Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Bibasic sequences and norming basic sequences


Authors: William J. Davis, David W. Dean and Bor Luh Lin
Journal: Trans. Amer. Math. Soc. 176 (1973), 89-102
MSC: Primary 46B15
MathSciNet review: 0313763
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that every infinite dimensional Banach space X contains a basic sequence $ ({x_n})$ having biorthogonal functionals $ ({f_n}) \subset {X^\ast}$ such that $ ({f_n})$ is also basic. If $ [{f_n}]$ norms $ [{x_n}]$ then $ ({f_n})$ is necessarily basic. If $ [{f_n}]$ norms $ [{x_n}]$ then $ [{x_n}]$ norms $ [{f_n}]$. In order that $ [{f_n}]$ norms $ [{x_n}]$ it is necessary and sufficient that the operators $ {S_n}x = \Sigma _1^n{f_i}(x){x_i}$ be uniformly bounded. If $ [{f_n}]$ norms $ [{x_n}]$ then $ {X^\ast}$ has a complemented subspace isomorphic to $ {[{x_n}]^\ast}$. Examples are given to show that $ ({f_n})$ need not be basic and, if $ ({f_n})$ is basic, still $ [{f_n}]$ need not norm $ [{x_n}]$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46B15

Retrieve articles in all journals with MSC: 46B15


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0313763-9
PII: S 0002-9947(1973)0313763-9
Keywords: Basis, Markushevich basis, reflexivity, projection, complemented subspace, pseudo-reflexive space
Article copyright: © Copyright 1973 American Mathematical Society