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
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].
![∑ ∑ ∑
ϕ[ |ai|r]1∕r ≦ ∥ aiei∥ ≦ Φ[ |ai|s]1∕s,](a150x.png)
