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## Studia Mathematica

1996 | 120 | 3 | 247-258
Tytuł artykułu

### Acyclic inductive spectra of Fréchet spaces

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We provide new characterizations of acyclic inductive spectra of Fréchet spaces which improve the classical theorem of Palamodov and Retakh. It turns out that acyclicity, sequential retractivity (defined by Floret) and further strong regularity conditions (introduced e.g. by Bierstedt and Meise) are all equivalent. This solves a problem that was folklore since around 1970. For inductive limits of Fréchet-Montel spaces we obtain even stronger results, in particular, Grothendieck's problem whether regular (LF)-spaces are complete has a positive solution in this case and we show that even the weakest regularity conditions already imply acyclicity. One of the main benefits from our results is an improvement in the theory of projective spectra of (DFM)-spaces. We prove the missing implication in a theorem of Vogt and thus obtain evaluable conditions for vanishing of the derived projective limit functor which have direct applications to classical problems of analysis like surjectivity of partial differential operators on various classes of ultradifferentiable functions (as was explained e.g. by Braun, Meise and Vogt).
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
247-258
Opis fizyczny
Daty
wydano
1996
otrzymano
1995-12-18
poprawiono
1996-05-09
Twórcy
autor
• FB IV-Mathematik, Universität Trier, D-54286 Trier, Germany
Bibliografia
• [1] K. D. Bierstedt, An introduction to locally convex inductive limits, in: Functional Analysis and Applications, Nice 1986, World Sci., Singapore, 1988, 35-133.
• [2] K. D. Bierstedt and J. Bonet, Weighted inductive limits of continuous functions, Math. Nachr. 165 (1994), 25-48.
• [3] K. D. Bierstedt and R. Meise, Bemerkungen über die Approximationseigenschaft lokalkonvexer Funktionenräume, Math. Ann. 209 (1974), 99-107.
• [4] J. Bonet and C. Fernández, Bounded sets in (LF)-spaces, Proc. Amer. Math. Soc. 123 (1995), 3717-3721.
• [5] R. W. Braun, Surjectivity of partial differential operators on Gevrey classes, in: Functional Analysis, Proceedings of the First Workshop at Trier University, S. Dierolf, S. Dineen and P. Domański (eds.), de Gruyter, to appear.
• [6] R. W. Braun, R. Meise and D. Vogt, Applications of the projective limit functor to convolutions and partial differential equations, in: Advances in the Theory of Fréchet Spaces, T. Terzioğlu (ed.), NATO Adv. Sci. Inst. Ser. C 287, Kluwer, Dordrecht, 1989, 22-46.
• [7] R. W. Braun, R. Meise and D. Vogt, Existence of fundamental solutions and surjectivity of convolution operators on classes of ultradifferentiable functions, Proc. London Math. Soc. 61 (1990), 344-370.
• [8] R. W. Braun, R. Meise and D. Vogt, Characterization of the linear partial differential operators with constant coefficients which are surjective on quasianalytic classes of Roumieu type on $ℝ^N$, Math. Nachr. 168 (1994), 19-54.
• [9] B. Cascales and J. Orihuela, Metrizability of precompact subsets in (LF)-spaces, Proc. Roy. Soc. Edinburgh Sect. A 103 (1986), 293-299.
• [10] C. Fernández, Regularity conditions on (LF)-spaces, Arch. Math. (Basel) 54 (1990), 380-383.
• [11] K. Floret, Bases in sequentially retractive limit spaces, Studia Math. 38 (1970), 221-226.
• [12] K. Floret, Folgenretraktive Sequenzen lokalkonvexer Räume, J. Reine Angew. Math. 259 (1973), 65-85.
• [13] K. Floret, Some aspects of the theory of locally convex inductive limits, in: Functional Analysis: Surveys and Recent Results II, K. D. Bierstedt and B. Fuchssteiner (eds.), North-Holland Math. Stud. 38, North-Holland, Amsterdam, 1980, 205-237.
• [14] L. Frerick, A splitting theorem for nuclear Fréchet spaces, in: Functional Analysis, Proceedings of the first Workshop at Trier University, S. Dierolf, S. Dineen and P. Domański (eds.), de Gruyter, to appear.
• [15] L. Frerick and J. Wengenroth, A sufficient condition for vanishing of the derived projective limit functor, Arch. Math. (Basel), to appear.
• [16] A. Grothendieck, Produits tensorielles topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955).
• [17] L. Hörmander, On the range of convolution operators, Ann. of Math. 76 (1962), 148-170.
• [18] L. Hörmander, On the existence of real analytic solutions of partial differential operators with constant coefficients, Invent. Math. 21 (1973), 152-182.
• [19] H. Neus, Über die Regularitätsbegriffe induktiver lokalkonvexer Sequenzen, Manuscripta Math. 25 (1978), 135-145.
• [20] V. P. Palamodov, The projective limit functor in the category of linear topological spaces, Mat. Sb. 75 (1968), 567-603 (in Russian); English transl.: Math. USSR-Sb. 4 (1968), 529-558.
• [21] V. P. Palamodov, Homological methods in the theory of locally convex spaces, Uspekhi Mat. Nauk 26 (1) (1971), 3-65 (in Russian); English transl.: Russian Math. Surveys 26 (1971), 1-64.
• [22] P. Pérez Carreras and J. Bonet, Barrelled Locally Convex Spaces, North-Holland Math. Stud. 131, North-Holland, 1987.
• [23] V. S. Retakh, Subspaces of a countable inductive limit, Dokl. Akad. Nauk SSSR 194 (1970), 1277-1279 (in Russian); English transl.: Soviet Math. Dokl. 11 (1970), 1384-1386.
• [24] W. Roelcke, On the finest locally convex topology agreeing with a given topology on sequence of absolutely convex sets, Math. Ann. 198 (1972), 57-80.
• [25] M. Valdivia, Topics in Locally Convex Spaces, North-Holland Math. Stud. 67, North-Holland, 1982.
• [26] D. Vogt, On the functors $Ext^1(E,F)$ for Fréchet spaces, Studia Math. 85 (1987), 163-197.
• [27] D. Vogt, Lectures on projective spectra of (DF)-spaces, Seminar lectures, AG Funktionalanalysis Düsseldorf/Wuppertal, 1987.
• [28] D. Vogt, Topics on projective spectra of (LB)-spaces, in: Advances in the Theory of Fréchet Spaces, T. Terzioğlu (ed.), NATO Adv. Sci. Inst. Ser. C 287, Kluwer, Dordrecht, 1989, 11-27.
• [29] D. Vogt, Regularity properties of (LF)-spaces, in: Progress in Functional Analysis, North-Holland Math. Stud. 170, North-Holland, 1992, 57-84.
• [30] J. Wengenroth, Retractive (LF)-spaces, Dissertation, Universität Trier, 1995.
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