Symmetric Sequence Subspaces of C(α), II

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.