Metrizable weak barrelledness and dimension

• Autorzy: Saxon, S. , Sanchez Ruiz, L.
• Języki publikacji: EN

Abstrakty

EN

The least infinite-dimensionality for Frechet spaces is c (Mazur), for metrizable barrelled spaces, b (Saxon and Sanchez Ruiz, 1996). For metrizable spaces with the yet weaker inductive property, it is the dimension Aleph[1] of the space chi spanned by any Aleph[1] scalar sequences of the form [(1, x, x^2, x^3, . . .)]. (A locally convex space is inductive if it is the inductive limit of each increasing covering sequence of subspaces). Indeed, chi is a non-barrelled subspace of the Frechet space omega, where the fundamental theorem of algebra at once proves density, dimension and inductivity. Moreover, if each |x| < 1, geometric series then put chi inside the Banach space [l^1], where the identity principle similarly proves the normable case.

Słowa kluczowe

PL

przestrzeń indukcyjna i pierwotna; przestrzeń metryzowalna; szereg geometryczny i potęgowy; zasadnicze twierdzenie algebry

EN

c[0] and [l^p]; fundamental theorem of algebra; geometric and power series; inductive and primitive spaces; least infinite-dimensionality; metrizable; omega; S[sigma]

• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2001
• Tom: Vol. 49, no 2
• Strony: 97--101
• Opis fizyczny: Bibliogr. 15 poz.

Twórcy

autor

Saxon, S.

• Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, FL 32611-8105, U.S.A.,

autor

Sanchez Ruiz, L.

• Euiti-Departamento de Matematica Aplicada, Universidad Politecnica de Valencia, E-46022 Valencia, Spain ,

Bibliografia

• [1] J. C. Ferrando, L. M. Sánchez Ruiz, On sequential barrelledness, Arch. Math. (Basel), 57 (1991) 597-605.
• [2] P. Perez Carreras, J. Bonet, Barrelled Locally Convex Spaces, Math. Studies 131, North-Holland, 1987.
• [3] L. M. Sánchez Ruiz, S. A. Saxon, Weak barrelledness conditions, in: Functional Analysis with Current Applications in Science, Technology and Industry, eds.: M. Brokate, A. H. Siddiqi, Pitman RNMS 377, Longman, (1998) 20-36.
• [4] S. A. Saxon, Metrizable barrelled countable enlargements, Bull. London Math, Soc., 31 (1999) 711-718.
• [5] S. A. Saxon, P. P. Narayanaswami, Metrizable [normable] (LF)-spaces and two classical problems in Fréchet [Banach] spaces, Studia Math., 93 (1989) 1-16.
• [6] S. A. Saxon, L. M. Sánchez Ruiz, Optimal cardinals for metrizable barrelled spaces, J. London Math. Soc., 51 (1995) 137-147.
• [7] S. A. Saxon, L. M. Sánchez Ruiz, Barrelled countable enlargements and the bounding cardinal, J. London Math. Soc., 53 (1996) 158-166.
• [8] S. A. Saxon, L. M. Sánchez Ruiz, Dual local completeness, Proc. Amer. Math. Soc., 125 (1997) 1063-1070.
• [9] S. A. Saxon, L. M. Sánchez Ruiz, Mackey weak barrelledness, Proc. Amer. Math. Soc., 126 (1998) 3279-3282.
• [10] S. A. Saxon, L. M. Sánchez Ruiz, Reinventing weak barrelledness, preprint.
• [11] S. A. Saxon, L. M. Sánchez Ruiz, Three-space weak barrelledness, submitted.
• [12] S. A. Saxon, L. M. Sánchez Ruiz, I. Tweddle, Countably enlarging weak barrelledness, Note Mat., 17 (1997) 217-233.
• [13] S. A. Saxon, I. Tweddle, Mackey Rp-barrelled spaces, Adv. Math., 145 (1999) 230-238.
• [14] S. A. Saxon, I. Tweddle, The fit and flat components of barrelled spaces, Bull. Austral. Math. Soc., 51 (1995) 521-528.
• [15] S. A. Saxon, A. Wilansky, The equivalence of some Banach space problems, Colloq. Math., 37 (1977) 217-226.