Schauder basis in a locally k-convex space and perfect sequence spaces

  • R. Ameziane Université Sidi Mohamed Ben Abdellah.
  • Abdelkhalek El Amrani Université Sidi Mohamed Ben Abdellah.
  • Mohammed Babahmed Université Moulay Ismaïl.
Palabras clave: Non archimedean analysis, Locally K− convex spaces, Schauder basis, The weak basis theorem, Compatible topologies, Perfect sequence spaces, K− barrelled spaces and G- spaces.

Resumen

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.

Citas

[1] M. G. Arsove, and R. E. Edwards, generalized bases in topological linear spaces. Studia math.19, pp. 95-113, (1960).

[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. PeÃlczynski, 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 PolAcad 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. XX∨I, 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).
Publicado
2011-12-10
Cómo citar
Ameziane, R., El Amrani, A., & Babahmed, M. (2011). Schauder basis in a locally k-convex space and perfect sequence spaces. Proyecciones. Journal of Mathematics, 30(3), 369-399. https://doi.org/10.4067/S0716-09172011000300008
Sección
Artículos