On the vector space ℓ∞ of bounded scalar sequences {{nowrap|{
xj }
j∈
N}}, let
Pn denote the
linear operator which sets to zero all coordinates
xj of
x for which
j ≤
n. A finite sequence \{x_n\}_{n=1}^N of vectors in ℓ∞ is called
block-disjoint if there are natural numbers \textstyle \{a_n, b_n\}_{n=1}^N so that a_1 \leq b_1 , and so that (x_n)_i=0 when i or i>b_n, for each
n from 1 to
N. The
unit ball B∞ of ℓ∞ is
compact and
metrizable for the topology of
pointwise convergence (the
product topology). The crucial step in the Tsirelson construction is to let
K be the
smallest pointwise closed subset of
B∞ satisfying the following two properties: :
a. For every integer
j in
N, the
unit vector ej and all multiples \lambda e_j, for |λ| ≤ 1, belong to
K. :
b. For any integer
N ≥ 1, if \textstyle (x_1,\ldots,x_N) is a block-disjoint sequence in
K, then \textstyle{{1\over2}P_N(x_1 + \cdots + x_N)} belongs to
K. This set
K satisfies the following stability property: :
c. Together with every element
x of
K, the set
K contains all vectors
y in ℓ∞ such that |
y| ≤ |
x| (for the pointwise comparison). It is then shown that
K is actually a subset of
c0, the Banach subspace of ℓ∞ consisting of scalar sequences tending to zero at infinity. This is done by proving that :
d: for every element
x in
K, there exists an integer
n such that 2
Pn(
x) belongs to
K, and iterating this fact. Since
K is pointwise compact and contained in
c0, it is
weakly compact in
c0. Let
V be the closed
convex hull of
K in
c0. It is also a weakly compact set in
c0. It is shown that
V satisfies
b,
c and
d. The Tsirelson space
T* is the Banach space whose
unit ball is
V. The unit vector basis is an
unconditional basis for
T* and
T* is reflexive. Therefore,
T* does not contain an isomorphic copy of
c0. The other
ℓ p spaces, 1 ≤
p < ∞, are ruled out by condition
b. == Properties ==