Compact Subsets of P<∞(n) with Applications to the Embedding of Symmetric Sequence Spaces into C(α)

Let P<∞(N) be the set of all finite subsets of N, endowed with the product topology. A description of the compact subsets of P<∞(N) is given. Two applications of this result to Banach space theory are shown : (1) a characterization of the symmetric sequence spaces which embed into C(ω), and (2) a characterization, in terms of the Orlicz function M , of the Orlicz sequence spaces hM which embed into C(K) for some countable compact Hausdorff space K. If A is an arbitrary set, denote its power set by P(A). Identifying P(A) with 2, and endowing it with the product topology, yields a compact Hausdorff topological space. The symbols P<∞(A) and P∞(A) stand for the subspaces consisting of all finite, respectively, all infinite subsets of A. In the first part of this paper, we study the compact subsets of P<∞(N). The fruit of this study is applied in the latter part to obtain some results concerning the embedding of symmetric Banach sequence spaces into C(K), where K is a countable compact Hausdorff space. The main result in §1 is Theorem 3, which gives a description of the compact subsets of P<∞(N) in terms of a certain hierarchy of subsets (Afβ) of P<∞(N) (see the definition in §1 below). The motivation for the family (Afβ) comes from the collection of “admissible sets” used in the definition of a classical counterexample (the Schrier space [8]) in Banach space theory. Indeed, if f is the identity function on N, then Af1 is precisely the collection of “admissible sets” used to define the Schrier space. Finite iterations of the construction appears in [7]. It takes a certain amount of care, however, to extend the definition to transfinite ordinals. Careful choice is needed so that we may obtain 1991 Mathematics Subject Classification. 03E10, 03E15, 46B03, 46B45, 46E15, 54G12.