We prove that the dual unit ball of a Banach space X is a Corson compactum provided that the dual unit ball with respect to every equivalent norm on X is a Valdivia compactum. As a corollary we show that the dual unit ball of a Banach space X of density ${\aleph }_1$ is Corson if (and only if) X has a projectional resolution of the identity with respect to every equivalent norm. These results answer questions asked by M. Fabian, G. Godefroy and V. Zizler and yield a converse to Amir-Lindenstrauss’ theorem.

We give very short and transparent proofs of extrapolation theorems of Yano type in the framework of Lorentz spaces. The decomposition technique developed in Edmunds-Krbec (2000) enables us to obtain known and new results in a unified manner.

The area of research of this paper goes back to a 1930 result of H. Auerbach showing that a scalar series is (absolutely) convergent if all its zero-density subseries converge. A series ${\sum }_n{x}_n$ in a topological vector space X is called ℒ-convergent if each of its lacunary subseries ${\sum }_k{x}_{n_k}$ (i.e. those with $n_{k+1}-n_k\to \infty$) converges. The space X is said to have the Lacunary Convergence Property, or LCP, if every ℒ-convergent series in X is convergent; in fact, it is then subseries convergent. The Zero-Density Convergence...

In this article we consider a theory of vector valued strongly singular operators. Our results include Lp, Hp and BMO continuity results. Moreover, as is well known, vector valued estimates are closely related to weighted norm inequalities. These results are developed in the first four sections of our paper. In section 5 we use our vector valued singular integrals to estimate the corresponding maximal operators. Finally in section 6 we discuss applications to weighted norm inequalities for pseudo-differential...

There exists an absolute constant $c_0$ such that for any n-dimensional Banach space E there exists a k-dimensional subspace F ⊂ E with k≤ n/2 such that $in{f}_{ellipsoid\epsilon \subset B_E}{(vol\left(B_E\right)/vol\left(\epsilon \right))}^{1/n}\le c_{0}in{f}_{zonoidZ\subset B_F}{(vol\left(B_F\right)/vol\left(Z\right))}^{1/k}$ . The concept of volume ratio with respect to ${\ell }_p$-spaces is used to prove the following distance estimate for $2\le q\le p<\infty$: $su{p}_{F\subset {\ell }_p,dimF=n}in{f}_{G\subset L_q,dimG=n}d(F,G){\sim }_{c_{pq}}{n}^{(q/2)(1/q-1/p)}$.

The geometry of random projections of centrally symmetric convex bodies in ${ℝ}^N$ is studied. It is shown that if for such a body K the Euclidean ball $B{₂}^N$ is the ellipsoid of minimal volume containing it and a random n-dimensional projection $B=P_H\left(K\right)$ is “far” from $P_H\left(B{₂}^N\right)$ then the (random) body B is as “rigid” as its “distance” to $P_H\left(B{₂}^N\right)$ permits. The result holds for the full range of dimensions 1 ≤ n ≤ λN, for arbitrary λ ∈ (0,1).