Total subspaces in dual Banach spaces which are not norming over any infinite dimensional subspace

Total subspaces in dual Banach spaces which are not norming over any infinite dimensional subspace Abstract. The main result: the dual of separable Banach space X contains a total subspace which is not norming over any infinite dimensional subspace of X if and only if X has a nonquasireflexive quotient space with the strictly singular quotient mapping. Let X be a Banach space and X * be its dual space. Let us recall some basic definitions. A subspace M of X * is said to be total if for every 0 = x ∈ X there is an f ∈ M such that f (x) = 0. A subspace M of X * is said to be norming over a subspace L ⊂ X if for some c > 0 we have (∀x ∈ L)(sup f ∈S(M) |f (x)| ≥ c||x||), where S(M) is the unit sphere of M. If L = X then M is called norming. The following natural questions arise: 1) How far could total subspaces be from norming ones? (Of course, there are many different concretizations of this question.) 2) What is the structure of Banach spaces, whose duals contain total " very " non-norming subspaces? 3) What is the structure of total subspaces? These questions was studied by many authors: The present paper is devoted to the following natural class of subspaces which are far from norming ones. A subspace M of X * is said to be nowhere norming if it is not norming over every infinite dimensional subspace of X. If X is such that X * contains a total nowhere norming subspace then we shall write X ∈ T NNS. This class was introduced by W.J.Davis and W.B.Johnson in [DJ], where the first example of a total nowhere norming subspace was constructed. In the same paper it was noted that J.C.Daneman proved that every infinite dimensional subspace of l 1 is norming over some infinite dimensional subspace of c 0. In [O2] a class of the spaces with T NNS property was discovered. A.A.Albanese [Al] proved that the spaces of type C(K) are not in T NNS. The problem of description of Banach spaces with T NNS property arises in a natural way. Our main result (Theorem 2.1) states that for a separable Banach space X we have X ∈ T NNS if and only if for some nonquasireflexive Banach space Y …