$ℒ_p$-Spaces and injective locally convex spaces

Domański, Paweł

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Tytuł książki

$ℒ_p$-Spaces and injective locally convex spaces

Autorzy

Paweł Domański

Seria

Rozprawy Matematyczne tom/nr w serii: 298 wydano: 1990

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Abstrakty

CONTENTS
Introduction.................................................................................................................................................................5
Conventions and notations.........................................................................................................................................8
1. Preliminaries from the theory of locally convex spaces.........................................................................................11
2. Ultraproducts of locally convex spaces.................................................................................................................16
3. The principle of local reflexivity for locally convex spaces.....................................................................................19
4. Complemented subspaces of products and direct sums of Banach $L_p$-spaces and L₁ -predual spaces........24
5. Definition, examples and basic properties of locally convex $ℒ_p$-spaces..........................................................29
6. Definition, examples and basic properties of $Dℒ_p$-spaces..............................................................................39
7. Local properties of complemented subspaces of products and direct sums of Banach $L_p$-spaces................42
8. Duality between $ℒ_p$ and $Dℒ_p$-spaces and a characterization of injective spaces.....................................46
9. Extension, lifting and factorization properties of $ℒ_p$ and $Dℒ_p$-spaces.......................................................56
10. Fréchet $ℒ_p$- and (DF)-&Dℒ_p$-spaces with applications to Fréchet injective spaces..................................62
11. Comments and open problems...........................................................................................................................71
References...............................................................................................................................................................73

Słowa kluczowe

Tematy

Wydawca

Instytut Matematyczny Polskiej Akademii Nauk

Miejsce publikacji

Warszawa

Seria

Rozprawy Matematyczne tom/nr w serii: 298

Liczba stron

Liczba rozdzia³ów

Opis fizyczny

Dissertationes Mathematicae, Tom CCXCVIII

Daty

wydano

1990

Twórcy

autor

Paweł Domański

Instytut Matematyki, Uniwersytet im. A. Mickiewicza, Institute of Mathematics, A. Mickiewicz University, Matejki 48/49, 60-769 Poznań, Poland

Bibliografia

[1] J. Barwise (ed.), Handbook of Mathematical Logic, North-Holland, Amsterdam 1977.
[2] E. Behrends, Normal operators and multipliers on complex Banach spaces, Israel J. Math. 47 (1984), 23-28.
[3] E. Behrends, A generalization of the Principle of Local Reflexivity, Rev. Roumaine Math. Pures Appl. 31 (1986), 293-296.
[4] E. Behrends, On the Principle of Local Reflexivity, Preprint.
[5] E. Behrends, S. Dierolf and P. Harmand, On a problem of Bellenot and Dubinsky, Math. Ann. 275 (1986), 337-339.
[6] S. F. Bellenot and Ed Dubinsky, Fréchet spaces with nuclear Köthe quotients, Trans. Amer. Math. Soc. 273 (1982), 579-594.
[7] C. Bessaga and A. Pelczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151-164.
[8] J. Bonet, Quojections and projective tensor products. Arch. Math. 45 (1985), 169-173.
[9] J. Bonet, S. Dierolf, On LB-spaces of Moscatelli type, Doga Tr. J. Math. 13 (1989), 9-33.
[10] J. Bonet, M. Maestre, G. Metafune, V. B. Moscatelli and D. Vogt, Every quojection is the quotient of a countable product of Banach spaces, in: Proc. NATO-ASI Workshop on "Fréchet Spaces", Istanbul, August 1988, T. Terzioǧlu (ed.), Kluwer, Dordrecht 1989, 355-356.
[11] J. Bourgain, New Classes of $ℒ_p$-spaces, Lecture Notes in Math. 889, Springer, Berlin 1981.
[12] C. C. Chang and H. J. Keisler, Model Theory, North-Holland, Amsterdam 1973.
[13] D. Dacunha-Castelle and J. L. Krivine, Applications des ultraproduits à l'étude des espaces et des algèbres de Banach, Studia Math. 41 (1972), 315-334.
[14] D. W. Dean, The equation L(E,X**) = L(E,X)** and the Principle of Local Reflexivity, Proc. Amer. Math. Soc. 40 (1973), 146-148.
[15] M. De Wilde, Sur le relèvement des parties bornées d'un quotient d'espaces vectoriels topologiques, Bull. Soc. Roy. Sci. Liège 43 (1974), 299-301.
[16] S. Dierolf and V. B. Moscatelli, A Fréchet space which has a continuous norm bul whose bidual does not, Math. Z. 191 (1986), 17-21.
[17] S. Dierolf and V. B. Moscatelli, A note on quojections, Funct. Approx. Comment. Math. 17 (1987), 131-138.
[18] S. Dierolf, D. N. Zarnadze, A note on strictly regular Fréchet spaces, Arch. Math. 42 (1984), 549-556.
[19] J. Diestel, Sequences and Series in Banach Spaces, Springer, New York 1984.
[20] P. Domański, Extensions and liftings of linear operators, Thesis, Poznań.
[21] P. Domański, Complemented subspaces of products of Hilbert spaces, Proc. Amer. Math. Soc. (to appear).
[22] P. Domański, The principle of local reflexivity for linear operators and quojections. Arch. Math, (to appear).
[23] P. Domański, Twisted Fréchet spaces of continuous functions, (to appear).
[24] P. Domański, A. Ortyński, Complemented subspaces of products of Banach spaces, Trans. Amer. Math. Soc. 316 (1989), 215-232.
[25] L. Drewnowski, Another note on Kalton's theorem, Studia Math. 52 (1975), 233-237.
[26] K. Floret and V. B. Moscatelli, On bases in strict inductive and projective limits of locally convex spaces, Pacific J. Math. 119 (1985), 103-113.
[27] K. Floret and V. B. Moscatelli, Unconditional bases in Fréchet spaces, Arch. Math. 47 (1986), 129-130.
[28] M. G. Garnir, M. De Wilde and J. Schmets, Analyse Fonctionnelle, Birkhäuser, Basel and Stuttgart 1968.
[29] W. A. Gejler, On extending and lifting continuous linear mappings in topological vector spaces, Studia Math. 62 (1978), 295-303.
[30] W. A. Gejler, I. I. Chuchaev, General Principle of Local Reflexivity and its applications in theory of duality of cones, Sibirsk. Mat. Zh. 23 (1982), 32-43 (in Russian).
[31] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955).
[32] S. Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72-104.
[33] S. Heinrich, Ultrapowers of locally convex spaces and applications I, Math. Nachr. 118 (1984), 285-315.
[34] S. Heinrich, Ultrapowers of locally convex spaces and applications II, ibid. 121 (1985), 211-229.
[35] C. W. Henson and L. C. Moore jr., The nonstandard theory of topological vector spaces, Trans. Amer. Math. Soc. 172 (1972), 405-135.
[36] R. Hollstein, A Hahn-Banach theorem for holomorphic mappings on locally convex spaces, Math. Z. 188 (1985), 549-557.
[37] R. Hollstein, An extension and lifting theorem for bounded linear mappings in locally convex spaces and some applications, Arch. Math. 47 (1986), 251-262.
[38] H. Jarchow, Locally Convex Spaces, B. G. Teubner, Stuttgart 1981.
[39] W. B. Johnson, H. P. Rosenthal and M. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488-506.
[40] M. I. Kadec and A. Pełczyński, Bases, lacunary sequences and complemented subspaces in the spaces $L_p$, Studia Math. 21 (1962), 161-176.
[41] N. J. Kalton, Locally complemented subspaces and $ℒ_p$-spaces for 0 < p < I, Math. Nachr. 115 (1984), 74-97.
[42] H. J. Keisler, Good ideals in fields of sets, Ann. Math. 79 (1964), 338-359.
[43] S. W. Kislyakov, On the spaces with "small" annihilators, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 65 (1976), 192-195 (in Russian).
[44] G. Köthe, Topological Vector Spaces, Vol. I, Springer, New York 1969.
[45] G. Köthe, Topological Vector Spaces, Vol. II, Springer, New York 1975.
[46] J. Krone, Existence of bases and the dual splitting relation for Fréchet spaces, Studia Math. 92 (1989), 37-48.
[47] J. Krone, D. Vogt, The splitting relation for Köthe spaces, Math. Z. 190 (1985), 387-400.
[48] K. Kunen, Ultrafilters and independent sets, Trans. Amer. Math. Soc. 172 (1972), 299-306.
[49] J. Lindenstrauss, Extensions of compact operators, Mem. Amer. Math. Soc. 48 (1964).
[50] J. Lindenstrauss, On complemented subspaces of m, Israel J. Math. 5 (1967), 153-156.
[51] J. Lindenstrauss, On certain subspaces of l₁, Bull. Acad. Polon. Sci. 12 (1964), 539-542.
[52] J. Lindenstrauss, A. Pełczyński, Absolutely summing operators in lfp-spaces and their applications, Studia Math. 29 (1968), 275-326.
[53] H. P. Rosenthal, The $ℒ_p$-spaces, Israel J. Math. 7 (1969), 325-349.
[54] H. P. Rosenthal, L. Tzafriri, Classical Banach Spaces, Lecture Notes in Math. 338, Springer, Berlin 1973.
[55] H. P. Rosenthal, L. Tzafriri, Classical Banach Spaces, Vol. I, Springer, Berlin 1977.
[56] H. P. Rosenthal, L. Tzafriri, Classical Banach Spaces, Vol. II, Springer, Berlin 1977.
[57] R. Meise and D. Vogt, A characterization of the quasi-normable Fréchet spaces, Math. Nachr. 122 (1983), 141-150.
[58] G. Metafune and V. B. Moscatelli, Nuclear Fréchet spaces with basis do not have the three space property, Arch. Math. 50 (1988), 369-370.
[59] G. Metafune and V. B. Moscatelli, A twisted Fréchet space with basis, Monatsh. Math. 105 (1988), 127-129.
[60] G. Metafune and V. B. Moscatelli, Complemented subspaces of sums and products of Banach spaces, Ann. Mat. Pura Appl. 159 (1988), 175-190.
[61] G. Metafune and V. B. Moscatelli, Another construction of twisted spaces, Proc. Roy. Irish Acad. Sect. A 87 (1987), 163-168.
[62] G. Metafune and V. B. Moscatelli, On exact sequences of quojections, preprint 1988.
[63] G. Metafune and V. B. Moscatelli, On twisted Fréchet and LB-spaces, Proc. Amer. Math. Soc. (to appear).
[64] G. Metafune and V. B. Moscatelli, Quojections and prequojections, in: Proc. NATO-ASI Workshop on "Fréchet spaces", Istanbul, August 1988, T. Terzioğlu (ed.), Kluwer, Dordrecht 1989, 235-254.
[65] G. Metafune and V. B. Moscatelli, Generalized prequojections and bounded maps, Resultate Math. 15 (1989), 172-178.
[66] G. Metafune and V. B. Moscatelli, Dense subspace with continuous norms in Fréchet spaces, preprint 1988.
[67] V. B. Moscatelli, Fréchet spaces without continuous norms and without bases, Bull. London Math. Soc. 12 (1980), 63-66.
[68] V. B. Moscatelli, Strict inductive and projective limits, twisted spaces and quojections, Suppl. Rend. Circ. Mat. Palermo Ser. II, 10(1985), 119-131, Proc. XIII Winter School on Abstract Analysis, Srni 1985.
[69] V. B. Moscatelli, On strongly non-norming subspaces, Note Mat. 7 (1987), 311-314.
[70] V. B. Moscatelli, Strongly nonnorming subspaces and prequojections, Studia Math. 95 (1990), 249-254.
[71] L. Nachbin, Some problems in extending and lifting continuous linear transformations, in: Proc. Inter. Symp. Linear Spaces, Jerusalem 1960, 340-350.
[72] S. Önal and T. Terzioğlu, Unbounded linear operators and nuclear Köthe quotients, Arch. Math. 53 (1989), (to appear).
[73] V. P. Palamadov, Homological methods in the theory of locally convex spaces, Uspekhi Mat. Nauk 26 (1) (1971), 3-66 (in Russian).
[74] A. Pełczyński, Some aspects of the present theory of Banach spaces, in: S. Banach, Oeuvres, PWN, Warszawa 1979, 222-302.
[75] A. Pietsch, Operator Ideals, Deutsch. Verlag Wiss., Berlin 1978.
[76] G. Pisier, Factorisation of linear operators and geometry of Banach spaces, CBMS 60, Amer. Math. Soc., Providence 1986.
[77] H. P. Rosenthal, On relatively disjoint families of measures with some applications to Banach space theory, Studia Math. 37 (1970), 13-36.
[78] H. P. Rosenthal, On injective Banach spaces and the spaces $L_∞(μ) for finite measures μ, Acta Math. 124 (1970), 205-248.
[79] H. H. Schaefer, Topological Vector Spaces, $3^{rd}$ ed., Springer, New York 1971.
[80] M. Schreiber, Quelques remarques sur les caractérisations des espaces $L_p$, 0 < p < 1, Ann. Inst. H. Poincaré 8 (1972), 83-90.
[81] Ch. Stegall, A proof of the Principle of Local Reflexivity, Proc. Amer. Math. Soc. 78 (1980), 154-156.
[82] J. Stern, Some applications of model theory in Banach space theory, Ann. Math. Logic 9 (1976), 49-121.
[83] E. W. Straeuli, On extension and lifting of operators, an approach within the theory of operator ideals. Thesis, Zürich 1985.
[84] T. Terzioğlu and D. Vogt, A Köthe space which has a continuous norm but whose bidual does not, preprint 1988.
[85] D. Vogt, Some results on continuous linear maps between Fréchet spaces, in: Functional Analysis: Surveys and Recent Results III, K. D. Bierstedt, B. Fuchsteiner (eds.), North-Holland Math. Stud. 90 (1984), 349-381.
[86] D. Vogt, On the functors Ext¹(E,F) for Fréchet spaces, Studia Math. 85 (1987), 163-197.
[87] D. Vogt, On two problems of Mityagin, Math. Nachr. 141 (1989), 13-25.
[88] D. N. Zarnadze, Strictly regular Fréchet spaces. Mat. Zametki 31 (1982), 454-458 (in Russian).
[89] M. Zippin, The separable extension problem, Israel J. Math. 26 (1977), 372-387.
[90] P. Domański, Another note on a problem of Bellenot and Dubinsky, preprint, 1989.
[91] A. Defant, A duality theorem for locally convex tensor products, Math. Z. 190 (1985), 45-53.
[92] S. F. Bellenot, Local reflexivity of normed spaces, operators and Fréchet spaces, J. Funct. Anal. 59 (1984), 1-11.
[93] K. D. Bierstedt and J. Bonet, Stefan Heinrich's density condition for Fréchet spaces and the characterization of the distinguished Köthe echelon spaces, Math. Nachr. 135 (1988). 149-180.
[94] J. Bonet and S. Dierolf, Fréchet spaces of Moscatelli type, preprint, 1988.
[95] J. C. Diaz, Continuous norms of Fréchet lattices, Arch. Math. (Basel) 52 (1989), 155-158.
[96] P. Domański, On the projective LB-spaces, (to appear).
[97] P. Domański, On the injective C(T)-spaces, (to appear).
[98] P. Domański, Fréchet injective spaces of continuous functions, (to appear).
[99] G. Köthe, Hebbare lokalkonvexe Räume, Math. Ann. 165 (1966), 181-195.
[100] E. Straeuli, On the Hahn-Banach extensions for certain operator ideals. Arch. Math. (Basel) 47 (1986), 49-54.

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Uwagi

1985 Mathematics Subject Classification:
Primary 46A05, 46A06, 46A12, 46A20, 46A22, 46A99, 46M10, 03C20;
Secondary 46E10, 46E15, 46E30, 46C99.

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0012-3862

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