Schauder basis in a locally k-convex space and perfect sequence spaces
DOI:
https://doi.org/10.4067/S0716-09172011000300008Keywords:
Non archimedean analysis, Locally K− convex spaces, Schauder basis, The weak basis theorem, Compatible topologies, Perfect sequence spaces, K− barrelled spaces and G- spaces.Abstract
In this work, we are dealing with the natural topology in a perfect sequence space and the transfert of topologies of a locally K — convex space E with a Schauder basis (ei )i to such Space. We are also interested with the compatible topologies on E for which the basis(ei )i is equicontinuous, and the weak basis problem. Finally, we give some applications to barrelled Spaces and G—Spaces.
References
[2] R. Ameziane Hassani and M. Babahmed, Topologies polaires compatibles avec une dualité séparante sur un corps valué nonarchimédien, Proyecciones. Vol. 20, No. 2, pp. 217-241, (2001).
[3] S. Banach, Théorie des op´erateurs linéaires, Chelsea, New York (1955).
[4] S. Bennet and J. B. Cooper, Weak basis in F and (LF)-spaces. J. London Math. Soc. 44, pp. 505-508, (1969).
[5] G. Bessaga and A. Pelczynski, Properties of bases in spaces of type B0. Prace Math. 3, pp. 123-142, (1959).
[6] N. Bourbaki, Espaces vectoriels topologiques, Chap.1 à 5, Paris, (1981).
[7] N. De Grande-De Kimpe, C-compactness in locally K-convex spaces, Indag. Math. 33, pp. 176-180, (1971).
[8] N. De Grande-De Kimpe, Perfect locally K-convex sequence spaces, Indag. Math. 33, pp. 471-482, (1971).
[9] N. De Grande-De Kimpe, On the structure of locally K-convex spaces with a Schauder basis, Indag. Math. 34, pp. 396-406, (1972).
[10] N. De Grande-De Kimpe, Equicontinuous Schauder basis and compatible locally convex topologies, Proc. Kond. Ned. Akad. V. Wet. A77(3), pp. 276-283 (1973).
[11] N. De Grande-De Kimpe, On a class of locally convex spaces with a Schauder basis, Proc. Kond. Ned.Akad. V. Wet. pp. 307-312, (1976).
[12] N. De Grande-De Kimpe, Structure theorems for locally K-convex spaces. Proc. Kond. Ned. Akad. Wet. 80: pp. 11-22, (1977).
[13] M. De Wilde, Reseaux dans les espaces linéaires a semi-normes. Mem. Soc. R. Liège. (1969).
[14] Dorleyn, M, Beschouwingen over coördinatenruimten, oneindige matrices en determinanten in een niet-archmedisch gewaardeerd lichaam. Thesis, Amsterdam, (1951).
[15] E. Dubinsky, JR. Retherford, Schauder bases in compatible topologies. Stud. Math. 28: pp. 221-226, (1967).
[16] T. A. Efimova, On weak basis in the inductive limits of barrelled normed spaces. Vestnik. Leningrad Uni. Math. Meb. Astronom.119, pp. 21-26, (1981).
[17] K. Floret, Bases in sequentially retractive limits spaces. Proc. Int. Coll.on Nuclear Spaces and Ideals in operators Algebras, Warsaw1969, Studia Math. 38, pp. 221-226, (1970).
[18] D. J. H. Garling, On topological sequence spaces, Proc. Camb. Phil. Soc. 63, pp. 997-1019, (1967).
[19] D. J. H. Garling, The β-and γ-duality of sequence spaces, Proc. Camb. Phil. Soc. 63, pp. 963-981, (1967).
[20] N. J. Kalton, On the weak-basis theorem. Compositio Mathematica, Vol. 27, Fasc. 2, pp. 213-215, (1973).
[21] J. K¸akol and T. Gilsdorf, On the weak basis theorems for p-adic locally convex spaces, p-adic functional analysis edited by J. K¸akol, N. De Grande-De Kimpe and C. Perez-Garcia. Marcel Dekker, Ink. New York, (1999).
[22] J. Kakol, C. Perez-Garcia and W. H. Schikhof, Cardinality and Mackey topologies of non-archimedean Banach and Fréchet spaces. Bull Pol Acad Sci Math 44: pp. 131-141, (1996).
[23] G. Köthe, Topologische lineaire Räume; Springer Verlag, (1960).
[24] C. W. McArthur, The weak basis theorem, Colloq. Math. 17, pp. 71-76, (1967).
[25] A. F. Monna, Espaces linéaires à une infinité dé nombrable de coordonnées. Proc. Ned. Akad. V. Wetensch. 53, pp. 1548-1559, (1950).
[26] A. F. Monna, Sur le théorème de Banach-Steinhaus, Proc. Kond. Ned. Akad. V. Wetensch. A66, pp. 121-31 (1963).
[27] J. Orihuela, On the equivalence of weak and Schauder bases, Arch. Math. Vol. 46, pp. 447-452, (1986).
[28] C. Perez-Garcia and W. H. Schikhof, The Orlicz-Pettis property in p-adic analysis, collect. Math. 43, 3, pp. 225-233, (1992).
[29] J. H. Shapiro, On the weak basis theorem in F-Spaces, can. J. Math. Vol. XXVI, No. 6, pp. 1294-1300, (1974).
[30] H. H. Schaefer, Topological vector spaces, Springer-Verlag New-York, herdlberg Berlin, (1971).
[31] W. H. Schikhof, Compact-like sets in non-archimedean fonctional analysis, Proc. of the conf´erence on p-adic analysis henglehoef, Belgium, pp. 137-147 (1986).
[32] W. H. Schikhof, The continuous linear image of p-adic compactoid. Proc Kon Ned Akad Wet 92: pp. 119-123, (1989).
[33] I. Singer, Weak*-bases in conjugate Banach spaces, Stud. Math. 21, (1961).
[34] T. A. Springer, Une notion de compacit´e dans la th´eorie des espaces vectoriels topologiques, Indag. Math. 27, pp. 182-189 (1965).
[35] W. J. Stiles, On properties of subspaces of lp, 0 ≺ p ≺ 1, Trans. mer. Math. Soc. 149, pp. 405-415 (1970).
[36] J. Van-tiel, Espaces localement K-convexes, I-III. Proc. Kon. Ned. Akad. van Wetensch. A68, pp. 249-289 (1965).
Published
How to Cite
Issue
Section
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.