TOTAL SUBSPACES WITH LONG CHAINS OF NOWHERE NORMING WEAK SEQUENTIAL CLOSURES M.I.Ostrovskii

If a separable Banach space X is such that for some nonquasireflexive Banach space Y there exists a surjective strictly singular operator T : X → Y then for every countable ordinal α the dual of X contains a subspace whose weak∗ sequential closures of orders less than α are not norming over any infinite-dimensional subspace of X and whose weak∗ sequential closure of order α+ 1 coincides with X∗ Let X be a Banach space, X be its dual space. The closed unit ball and the unit sphere of X are denoted by B(X) and S(X) respectively. The term “operator” means a bounded linear operator. Let us recall some definitions. A subspace M of X is said to be total if for every 0 6= x ∈ X there is an f ∈ M such that f(x) 6= 0. A subspace M of X is said to be norming over a subspace L of X if for some c > 0 we have (∀x ∈ L)( sup f∈S(M) |f(x)| ≥ c||x||). A subspace M of X is said to be norming if it is norming over X . If M is not norming over any infinite dimensional subspace of X then we shall say that M is nowhere norming. The set of all limits of weak convergent sequences in a subset M of X is called the weak sequential closure of M and is denoted by M(1). If M is a subspace then M(1) is also a subspace. This subspace need not be closed and all the more need not be weak closed [M]. In this connection S.Banach introduced [B, p. 208, 213] the weak sequential closures (S.Banach used the term ”dérivé faible”) of other orders, including transfinite ones. For ordinal α the weak sequential closure of order α of a subset M of X is the set M(α) = ∪β<α(M(β))(1). It should be noted that for separableX the notion of the weak sequential closure of order α coincides with the notion of the derived set of order α considered in [A], [M1], [M2]. For the chain of the weak sequential closures we have M(1) ⊂ M(2) ⊂ . . . ⊂ M(α) ⊂ M(α+1) ⊂ . . . . If we have M(α) = M(α+1) then all subsequent closures coincide with M(α). The least ordinal α for which M(α) = M(α+1) is called the order of M . 1991 Mathematics Subject Classification. Primary 46B20.