It is shown that every Banach space with a norming Markushevich basis has a strong Markushevich basis. It is also shown that the converse is false.

One of the main open problems in the theory of Asplund spaces is whether every Asplund space admits a Fréchet differentiable bump function. This problem is also open for C(K) Asplund spaces, where it is unknown even for C∞-Fréchet smooth bump (a general Asplund space does not always admit C2-Fréchet smooth bump – it suffices to consider ℓ3/2[DGZ2]).

It is shown that Lipschitz quotient mappings between finite dimensional spaces behave nicely (e.g., are bijective in the case of equal dimensions) if the Lipschitz and co-Lipschitz constants are close to each other. For Lipschitz quotient mappings of the plane, a bound for the cardinality of the pre-image of a point in terms of the ratio of the constants is obtained.

The main purpose of this work is to study and apply generalized contact distributions of (inhomogeneous) Boolean models Z with values in the extended convex ring. Given a convex body L ⊂ ℝd and a gauge body B ⊂ ℝd , such a generalized contact distribution is the conditional distribution of the random vector (d B (L,Z),u B (L,Z),p B (L,Z),l B (L,Z)) given that Z∩L = ∅, where Z is a Boolean model, d B (L,Z) is the distance of L from Z with respect to B, p B (L,Z) is the boundary point in L realizing this distance (if it exists uniquely), u B (L,Z) is the corresponding boundary point of B (if it exists uniquely) and l B (L,·) may be taken from a large class of locally defined functionals. In particular, we pursue the question of the extent to which the spatial density and the grain distribution underlying an inhomogeneous Boolean model Z are determined by the generalized contact distributions of Z.

It is shown that it is possible to extend α Hölder maps from subsets of Lp to Lq (1 < p, q ≤ 2) isometrically if and only if α≤p/q*, and isomorphically if and only if α≤p/2. It is also proved that the set of αs which allow an isomorphic extension for α Hölder maps from subsets of X to Y is monotone when Y is a dual Banach space. Finally, the isometric and isomorphic extension problems for Hölder functions between Lp and Lq is studied for general p, q ≥ 1, and a question posed by K. Ball is solved by showing that it is not true that all Lipschitz maps from subsets of Hilbert space into normed spaces extend to the whole of Hilbert space.

Let ℳ be the collection of all intersections of balls, considered as a subset of the hyperspace ℳ of all closed, convex and bounded sets of a Banach space, furnished with the Hausdorff metric. It is proved that ℳ is uniformly very porous if and only if the space fails the Mazur intersection property.

In this paper, it is proved that, for any m unit vectors. x1…, xm in any n-dimensional real Hilbert space, there exists a unit vector x0 such that

for any y∈Sn−1. The exact value of the above integral is calculated, and these results used to improve some lower bounds for multilinear forms on real Hilbert spaces. An integral expression is also given for the complex case.

In [4], we investigated the spaces of continuous functions on countable products of compact Hausdorff spaces. Our main object here is to extend the discussion to arbitrary products of compact Hausdorff spaces. We prove the following theorems in Section 3.

Given a Banach space X and a norming subspace Z⊂X*, a geometrical method is introduced to characterize the existence of an equivalent σ(X, Z)-lsc LUR norm on X. A new simple proof of the Theorem of Troyanski: every rotund space with a Kadec norm is LUR renormable, and a generalization of the Moltó, Orihuela and Troyanski characterization of the LUR renormability, are provided without probability arguments. Among other applications, it is shown that a dual Banach space with a w*-Kadec norm admits a dual LUR norm.

Let X be a Hausdorff topological space and let ρ be a metric on it, not necessarily related to the topology. The space X is said to be fragmented by the metric ρ if each nonempty set in X has nonempty relatively open subsets of arbitrary small ρ-diameter. This concept was introduced by Jayne and Rogers (see [2]) while they studied the existence of Borel selections for upper semicontinuous set-valued maps.

Let X be a Banach space and Y its closed subspace having property U in X. We use a net (Aα) of continuous linear operators on X such that ‖ Aα ‖ ≤ 1, Aα (X) ⊂ Y for all α, and limαg(Aαy) = g(y), y ∈ Y, g∈Y* to obtain equivalent conditions for Y to be an HB-subspace, u-ideal or h-ideal of X. Some equivalent renormings of c0 and l2 are shown to provide examples of spaces X for which K(X) has property U in L(X) without being an HB-subspace. Considering a generalization of the Godun set [3], we establish some relations between Godun sets of Banach spaces and related operator spaces. This enables us to prove e.g., that if K(X) is an HB-subspace of L(X), then X is an HB-subspace of X**—the result conjectured to be true by Å. Lima [9].

This paper is a contribution to the general problem of differentiability of Lipschitz functions between Banach spaces. We establish here a result concerning the existence of derivatives which are in some sense between the notions of Gâteaux and Frechet differentiability.

We show that if the derivative of a convex function on c0 is locally uniformly continuous, then every point x ∈ c0, has a neighbourhood O such that f′(O) is relatively compact in ℓ1.

We present a new characterization of σ-fragmentability and illustrate its usefulness by proving some results relating analyticity and crfragmentability. We show, for instance, that a Banach space with the weak topology is σ-fragmented if, and only if, it is almost Čech-analytic and that an almost Čech-analytic topological space is σ-fragmented by a lower-semicontinuous metric if, and only if, each compact subset of the space is fragmented by the metric.

Many positive results are known to hold for the class of Banach spaces known as Asplund spaces and it was for a time conjectured that Asplund spaces should admit equivalent norms with good smoothness and strict convexity properties. A counterexample to these conjectures, in the form of a space of continuous real-valued functions on a suitably chosen tree, was presented in [5]. In this paper we show that the bad behaviour of that example is shared by a wider class of Banach spaces, associated with a wider class of trees. The immediate aim of this extension of the original result is to answer a question posed by Deville and Godefroy [3]. They introduced and studied a subclass of Asplund spaces, those with Corson compact bidual balls, and asked whether this additional assumption is enough to guarantee the existence of nice renormings. We show that it is not.

It is proved that C(K) has no equivalent uniformly Gâteaux differentiable norm (UGD) when K-is an uncountable separable scattered com¬pact space. This result is applied to obtain an example of scattered compact K such that K′″ = ∅ and C(K) has no UGD renorming.

We shall study the properties of typical n-dimensional subspaces of , or equivalently, -typical n-dimensional quotients of , where the meaning of what is typical and what is not is defined in terms of the Haar measure μn,N on the Grassmann manifold Gn,N of all n-dimensional subspaces of .

We prove theorems relating descriptive set theory to nonreflexive Banach spaces. In Theorems 1, 2, and 3 X denotes a Banach space that is separable, but is not reflexive. JX denotes the cannonical embedding of X in X**.

Norms with moduli of smoothness of power type are constructed on spaces with the Radon-Nikodym property that admit pointwise Lipschitz bump functions with pointwise moduli of smoothness of power type. It is shown that no norms with pointwise moduli of rotundity of power type can exist on nonsuperreflexive spaces. A new smoothness characterization of spaces isomorphic to Hilbert spaces is given.

§1. Introduction. Let X be a Hausdorff space and let ρ be a metric, not necessarily related to the topology of X. The space X is said to be fragmented by the metric ρ if each non-empty set in X has non-empty relatively open subsets of arbitrarily small ρ-diameter. The space X is said to be a σ-fragmented by the metric ρ if, for each ε>0, it is possible to write

where each set Xi, i≥1, has the property that each non-empty subset of Xi, has a non-empty relatively open subset of ρ-diameter less than ε. If is any family of subsets of X, we say that X is σ-fragmented by the metric ρ, using sets from, if, for each ε>0, the sets Xi, i ≥ 1, in (1.1) can be taken from