. FA ] 2 9 M ar 1 99 3 TOPOLOGIES ON THE SET OF ALL SUBSPACES OF A BANACH SPACE AND RELATED QUESTIONS OF BANACH SPACE GEOMETRY

For a Banach space X we shall denote the set of all closed subspaces of X by G(X). In some kinds of problems it turned out to be useful to endow G(X) with a topology. The main purpose of the present paper is to survey results on two the most common topologies on G(X). The organization of this paper is as follows. In section 2 we introduce some definitions and notation. In sections 3 and 4 we introduce two topologies on G(X). Section 5 is devoted to the problem of comparison of these topologies. In section 6 we investigate the following general problem: How close should be the structure of the subspaces which are close with respect to the natural metrics, which generate introduced topologies? (It should be mentioned that both introduced topologies are metrizable.) In section 7 we survey those results on introduced topologies and related quantities which were not discussed in previous sections. Here we also try to describe known applications of introduced topologies and related quantities. This section is nothing more than guide to the literature. If x is a vector of a Banach space X and A, D are subsets of X then we shall denote the value inf a∈A ||x − a|| by dist(x, A) and the value inf a∈A dist(a, D) by dist(A, D). The closed unit ball and the unit sphere of a Banach space X are denoted by B(X) and S(X) respectively. For a subset A of a Banach space X by A ⊥ , lin(A), conv(A) and cl(A) we shall denote, respectively, the set {x * ∈ X * : (∀x ∈ A)(x * (x) = 0)}, the set of all finite linear combinations of vectors of A , the set of all convex combinations of vectors of A and the closure of A in the strong topology. For a subset A of a dual Banach space X * we shall denote the set {x ∈ X : (∀x * ∈ A)(x * (x) = 0)} by A ⊤. Let Y and Z be Banach spaces. For 1 ≤ p ≤ ∞ we shall denote by Y ⊕ p Z the Banach space of all pairs (y, z), y ∈ Y, z ∈ Z, with the norm ||(y, z)|| = (||y|| p + ||z|| p) 1/p (or max{||y||, ||z||}, if p = ∞). It is clear that all these norms define …