Summary: On Banach Spaces containing lp or c0
George Androulakis Nigel Kalton
February 20, 2009
We use the Gowers block Ramsey theorem to characterize Banach spaces
containing isomorphs of p (for some 1 p < ) or c0.
A result of Zippin [Z] gives a characterization of the unit vector basis of c0 and lp.
He showed that a normalized basis of a Banach space such that all normalized block
bases are equivalent, must be equivalent to the unit vector basis of c0 or lp for some
1 p < . Let 1 p A Banach space X with a basis (xi)i is called asymptotic-
lp (asymptotic-c0 if p = ) [M-TJ] if there exists K > 0 and an increasing function
f : N N such that, for all n, if (yi)n
i=1 is a normalized block basis of (xi)
i=1 is equivalent to the unit vector basis of ln
p . In [F-F-K-R] Figiel, Frankiewicz,
Komorowski and Ryll-Nardzewski gave necessary and sufficient conditions for finding