Symmetric Sequence Subspaces of C(α)

Let α be an ordinal. Then α­1, the set of all ordinals equal to or preceding α, is a compact Hausdorff topological space. The space of all real-valued continuous functions on α­1 is commonly denoted by C(α). In this paper, we study the symmetric sequence spaces which are isomorphic to subspaces of C(α) for some countable ordinal α. In the next section, we prove a general result which shows that certain symmetric sequence spaces can be embedded into C(ω). In particular, the result is applicable to every Marcinkiewicz sequence space and every Orlicz sequence space h M such that lim t!! M(ηt)}M(t) ̄ 0 for some η" 0. The remaining sections are devoted to proving the converses of the results in §1. First, it is shown that the problem can be reduced to analysing certain collections of finite subsets of .. To describe these collections of sets, we introduce a hierarchy which is largely similar to the Schreier families that have been considered by many authors [1–4, 13]. With the help of these families, we obtain a characterisation of the symmetric Banach sequence spaces which embed into C(ω) (Theorems 1 and 15), and a characterisation, in terms of the Orlicz function M, of the Orlicz sequence spaces h M which embed into C(α) for some countable ordinal α (Theorem 18). For terms and notation concerning ordinal numbers and general topology, we refer to [5]. The first infinite ordinal, respectively the first uncountable ordinal, is denoted by ω, respectively ω " . Any ordinal is either 0, a successor, or a limit. If α is a successor ordinal, denote its immediate predecessor by α®1. Following common practice, an ordinal α is identified with the set 2β :β!α ́ of all its predecessors. An ordinal is a Hausdorff topological space when endowed with the other topology. It is compact if and only if it is not a limit ordinal. 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 s f s ̄ sup t`K r f(t)r. In particular, if α is an ordinal, we write C(α) for the space of all continuous real-valued functions on the compact Hausdorff space α­1. (The notation is inconsistent, but commonly used.) Detailed discussions of such spaces can be found in [14]. It is worth pointing out that every countable compact Hausdorff space is homeomorphic to some countable compact ordinal [11]. Thus