Wee-Kee Tang

The Bourgain ℓ1-index of mixed Tsirelson space

Abstract Suppose that ( F n ) n=0 ∞ is a sequence of regular families of finite subsets of N such that F 0 contains all singletons, and (θn)n=1∞ is a nonincreasing null sequence in (0,1). The mixed Tsirelson space T( F 0 ,(θ n , F n ) n=1 ∞ ) is the completion of c00 with respect to the implicitly defined norm ||x||= max ||x|| F 0 , sup n∈ N sup θ n ∑ i=1 k ||E i x|| , where ||x|| F 0 = sup F∈ F 0 ||Fx|| l 1 and the last supremum is taken over all sequences (Ei)i=1k in [ N ] such that max E i min E i+1 and { min E i :1⩽i⩽k}∈ F n . In this paper, we compute the Bourgain l1-index of the space T( F 0 ,(θ n , F n ) n=1 ∞ ) . As a consequence, it is shown that if η is a countable ordinal not of the form ωξ for some limit ordinal ξ, then there is a Banach space whose l1-index is ωη.

Uniformly differentiable bump functions

We present a construction of uniformly smooth norms from uniformly smooth bumb functions without making use of the Implicit Function Theorem.

On the extension of rotund norms

We constructed an extension of norms from a closed subspace of a Banach space to the whole space that preserves various types of rotundity possessed by the subspace norms. We also constructed a strictly convex norm such that a prescribed set of points lies on the unit sphere of this norm.

The ℓ¹-indices of Tsirelson type spaces

If a and β are countable ordinals such that β ¬= 0, denote by T α,β the completion of c 00 with respect to the implicitly defined norm ||x|| = max{||x||S α , 1/2 sup Σ ||E i x||}, where the supremum is taken over all finite subsets E 1 ,...,E j of N such that E 1 a = ω α1 .m 1 + … +ω αn .m n in Cantor normal form and α n is not a limit ordinal, then there exists a Banach space whose l 1 -index is ω α .

Semilattice structures of spreading models

Given a Banach space X, denote by SP w (X) the set of equivalence classes of spreading models of X generated by normalized weakly null sequences in X. It is known that SP w (X) is a semilattice, i.e., it is a partially ordered set in which every pair of elements has a least upper bound. We show that every countable semilattice that does not contain an infinite increasing sequence is order isomorphic to SP w (X) for some separable Banach space X.

Symmetric sequence subspaces of

If α is an ordinal, then the space of all ordinals less than or equal to α is a compact Hausdorff space when endowed with the order topology. Let C(α) be the space of all continuous real-valued functions defined on the ordinal interval [0, α]. We characterize the symmetric sequence spaces which embed into C(α) for some countable ordinal α. A hierarchy (Eα) of symmetric sequence spaces is constructed so that, for each countable ordinal α, Eα embeds into C(ωω α ), but does not embed into C(ωω β ) for any β < α. Let α be an ordinal. The ordinal interval [0, α] is a compact Hausdorff space in the order topology. The space of all continuous real-valued functions on [0, α] is commonly denoted by C(α). In [4], the symmetric sequence spaces which embed into C(ω) are characterized. This paper, which is a continuation of [4], gives a characterization of the symmetric sequence spaces which embed into C(α) for some countable ordinal α. In [4], it is shown that any Orlicz sequence space which embeds into C(α) for some countable ordinal α already embeds into C(ω). Here, we construct a hierarchy of symmetric sequence spaces (Eα)α<ω1 such that, for each countable ordinal α, Eα embeds into C(ω ωα), but does not embed into C(ω β ) for any β < α. Since, according to Bessaga and Pełczynski [2], if α < β are countable infinite ordinals, then C(α) and C(β) are isomorphic if and only if β < α, (Eα) is a full hierarchy of mutually non-isomorphic symmetric sequence spaces which embed into C(α) for some countable ordinal α. The authors thank the referee for pointing out some errors in an earlier version of the paper, and for various suggestions for improving the exposition. For terms and notation concerning ordinal numbers and general topology, we refer to [3]. The first infinite ordinal, respectively, the first uncountable ordinal, is denoted by ω, respectively, ω1. Any ordinal is either 0, a successor, or a limit. If α is a successor ordinal, denote its immediate predecessor by α − 1. If K is a compact Hausdorff space, C(K) denotes the space of all continuous real-valued functions on K. It is a Banach space under the norm ‖ f ‖ = supt∈K | f (t)|. If K is a topological space, its derived set K (1) is the set of all of its limit points. A transfinite sequence of derived sets may be defined as follows. Let K(0) = K. If α is an ordinal, let K(α+1) = (K(α))(1). Finally, for a limit ordinal α, we define K(α) = ⋂ β<α K (β). The cardinality of a set A is denoted by |A|. By P∞(N), respectively, P<∞(N), we mean the collection of all infinite, respectively, finite, subsets of N. These are subsets of 2, and consequently inherit the product topology. If A and B are nonempty subsets of N, we say that A < B if max A < min B. We also allow that ∅ < A and A < ∅ for any A ⊆ N. We follow standard Banach space terminology, as may be found in the book [5]. We say that a Banach space is a sequence space if it is a vector subspace of the space of all real sequences. Such is the case, for instance, when a Banach space E has a (Schauder) basis (ek), Received by the editors December 15, 1997; revised July 16, 1998. AMS subject classification: 03E13, 03E15, 46B03, 46B45, 46E15, 54G12. c ©Canadian Mathematical Society 1999.

C(\alpha)

Approximation by smooth convex functions and questions on the Smooth Variational Principle for a given convex function f on a Banach space are studied in connection with majorising f by C 1 -smooth functions.