On duals of weakly acyclic \((LF)\)-spaces. (English) Zbl 0883.46001
Summary: For countable inductive limits of Fréchet spaces (\((LF)\)-spaces) the property of being weakly acyclic in the sense of Palamodov (or, equivalently, having condition \((M_{0})\) in the terminology of Retakh) is useful to avoid some important pathologies and in relation to the problem of well-located subspaces. In this note we consider if weak acyclicity is enough for a \((LF)\)-space \(E:= \text{ind} E_{n}\) to ensure that its strong dual is canonically homeomorphic to the projective limit of the strong duals of the spaces \(E_{n}\). First we give an elementary proof of a known result by Vogt and obtain that the answer to this question is positive if the steps \(E_{n}\) are distinguished or weakly sequentially complete. Then we construct a weakly acyclic \((LF)\)-space for which the answer is negative.
MSC:
| 46A13 | Spaces defined by inductive or projective limits (LB, LF, etc.) |
| 46A08 | Barrelled spaces, bornological spaces |
Keywords:
countable inductive limits; Fréchet spaces; weakly acyclic in the sense of Palamodov; well-located subspaces; projective limit of the strong duals; distinguished; weakly sequentially complete; acyclic \((LF)\)-spaceReferences:
| [1] | I. Amemiya, Some examples of \((F)\) and \((DF)\) spaces,, Proc. Japan Acad. 33 (1957), 169-171. · Zbl 0078.28702 |
| [2] | Klaus D. Bierstedt, An introduction to locally convex inductive limits, Functional analysis and its applications (Nice, 1986) ICPAM Lecture Notes, World Sci. Publishing, Singapore, 1988, pp. 35 – 133. · Zbl 0786.46001 · doi:10.1007/s13116-009-0018-2 |
| [3] | Klaus D. Bierstedt and José Bonet, A question of D. Vogt on (\?\?)-spaces, Arch. Math. (Basel) 61 (1993), no. 2, 170 – 172. · Zbl 0781.46004 · doi:10.1007/BF01207465 |
| [4] | José Bonet and Susanne Dierolf, A note on biduals of strict (LF)-spaces, Results Math. 13 (1988), no. 1-2, 23 – 32. · Zbl 0653.46006 · doi:10.1007/BF03323393 |
| [5] | José Bonet and Susanne Dierolf, On distinguished Fréchet spaces, Progress in functional analysis (Peñíscola, 1990) North-Holland Math. Stud., vol. 170, North-Holland, Amsterdam, 1992, pp. 201 – 214. · Zbl 0785.46003 · doi:10.1016/S0304-0208(08)70320-7 |
| [6] | J. Bonet, S. Dierolf, and C. Fernández, On two classes of LF-spaces, Portugal. Math. 49 (1992), no. 1, 109 – 130. · Zbl 0804.46008 |
| [7] | Juan Carlos Díaz, Two problems of Valdivia on distinguished Fréchet spaces, Manuscripta Math. 79 (1993), no. 3-4, 403 – 410. · Zbl 0799.46003 · doi:10.1007/BF02568354 |
| [8] | Susanne Dierolf, On two questions of A. Grothendieck, Bull. Soc. Roy. Sci. Liège 50 (1981), no. 5-8, 282 – 286. · Zbl 0483.46001 |
| [9] | Klaus Floret, Some aspects of the theory of locally convex inductive limits, Functional analysis: surveys and recent results, II (Proc. Second Conf. Functional Anal., Univ. Paderborn, Paderborn, 1979) North-Holland Math. Stud., vol. 38, North-Holland, Amsterdam-New York, 1980, pp. 205 – 237. |
| [10] | A. Grothendieck, Sur les espaces \((F)\) et \((DF)\), Summa Brasil Math. 3 (1954), 57-123. · Zbl 0058.09803 |
| [11] | A. Grothendieck, Topological vector spaces, Gordon and Breach Science Publishers, New York-London-Paris, 1973. Translated from the French by Orlando Chaljub; Notes on Mathematics and its Applications. · Zbl 0275.46001 |
| [12] | John Horváth, Topological vector spaces and distributions. Vol. I, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. |
| [13] | Gottfried Köthe, Topological vector spaces. I, Translated from the German by D. J. H. Garling. Die Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag New York Inc., New York, 1969. · Zbl 0179.17001 |
| [14] | Reinhard Mennicken and Manfred Möller, Well located subspaces of LF-spaces, Functional analysis, holomorphy and approximation theory (Rio de Janeiro, 1980) North-Holland Math. Stud., vol. 71, North-Holland, Amsterdam-New York, 1982, pp. 287 – 298. · Zbl 0533.46001 |
| [15] | S. Müller, S. Dierolf, and L. Frerick, On acyclic inductive sequences of locally convex spaces, Proc. Roy. Irish Acad. Sect. A 94 (1994), no. 2, 153 – 159. · Zbl 0848.46002 |
| [16] | Pedro Pérez Carreras and José Bonet, Barrelled locally convex spaces, North-Holland Mathematics Studies, vol. 131, North-Holland Publishing Co., Amsterdam, 1987. Notas de Matemática [Mathematical Notes], 113. · Zbl 0614.46001 |
| [17] | M. Valdivia, Fréchet spaces with no subspaces isomorphic to \?\(_{1}\), Math. Japon. 38 (1993), no. 3, 397 – 411. · Zbl 0778.46001 |
| [18] | D. Vogt, Lectures on projective spectra of \((DF)\)-spaces, Seminar lectures, AG Funktionalanalysis, Düsseldorf, Wuppertal, 1987. |
| [19] | Dietmar Vogt, Regularity properties of (LF)-spaces, Progress in functional analysis (Peñíscola, 1990) North-Holland Math. Stud., vol. 170, North-Holland, Amsterdam, 1992, pp. 57 – 84. · Zbl 0779.46005 · doi:10.1016/S0304-0208(08)70311-6 |
| [20] | J. Wengenroth, Retractive (LF)-spaces, Dissertation, Universität, Trier, July 1995. · Zbl 0974.46500 |
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