Related papers: A note on symmetric separation in Banach spaces

We investigate the following property for Banach spaces. A Banach space $X$ satisfies the Uniform Approximation on Large Subspaces (UALS) if there exists $C>0$ with the following property: for any $A\in\mathcal{L}(X)$ and convex compact…

Some new examples of K-monotone couples of the type (X, X(w)), where X is a symmetric space on [0, 1] and w is a weight on [0, 1], are presented. Based on the property of the w-decomposability of a symmetric space we show that, if a weight…

This paper systematically investigates a new geometric constant associated with isosceles orthogonality in Banach spaces. By establishing the connection between this new constant and a classical function, sharp upper and lower bounds for…

We construct infinitely differentiable norms and partitions of unity for a class of Banach spaces which includes all spaces $\C(K)$ with $K$ a countable compact space, and all spaces $\C_0[0,\Omega )$ with $\Omega $ an ordinal.

Given a Banach space we consider the $\sigma$-ideal of all of its subsets which are covered by countably many hyperplanes and investigate its standard cardinal characteristics as the additivity, the covering number, the uniformity, the…

Let $X$ be a Banach space and let $(\xi_j)_{j\ge 1}$ be an i.i.d. sequence of symmetric random variables with finite moments of all orders. We prove that the following assertions are equivalent: (1). There exists a constant $K$ such that $$…

Relatively recently it was proved that if $\Gamma$ is an arbitrary set, then any equivalent norm on $c_0(\Gamma)$ can be approximated uniformly on bounded sets by polyhedral norms and $C^\infty$ smooth norms, with arbitrary precision. We…

In the first part of the paper we study the structure of Banach spaces with a conditional spreading basis. The geometry of such spaces exhibit a striking resemblance to the geometry of James' space. Further, we show that the averaging…

We present an isometric version of the complementably universal Banach space $\mathcal{B}$ with a monotone Schauder basis. The space $\mathcal{B}$ is isomorphic to Pe{\l}czy\'nski's space with a universal basis as well as to Kadec'…

We give a self-contained treatment of symmetric Banach sequence spaces and some of their natural properties. We are particularly interested in the symmetry of the norm and the existence of symmetric linear functionals. Many of the presented…

In this working note we study the one-sided positive Banach-Mazur distance between some pairs of $C(K)$ Banach spaces. Building on methods developed in [4], we solve, in particular, one of the problems posed in [2].

We show that norms on certain Banach spaces $X$ can be approximated uniformly, and with arbitrary precision, on bounded subsets of $X$ by $C^{\infty}$ smooth norms and polyhedral norms. In particular, we show that this holds for any…

It is a longstanding problem whether every contractible Banach algebra is necessarily finite-dimensional. In this note, we confirm this for Banach algebras acting on Banach spaces with the uniform approximation property. This generalizes a…

For every couple of Hausdorff functions $ \psi$ and $\varphi $ verifying some mild assumptions, there exists a compact subset $ K $ of the Baire space such that the $ \varphi$-Hausdorff measure and the $ \psi$-packing measure on $ K$ are…

We investigate relations between symmetrizations of quasi-Banach function spaces and constructions such as Calderon-Lozanovskii spaces, pointwise product spaces and pointwise multipliers. We show that under reasonable assumptions the…

We present a necessary condition for a pair of $\mathcal{C}(K)$ spaces to be isomorphic in terms of topological properties of Cantor-Bendixon derivatives of $K$. This in particular gives a completely new information about the perfect…

~This paper presents a general result that allows for establishing a link between the Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers and Feller's strong law of large numbers in a Banach space setting. Let $\{X, X_{n}; n \geq…

We investigate the local preservation of Birkhoff-James orthogonality at a point by a linear operator on a finite-dimensional Banach space and illustrate its importance in understanding the action of the operator in terms of the geometry of…

We show that for every separable Banach space $X$, either $\spw(X)$ (the set of all spreading models of $X$ generated by weakly-null sequences in $X$, modulo equivalence) is countable, or $\spw(X)$ contains an antichain of the size of the…

We prove that in every separable Banach space $X$ with a Schauder basis and a $C^k$-smooth norm it is possible to approximate, uniformly on bounded sets, every equivalent norm with a $C^k$-smooth one in a way that the approximation is…