Vol. 41, No. 2, 1972

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 294: 1
Vol. 293: 1  2
Vol. 292: 1  2
Vol. 291: 1  2
Vol. 290: 1  2
Vol. 289: 1  2
Vol. 288: 1  2
Vol. 287: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
Super-reflexive spaces with bases

Robert Clarke James

Vol. 41 (1972), No. 2, 409–419
Abstract

Super-reflexivity is defined in such a way that all superreflexive Banach spaces are reflexive and a Banach space is super-reflexive if it is isomorphic to a Banach space that is either uniformly convex or uniformly non-square. It is shown that, if 0 < 2ϕ < 𝜖 1 < Φ and B is super-reflexive, then there are numbers r and s for which 1 < r < , 1 < s < and, if {ei} is any normalized basic sequence in B with characteristic not less than 𝜖, then

  ∑            ∑           ∑
ϕ[   |ai|r]1∕r ≦ ∥  aiei∥ ≦ Φ[  |ai|s]1∕s,

for all numbers {ai} such that Σaiei is convergent. This also is true for unconditional basic subsets in nonseparable super-reflexive Banach spaces. Gurariǐ and Gurariǐ recently established the existence of ϕ and r for uniformly smooth spaces, and the existence of Φ and s for uniformly convex spaces [Izv. Akad. Nauk SSSR Ser. Mat., 35 (1971), 210-215].

Mathematical Subject Classification 2000
Primary: 46B15
Milestones
Received: 20 July 1971
Published: 1 May 1972
Authors
Robert Clarke James