Weak bases in p-adic spaces
De Grande-De Kimpe, N. ; Kąkol, J. ; Perez-Garcia, C. ; Schikhof, W. H.
Bollettino dell'Unione Matematica Italiana, Tome 5-A (2002), p. 667-676 / Harvested from Biblioteca Digitale Italiana di Matematica
Access to full text
Full (PDF)
Full (DjVu)

We study polar locally convex spaces over a non-archimedean non-trivially valued complete field with a weak topological basis. We prove two completeness theorems and a Hahn-Banach type theorem for locally convex spaces with a weak Schauder basis.

Si studiano spazi polari localmente convessi su un non trivialmente valutato campo completo non archimedeo con una debole base topologica. Dimostriamo due teoremi di completezza e un teorema tipo Hahn-Banach per spazi localmente convessi con una debole base di Schauder.

Publié le : 2002-10-01
@article{BUMI_2002_8_5B_3_667_0,
     author = {N. De Grande-De Kimpe and J. K\k akol and C. Perez-Garcia and W. H. Schikhof},
     title = {Weak bases in $p$-adic spaces},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {5-A},
     year = {2002},
     pages = {667-676},
     zbl = {1072.46051},
     mrnumber = {1934373},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2002_8_5B_3_667_0}
}

De Grande-De Kimpe, N.; Kąkol, J.; Perez-Garcia, C.; Schikhof, W. H. Weak bases in $p$-adic spaces. Bollettino dell'Unione Matematica Italiana, Tome 5-A (2002) pp. 667-676. http://gdmltest.u-ga.fr/item/BUMI_2002_8_5B_3_667_0/

[1] De Grande-De Kimpe, N., On the structure of locally K-convex spaces with a Schauder base, Indag. Math., 34 (1972), 396-406. | MR 320689 | Zbl 0244.46005

[2] De Grande-De Kimpe, N.-Perez-Garcia, C., Weakly closed subspaces and the Hahn-Banach extension property in p-adic analysis, Indag. Math., 91 (1988), 253-261. | MR 964832 | Zbl 0678.46055

[3] De Grande De Kimpe, N.-Kąkol, J.-Perez Garcia, C., Schikhof, W. H., Orthogonal sequences in non-archimedean locally convex spaces, Indag. Math. N.S., 11 (2000), 187-195. | MR 1813159 | Zbl 0980.46056

[4] Dierolf, S., The barrelled space associated with a bornological space need not be bornological, Bull. London Math. Soc., 12 (1980), 60-62. | MR 565486 | Zbl 0402.46002

[5] Gilsdorf, T.-Kąkol, J., On some non-archimedean closed graph theorems, In: Proc 4th Intern. Conf. on p-adic Functional Analysis, Nijmegen, The Netherlands, Marcel Dekker, 192 (1997), 153-159. | MR 1459210 | Zbl 0889.46064

[6] Kalton, N. J., Schauder decomposition and completeness, Bull. London Math. Soc., 2 (1970), 34-36. | MR 259547 | Zbl 0196.13601

[7] Kamthan, P. K.-Gupta, M., Weak Schauder bases and completeness, Proc. Roy. Irish Ac., 78 (1978), 51-54. | MR 506956 | Zbl 0388.46002

[8] Kąkol, J., The weak basis theorem for K-Banach spaces, Bull. Soc. Math. Belg., 45 (1993), 1-4. | MR 1314928 | Zbl 0728.46003

[9] Kąkol, J.-Gilsdorf, T., On the weak basis theorems for p-adic locally convex spaces, In: Proc. 5th Intern. Conf. on p-adic Functional Analysis, Poznań, Poland, Marcel Dekker, 207 (1999), 149-167. | MR 1702053 | Zbl 0949.46039

[10] Perez-Garcia, C.-Schikhof, W. H., p-Adic barrelledness and spaces of countable type, Indian J. Pure Appl. Math., 29 (1998), 1099-1109. | MR 1658689 | Zbl 0926.46067

[11] Perez-Garcia, C.-Schikhof, W. H., The Orlicz-Pettis property in p-adic analysis, Collect. Math., 43 (1992), 225-233. | MR 1252732 | Zbl 0788.46078

[12] Van Rooij, A. C. M., Non-archimedean functional analysis, Marcel Dekker, New York (1978). | MR 512894 | Zbl 0396.46061

[13] Schikhof, W. H., Locally convex spaces over nonspherically complete valued fields, Bull. Soc. Math. Belg., 38 (1986), 187-224. | MR 871313 | Zbl 0615.46071

[14] Śliwa, W., Every infinite-dimensional non-archimedean Fréchet space has an orthogonal basic sequence (to appear in Indag. Math.). | Zbl 0980.46057

[15] Van Tiel, J., Espaces localement K-convexes, Indag. Math, 27 (1965), 249-289. | MR 179593 | Zbl 0133.06502

[16] Webb, J. H., Schauder decompositions in locally convex spaces, Camb. Phil. Soc., 76 (1974), 145-152. | MR 350370 | Zbl 0283.46003