Weak * Sequential Closures in Banach Space Theory and Their Applications

Let X be a (real or complex) Banach space, its dual Banach space will be denoted by X *. We use standard notation and terminology of Banach space theory, see J.Lindenstrauss and L.Tzafriri [LT]. By a subspace we mean a linear, but not necessarily closed, subspace. We also assume some knowledge of general topology and ordinal numbers, see P.S.Aleksandrov[A]. Definition 1.1. Let A be a subset of X *. The set of all limits of weak *-convergent sequences in A is called the weak * sequential closure of A and is denoted by A (1). S.Banach asked the following question (see [Maz]). Question. Let X be separable Banach space and A be a subspace of X *. Whether (A (1)) (1) = A (1) ? This question was answered in negative by S.Mazurkiewicz [Maz]. The result of S.Mazurkiewicz makes it natural to introduce the following definition. (It was done by S.Banach [B2, p. 208, 213]. S.Banach used the term " dérivé faible " .) Definition 1.2. For an ordinal α > 1 the weak * sequential closure of order α of A is the set A (α) = β<α (A (β)) (1). Weak * sequential closures were studied by S.Banach in his book [B2] (see, also, [B3] and [B4]). He proved the following results. Let X be a separable Banach space and let A be a subspace in X *. Then A = A (1) if and only if for every f ∈ X * \A there exists x ∈ X such that f (x) = 1 but a(x) = 0 for every a ∈ A. In modern terminology this result can be stated as: a subspace in the dual of a separable Banach space is weak * closed if and only if it is weak * sequentially closed.