×

Principle of local reflexivity for operators and quojections. (English) Zbl 0673.46002

The principle of local reflexivity for linear operators between Banach spaces is proved. Then it is applied in order to obtain the principle for locally convex (especially Fréchet) spaces X whenever X is a quojection. A special dual form of the principle for strict LB-spaces is also obtained. The results of the paper are applied in order to generalize the theory of \({\mathcal L}_ p\)-spaces to the classes of Fréchet and strict LB-spaces in the author’s paper: “\({\mathcal L}_ p\)- and injective locally convex spaces”, Dissertationes Math., to appear.
Reviewer: P.Domaǹski

MSC:

46A04 Locally convex Fréchet spaces and (DF)-spaces
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.)
46A20 Duality theory for topological vector spaces
46B10 Duality and reflexivity in normed linear and Banach spaces
46A03 General theory of locally convex spaces
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] E. Behrends, A generalization of the principle of local reflexivity. Rev. Roum. Math. PuresAppl.31, 293–296 (1986). · Zbl 0609.46006
[2] E.Behrends, On the principle of local reflexivity. Preprint. · Zbl 0757.46018
[3] E. Behrends, Normal operators and multipliers on complex Banach spaces. Israel J. Math.47, 23–28 (1984). · Zbl 0544.47018 · doi:10.1007/BF02760559
[4] E. Behrends, S. Dierolf andP. Harmand, On a problem of Bellenot and Dubinsky. Math. Ann.275, 337–339 (1986). · Zbl 0586.46001 · doi:10.1007/BF01458608
[5] S. F. Bellenot andE. Dubinsky, Fréchet spaces with nuclear Köthe quotients. Trans. Amer. Math. Soc.273, 579–594 (1982).
[6] I. I. Chuchaev andW. A. Gejler, General principle of local reflexivity and its applications in theory of duality of cones (in Russian). Sibirsk. Mat. Zh.23, 32–43 (1982). · Zbl 0503.46006
[7] S. Dierolf andV. B. Moscatelli, A note on quojections. Funct. Approx. Comment. Math.17, 131–138 (1987). · Zbl 0617.46006
[8] S. Dierolf andD. N. Zarnadze, A note on strictly regular Frechet spaces. Arch. Math.42, 549–556 (1984). · Zbl 0525.46004 · doi:10.1007/BF01194053
[9] P.Domański, p- and injective locally convex spaces. Diss. Math., to appear.
[10] P.Domanski, Twisted Fréchet spaces of continuous functions. To appear.
[11] P.Domański, Duals and preduals of injective Fréchet spaces andD p -spaces. Preprint.
[12] G. Godefroy etP. Saab, Quelques espaces de Banach ayant les propriétés (V) ou (V*) de A. Pełczyński. Comp. Rend. Acad. Sci. Paris (11)303-1, 503–506 (1986).
[13] G.Godefroy and P.Saab, Weakly unconditionally convergent series inM-ideals. To appear. · Zbl 0676.46006
[14] W. B. Johnson, H. P. Rosenthal andM. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces. Israel J. Math.9, 488–506 (1971). · Zbl 0217.16103 · doi:10.1007/BF02771464
[15] J. Lindenstrauss andH. P. Rosenthal, The p-spaces. Israel J. Math.7, 325–349 (1969). · Zbl 0205.12602 · doi:10.1007/BF02788865
[16] G. Metafune andV. B. Moscatelli, Another construction of twisted spaces. Proc. Roy. Irish Acad.87A, 163–168 (1987). · Zbl 0618.46002
[17] V. B. Moscatelli, Fréchet spaces without continuous norms and without bases. Bull. London Math. Soc.12, 63–66 (1980). · Zbl 0417.46004 · doi:10.1112/blms/12.1.63
[18] V. B.Moscatelli, Strict inductive and projective limits, twisted spaces and quojections. Proc. 13th Winter School on Abstract Analysis, Srni 1985; Rend. Circ. Mat. Palermo (2) Suppl.10, 119–131 (1985). · Zbl 0619.46067
[19] H. H.Schaeffer, Topological Vector Spaces, 3rd ed. NewYork 1971.
[20] D. N. Zarnadze, Strictly regular Fréchet spaces (in Russian). Mat. Zametki31, 454–458 (1982). · Zbl 0562.46001
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.