Positivity 5: 75–94, 2001.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
Finite Representability of
in Quotients of
and NICOLE TOMCZAK-JAEGERMANN
CVUT, Feld, Department of Mathematics, Technická 2, Prague, 166 27, Czech Republic.
Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G
2G1. E-mail: firstname.lastname@example.org
(Accepted 15 November 1999)
Abstract. A typical result of the paper states that if X is a Banach space with a basis and for some
1 p q ∞, the spaces
are ﬁnitely block representable in every block subspace
of X, then every block subspace of X admits a block quotient Z such that for every r ∈[p, q],
is ﬁnitely block representable in Z. Results of a similar nature are also established for
-block-sequences and asymptotic spaces.
Mathematics Subject Classiﬁcation (2000): 46B20, 46B07
Questions on what kind of ‘nice’ linear substructure must exist in every Banach
space are very natural and have been studied in many forms. The classical can-
didates for a nice linear substructure are the spaces
for 1 p<∞ and c
Depending on whether one asks about ﬁnite- or inﬁnite-dimensional spaces
answers are very different.
In this paper we will be interested in essentially ﬁnite-dimensional phenom-
ena that appear in a certain stabilized manner. We shall concentrate on general
problems of the following type: For which values of 1 p ∞ does a given
Banach space X with a basis contain copies of
on blocks for every N? One of
fundamental theorems of the local theory of Banach spaces addresses this question:
Krivine’s theorem () states that some
(with 1 p<∞) or c
is ﬁnitely block
representable in X. It also implies that the space
is ‘present’ in X in other ways:
-block-sequences (which is a stronger version of ﬁnite block representability,
see below), and as asymptotic spaces. Among these notions only asymptotic spaces
are relatively recent. They are directly connected to a new asymptotic point of
During the work on this paper, this author held a Post Doctoral position at the University of
Texas at Austin. He was also a Young Investigator at the NSF workshop in Linear Analysis and
Probability at Texas A&M University in August 1997.
This author held Canada Council Killam Research Fellowship in 1997/99.