Linear structure of sets of divergent sequences and series

Thumbnail Image
Full text at PDC
Publication Date
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Elsevier Science Inc
Google Scholar
Research Projects
Organizational Units
Journal Issue
We show that there exist infinite dimensional spaces of series, every non-zero element of which, enjoys certain pathological property. Some of these properties consist on being (i) conditional convergent, (ii) divergent, or (iii) being a subspace of l(infinity) of divergent series. We also show that the space 1(1)(omega)(X) of all weakly unconditionally Cauchy series in X has an infinite dimensional vector space of non-weakly convergent series, and that the set of unconditionally convergent series on X contains a vector space E, of infinite dimension, so that if x is an element of E \ {0} then Sigma(i) parallel to x(i)parallel to = infinity.
Unesco subjects
R. Aron, V. Gurariy, J.B. Seoane-Sepúlveda, Lineability and spaceability of sets of functions on R, Proc. Amer. Math.Soc. 133 (2005) 795–803. V. Bessaga, A. Pełczy´nski, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958) 151–164. J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Mathematics, Springer-Verlag, New York, 1984. A. Dvoretzky, C.A. Rogers, Absolute and unconditional convergence in normed linear spaces, Proc. Natl. Acad. Sci.USA 36 (1950) 192–197. V. Fonf, V. Gurariy, V. Kadec, An infinite dimensional subspace of C[0, 1] consisting of nowhere differentiable functions, C. R. Acad. Bulgare Sci. 52 (1999) 11–12, 13–16. C.W. McArthur, On relationships amongst certain spaces of sequences in an arbitrary Banach space, Canad. J. Math.8 (1956) 192–197. L. Rodrıguez-Piazza, Every separable Banach space is isometric to a space of continuous nowhere differentiable functions, Proc. AMS 123 (12) (1995) 3649–3654.