We construct an example of a real Banach space whose group of surjective isometries reduces to ± Id, but the group of surjective isometries of its dual contains the group of isometries of a separable infinite-dimensional Hilbert space as a subgroup. To do so, we present examples of extremely non-complex Banach spaces (i.e. spaces X such that ‖ Id+T ‖ = 1+‖T ‖ for every bounded linear operator T...

A Banach space X is asymptotically symmetric (a.s.) if for some C <∞, for all m ∈ N, for all bounded sequences (xj)j=1 ⊆ X, 1 ≤ i ≤ m, for all permutations σ of {1, . . . ,m} and all ultrafilters U1, . . . ,Um on N, lim n1,U1 . . . lim nm,Um ∥∥∥∥ m ∑ i=1 xini ∥∥∥∥ ≤ C lim nσ(1),Uσ(1) . . . lim nσ(m),Uσ(m) ∥∥∥∥ m ∑

We survey the recent developments concerning fixed point properties for group actions on Banach spaces. In the setting of Hilbert spaces such fixed point properties correspond to Kazhdan’s property (T). Here we focus on the general, non-Hilbert case, we discuss the methods, examples and several applications. 2010 Mathematics Subject Classification: 20J06, 18H10, 46B99

Greedy algorithms which use only function evaluations are applied to convex optimization in a general Banach space X . Along with algorithms that use exact evaluations, algorithms with approximate evaluations are treated. A priori upper bounds for the convergence rate of the proposed algorithms are given. These bounds depend on the smoothness of the objective function and the sparsity or compre...

The first representation theorem establishes a correspondence between positive, self-adjoint operators and closed, positive forms on Hilbert spaces. The aim of this paper is to show that some of the results remain true if the underlying space is a reflexive Banach space. In particular, the construction of the Friedrichs extension and the form sum of positive operators can be carried over to thi...

A Banach space E is c0-saturated if every closed infinite dimensional subspace of E contains an isomorph of c0. A c0-saturated Banach space with an unconditional basis which has a quotient space isomorphic to l2 is constructed. A Banach space E is c0-saturated if every closed infinite dimensional subspace of E contains an isomorph of c0. In [2] and [3], it was asked whether all quotient spaces ...

Every isometric property of Banach spaces preserved by real or complex interpolation is subspace-hereditary, and every isomorphic property of separable Banach spaces so preserved is quotient-hereditary. Introduction Many properties of Banach spaces are known to pass to interpolation spaces obtained from the real k-method of interpolation or from the complex method, completeness itself being the...

We prove a geometric characterization of Banach space stability. We show that a Banach space X is stable if and only if the following condition holds. Whenever X̂ is an ultrapower of X and B is a ball in X̂, the intersection B ∩X can be uniformly approximated by finite unions and intersections of balls in X; furthermore, the radius of these balls can be taken arbitrarily close to the radius of B,...