Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-27T16:09:36.174Z Has data issue: false hasContentIssue false
  • English
  • Français

Necessary and Sufficient Conditions for the Equality of L(f) and l1

Published online by Cambridge University Press:  20 November 2018

Waleed Deeb
Waleed Deeb*
Affiliation:
University of Jordan, Amman, Jordan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Introduction. Let f be a modulus, ei = (δij) and E = {ei, i = 1, 2, …}. The L(f) spaces were created (to the best of our knowledge) by W. Ruckle in [2] in order to construct an example to answer a question of A. Wilansky. It turned out that these spaces are interesting spaces. For example lp, 0 < p ≦ 1 is an L(f) space with f(x) = xp, and every FK space contains an L(f) space [2]. A natural question is: For which f is L(f) a locally convex space? It is known that L(f) ⊆ l1, for all f modulus (see [2]), and l1 is the smallest locally convex FK space in which E is bounded (see [1]). Thus the question becomes: For which f does L(f) equal l1? In this paper we characterize such f. (An FK space need not be locally convex here.) We also characterize those f for which L(f) contains a convex ball. The final result of this paper is to show that if f satisfies f(x · y) ≦ f(x) · f(y) and L(f) ≠ l1 then L(f) contains no infinite dimensional subspace isomorphic to a Banach space.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 2 , 01 April 1982 , pp. 406 - 410
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Bennett, G., Some inclusion theorems for sequence spaces, Pacific J. Math. 46 (1973), 17–30.Google Scholar
2. Ruckle, W., FK spaces in which the sequence of coordinate vectors is bounded. Can. J. Math. 25 (1973), 973–978.Google Scholar
3. Stiles, W., On properties of sub spaces of lp, 0 < p < 1, Trans. Amer. Math. Soc. 149 (1970), 405–415.Google Scholar
4. Wilansky, A., Functional analysis (Blaisdell, New York, 1964).Google Scholar