A nonreflexive Banach space may have a weakly uniformly rotund dual. The aim of this paper is to determine alternative characterisations and study further implications of this property in higher duals.

We use the best constants in the Khintchine inequality to generalise a theorem of Kato [‘Similarity for sequences of projections’, Bull. Amer. Math. Soc.73(6) (1967), 904–905] on similarity for sequences of projections in Hilbert spaces to the case of unconditional Schauder decompositions in $\ell _{p}$ spaces. We also sharpen a stability theorem of Vizitei [‘On the stability of bases of subspaces in a Banach space’, in: Studies on Algebra and Mathematical Analysis, Moldova Academy of Sciences (Kartja Moldovenjaska, Chişinău, 1965), 32–44; (in Russian)] in the case of unconditional Schauder decompositions in any Banach space.

This article explores the properties of fractal interpolation functions with variable scaling parameters, in the context of smooth fractal functions. The first part extends the Barnsley–Harrington theorem for differentiability of fractal functions and the fractal analogue of Hermite interpolation to the present setting. The general result is applied on a special class of iterated function systems in order to develop differentiability of the so-called $\boldsymbol{{\it\alpha}}$-fractal functions. This leads to a bounded linear map on the space ${\mathcal{C}}^{k}(I)$ which is exploited to prove the existence of a Schauder basis for ${\mathcal{C}}^{k}(I)$ consisting of smooth fractal functions.

We investigate questions of maximal symmetry in Banach spaces and the structure of certain bounded non-unitarisable groups on Hilbert space. In particular, we provide structural information about bounded groups with an essentially unique invariant complemented subspace. This is subsequently combined with rigidity results for the unitary representation of $\text{Aut}(T)$ on $\ell _{2}(T)$, where $T$ is the countably infinite regular tree, to describe the possible bounded subgroups of $\text{GL}({\mathcal{H}})$ extending a well-known non-unitarisable representation of $\mathbb{F}_{\infty }$.

As a related result, we also show that a transitive norm on a separable Banach space must be strictly convex.

We continue the study of the boundedness of the operator

on the set of decreasing functions in Lp(w). This operator was first introduced by Braverman and Lai and also studied by Andersen, and although the weighted weak-type estimate was completely solved, the characterization of the weights w such that is bounded is still open for the case in which p > 1. The solution of this problem will have applications in the study of the boundedness on weighted Lorentz spaces of important operators in harmonic analysis.

We give a positive answer to the question of Bouras [‘Almost Dunford–Pettis sets in Banach lattices’, Rend. Circ. Mat. Palermo (2) 62 (2013), 227–236] concerning weak compactness of almost Dunford–Pettis sets in Banach lattices. That is, every almost Dunford–Pettis set in a Banach lattice $E$ is relatively weakly compact if and only if $E$ is a $\mathit{KB}$-space.

In this paper, we introduce the concept of a semi-parallelogram and obtain some results for the Aleksandrov–Rassias problem using this concept. In particular, we resolve an important case of this problem for mappings preserving two distances with a nonintegral ratio.

We show that the dual to any subspace of $c_{0}({\rm\Gamma})$ (${\rm\Gamma}$ is an arbitrary index set) has the strongest possible quantitative version of the Schur property. Further, we establish a relationship between the quantitative Schur property and quantitative versions of the Dunford–Pettis property. Finally, we apply these results to show, in particular, that any subspace of the space of compact operators on $\ell _{p}$ ($1<p<\infty$) with the Dunford–Pettis property automatically satisfies both its quantitative versions.

We give continuous separation theorems for convex sets in a real linear space equipped with a norm that can assume the value infinity. In such a space, it may be impossible to continuously strongly separate a point $p$ from a closed convex set not containing $p$, that is, closed convex sets need not be weakly closed. As a special case, separation in finite-dimensional extended normed spaces is considered at the outset.

In a recent paper, topological spaces $(X,{\it\tau})$ that are fragmented by a metric that generates the discrete topology were investigated. In the present paper we shall continue this investigation. In particular, we will show, among other things, that such spaces are ${\it\sigma}$-scattered, that is, a countable union of scattered spaces, and characterise the continuous images of separable metrisable spaces by their fragmentability properties.

Let $(X,+)$ be an Abelian group and $E$ be a Banach space. Suppose that $f:X\rightarrow E$ is a surjective map satisfying the inequality

$$\begin{eqnarray}|\,\Vert f(x)-f(y)\Vert -\Vert f(x-y)\Vert \,|\leq {\it\varepsilon}\min \{\Vert f(x)-f(y)\Vert ^{p},\Vert f(x-y)\Vert ^{p}\}\end{eqnarray}$$

${\it\varepsilon}>0$

$p>1$

$x,y\in X$

$f$

$0<p\leq 1$

$f$

$E$

$F$

${\it\epsilon}>0$

$p>1$

$$\begin{eqnarray}|\,\Vert f(x)-f(y)\Vert -\Vert f(u)-f(v)\Vert \,|\leq {\it\epsilon}\min \{\Vert f(x)-f(y)\Vert ^{p},\Vert f(u)-f(v)\Vert ^{p}\}\end{eqnarray}$$

$\Vert x-y\Vert =\Vert u-v\Vert$

$f$

$\dim E\geq 2$

$K\neq 0$

$Kf$

Given a finite-dimensional Banach space $X$ and an Auerbach basis $\{(x_{k},x_{k}^{\ast }):1\leqslant k\leqslant n\}$ of $X$, it is proved that there exist $n+1$ linear combinations $z_{1},\ldots ,z_{n+1}$ of $x_{1},\ldots ,x_{n}$ with coordinates $0,\pm 1$, such that $\Vert z_{k}\Vert =1$, for $k=1$, $2,\ldots ,n+1$ and $\Vert z_{k}-z_{l}\Vert >1$, for $1\leqslant k<l\leqslant n+1$.

In this paper we consider the stationary Poisson Boolean model with spherical grains and propose a family of nonparametric estimators for the radius distribution. These estimators are based on observed distances and radii, weighted in an appropriate way. They are ratio unbiased and asymptotically consistent for a growing observation window. We show that the asymptotic variance exists and is given by a fairly explicit integral expression. Asymptotic normality is established under a suitable integrability assumption on the weight function. We also provide a short discussion of related estimators as well as a simulation study.

We introduce an approximation property (${\mathcal{K}}_{\mathit{up}}$-AP, $1\leq p<\infty$), which is weaker than the classical approximation property, and discover the duality relationship between the ${\mathcal{K}}_{\mathit{up}}$-AP and the ${\mathcal{K}}_{p}$-AP. More precisely, we prove that for every $1<p<\infty$, if the dual space $X^{\ast }$ of a Banach space $X$ has the ${\mathcal{K}}_{\mathit{up}}$-AP, then $X$ has the ${\mathcal{K}}_{p}$-AP, and if $X^{\ast }$ has the ${\mathcal{K}}_{p}$-AP, then $X$ has the ${\mathcal{K}}_{\mathit{up}}$-AP. As a consequence, it follows that every Banach space has the ${\mathcal{K}}_{u2}$-AP and that for every $1<p<\infty$, $p\neq 2$, there exists a separable reflexive Banach space failing to have the ${\mathcal{K}}_{\mathit{up}}$-AP.

This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalizations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the Sobolev spaces $H^{s}({\rm\Omega})$ and $\widetilde{H}^{s}({\rm\Omega})$, for $s\in \mathbb{R}$ and an open ${\rm\Omega}\subset \mathbb{R}^{n}$. We exhibit examples in one and two dimensions of sets ${\rm\Omega}$ for which these scales of Sobolev spaces are not interpolation scales. In the cases where they are interpolation scales (in particular, if ${\rm\Omega}$ is Lipschitz) we exhibit examples that show that, in general, the interpolation norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio of these two norms can be arbitrarily large.

A Chebyshev set is a subset of a normed linear space that admits unique best approximations. In the first part of this paper we present some basic results concerning Chebyshev sets. In particular, we investigate properties of the metric projection map, sufficient conditions for a subset of a normed linear space to be a Chebyshev set, and sufficient conditions for a Chebyshev set to be convex. In the second half of the paper we present a construction of a nonconvex Chebyshev subset of an inner product space.

We make some comments on the existence, uniqueness and integrability of the scalar derivatives and approximate scalar derivatives of vector-valued functions. We are particularly interested in the connection between scalar differentiation and the weak Radon–Nikodým property.

Many characterizations of fragmentability of topological spaces have been investigated. In this paper we deal with some properties of weak-fragmentability of Banach spaces.

Recently, H’michane et al. [‘On the class of limited operators’, Acta Math. Sci. (submitted)] introduced the class of weak$^*$ Dunford–Pettis operators on Banach spaces, that is, operators which send weakly compact sets onto limited sets. In this paper, the domination problem for weak$^*$ Dunford–Pettis operators is considered. Let $S, T:E\to F$ be two positive operators between Banach lattices $E$ and $F$ such that $0\leq S\leq T$. We show that if $T$ is a weak$^{*}$ Dunford–Pettis operator and $F$ is $\sigma $-Dedekind complete, then $S$ itself is weak$^*$ Dunford–Pettis.

We study the metric entropy of the metric space ${\mathcal{B}}_{n}$ of all $n$-dimensional Banach spaces (the so-called Banach–Mazur compactum) equipped with the Banach–Mazur (multiplicative) “distance” $d$. We are interested either in estimates independent of the dimension or in asymptotic estimates when the dimension tends to $\infty$. For instance, we prove that, if $N({\mathcal{B}}_{n},d,1+{\it\varepsilon})$ is the smallest number of “balls” of “radius” $1+{\it\varepsilon}$ that cover ${\mathcal{B}}_{n}$, then for any ${\it\varepsilon}>0$ we have

$$\begin{eqnarray}0<\liminf _{n\rightarrow \infty }n^{-1}\log \log N({\mathcal{B}}_{n},d,1+{\it\varepsilon})\leqslant \limsup _{n\rightarrow \infty }n^{-1}\log \log N({\mathcal{B}}_{n},d,1+{\it\varepsilon})<\infty .\end{eqnarray}$$

$n$

$d_{N}$

$N\times N$

$N$

$N$

$N=1$

$n$

$N\times N$