Positivity 7: 347–353, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
Two Characterizations of the Standard Unit Vector
Basis of l
University of Texas at Austin, Austin, TX, USA
Received 26 September 2000; accepted 1 July 2002
Abstract. We show that for a sequence in a Banach space, the property of being stable under large
perturbations characterizes the property of being equivalent to the unit vector basis of l
that a normalized unconditional basic sequence in l
that is semi-normalized in l
is equivalent to
the standard unit vector basis of l
2000 Mathematics Subject Classiﬁcation: 46B45, 46B15.
Key words: Banach space, unconditional basic sequence, unit vector basis of l
In this note, we present two results concerning the standard unit vector basis of l
The ﬁrst result, Theorem 1, characterizes, up to equivalence, the standard unit vec-
tor basis of l
as the unique sequence satisfying a certain stability property. This
result is a generalization of [6, Theorem II.11.5]. Our second result, Theorem 2,
characterizes, among the sequences in l
that are semi-normalized in both l
, those that are unconditionally basic as simply those that are equivalent to the
standard unit vector basis of l
. In contrast to Theorem 2, we present Proposition 1,
which shows that not every copy of the standard unit vector basis of l
semi-normalized in l
Throughout this note, we shall use the following notations: X and Y denote
inﬁnite dimensional, real Banach spaces; the symbol I
denotes the formal identity
operator from l
; we denote the standard unit vector basis of l
ﬁnally, we denote the i
-coordinate of a sequence of real numbers x by x(i).All
other notation and terminology, not otherwise explained, are as in .
We now recall the deﬁnitions pertinent to the foregoing discussion.
DEFINITION 1. The unconditional basis constant of an unconditional basic se-
in X is the smallest constant K
satisfying, for any sequence of
This work is a subset of the results contained in the author’s dissertation, Sequences that are
unconditionally basic in both l
, which was written under the direction of Professor Maria