’analyse complexe dans les espaces de Fréchet , C.R. Acad. Sc., Paris, Vol. 287, 1978 , pp. 1097-1100. MR 80b:32019 | Zbl 0403.46003 · Zbl 0403.46003 |
| [13] | LELONG (P.) . - A class of Fréchet complex spaces in which the bounded sets are C-polar , Functional Analysis, Holomorphy and Approximation Theory, J. A. BARROSO (Ed.), North Holland Mathematics Studies, Vol. 71, 1982 , pp. 255-272. MR 85d:46004 | Zbl 0511.46046 · Zbl 0511.46046 |
| [14] | KISELMAN (C. O.) . - Croissance des fonctions plurisousharmoniques en dimension infinie , to appear in Ann. Inst. Fourier, Vol. 34, 1984 . Numdam | MR 85i:32028 | Zbl 0523.32012 · Zbl 0523.32012 · doi:10.5802/aif.955 |
| [15] | MEISE (R.) and VOGT (D.) . - Structure of spaces of holomorphic functions on infinite dimensional polydiscs , Studia Math., Vol. 75, 1983 , pp. 235-252. Article | MR 85b:46052 | Zbl 0527.46019 · Zbl 0527.46019 |
| [16] | MEISE (R.) and VOGT (D.) . - Holomorphic functions of uniformly bounded type on nuclear Fréchet spaces , preprint. · Zbl 0657.46003 |
| [17] | NOVERRAZ (P.) . - Pseudoconvexité, convexité polynomial et domaines d’holomorphie en dimension infinie , North Holland Mathematics Studies, Vol. 3, 1973 . MR 50 #10814 | Zbl 0251.46049 · Zbl 0251.46049 |
| [18] | NOVERRAZ (P.) . - Pseudoconvex completions of locally convex topological vector spaces , Math. Ann., Vol. 208, 1974 , pp. 59-69. MR 49 #5842 | Zbl 0264.31009 · Zbl 0264.31009 · doi:10.1007/BF01432520 |
| [19] | PIETSCH (A.) . - Nuclear locally convex spaces , Ergebnisse der Math., Vol. 66, Springer, 1972 . MR 50 #2853 | Zbl 0236.46001 · Zbl 0236.46001 |
| [20] | SCHAEFER (H. H.) . - Topological vector spaces , Springer, 1971 . MR 49 #7722 | Zbl 0217.16002 · Zbl 0217.16002 |
| [21] | SCHOTTENLOHER (M.) . - The Levi problem for domains spread over locally convex spaces with a finite dimensional Schauder decomposition , Ann. Inst. Fourier, Vol. 26, 1976 , pp. 207-237. Numdam | MR 58 #1262 | Zbl 0309.32013 · Zbl 0309.32013 · doi:10.5802/aif.638 |
| [22] | SCHOTTENLOHER (M.) . - Polynomial approximation on compact sets , Infinite Dimensional Holomorphy and Applications, M. C. MATOS (Ed.), North Holland Mathematics Studies, Vol. 12, 1977 , pp. 379-395. MR 57 #3453 | Zbl 0397.46044 · Zbl 0397.46044 |
| [23] | VOGT (D.) . - Eine Charakterisierung der Potenzreihenräume von endlichem Typ und ihre Folgerungen , Manuscripta math., Vol. 37, 1982 , pp. 269-301. Article | MR 83i:46017 | Zbl 0512.46003 · Zbl 0512.46003 · doi:10.1007/BF01166224 |
| [24] | VOGT (D.) . - Sequence space representations of spaces of test functions and distributions , Functional Analysis, Holomorphy and Approximation Theory, G. ZAPATA (Ed.), Marcel Dekker, Lect. Notes in Pure and Appl. Math., Vol. 83, 1983 , pp. 405-443. MR 84f:46048 | Zbl 0519.46044 · Zbl 0519.46044 |
| [25] | VOGT (D.) . - Frécheträume, zwischen denen jede stetige lineare Abbildung beschränkt ist , J. reine angew. Math., Vol. 345, 1983 , pp. 182-200. MR 85h:46007 | Zbl 0514.46003 · Zbl 0514.46003 · doi:10.1515/crll.1983.345.182 |
| [26] | VOGT (D.) . - An example of a nuclear Fréchet space without the bounded approximation property , Math. Z., Vol. 182, 1983 , pp. 265-267. Article | MR 84m:46009 | Zbl 0488.46002 · Zbl 0488.46002 · doi:10.1007/BF01175629 |
| [27] | VOGT (D.) und WAGNER (M. J.) . - Charakterisierung der Quotientenräume von s und eine Vermutung von Martineau , Studia Math., Vol. 67, 1980 , pp. 225-240. Article | MR 81k:46002 | Zbl 0464.46010 · Zbl 0464.46010 |
| [28] | WAGNER (M. J.) . - Quotientenräume von stabilen Potenzreihenräumen endlichen Typs , Manuscripta math., Vol. 31, 1980 , pp. 97-109. MR 81m:46021 | Zbl 0453.46010 · Zbl 0453.46010 · doi:10.1007/BF01303269 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
