Abstract. We show that a function f : R → R of bounded variation satisfieswhere M f is the centered Hardy-Littlewood maximal function of f . Consequently, the operatorThis answers a question of Hajłasz and Onninen in the one-dimensional case.

ABSTRACT. Employing a construction of Tsirelson-like spaces due to Argyros and Deliyanni, we show that the class of all Banach spaces which are isomorphic to a subspace of c 0 is a complete analytic set with respect to the Effros Borel structure of separable Banach spaces. Moreover, the classes of all separable spaces with the Schur property and of all separable spaces with the DunfordPettis property are Π 1 2 -complete.

ABSTRACT. It was proved by Argyros and Dodos that, for many classes C of separable Banach spaces which share some property P, there exists an isomorphically universal space that satisfies P as well. We introduce a variant of their amalgamation technique which provides an isometrically universal space in the case that C consists of spaces with a monotone Schauder basis. For example, we prove that if C is a set of separable Banach spaces which is analytic with respect to the Effros-Borel structure and every X ∈ C is reflexive and has a monotone Schauder basis, then there exists a separable reflexive Banach space that is isometrically universal for C.

We present and thoroughly study natural Polish spaces of separable Banach spaces. These spaces are defined as spaces of norms, respectively pseudonorms, on the countable infinite-dimensional rational vector space. We provide an exhaustive comparison of these spaces with admissible topologies recently introduced by Godefroy and Saint-Raymond and show that Borel complexities differ little with respect to these two topological approaches. We investigate generic properties in these spaces and compare them with those in admissible topologies, confirming the suspicion of Godefroy and Saint-Raymond that they depend on the choice of the admissible topology.

The paper concerns the problem of whether a nonseparable C(K) space must contain a set of unit vectors whose cardinality equals the density of C(K), and such that the distances between every two distinct vectors are always greater than one. We prove that this is the case if the density is at most continuum, and that for several classes of C(K) spaces (of arbitrary density) it is even possible to find such a set which is 2-equilateral; that is, the distance between every two distinct vectors is exactly 2.