We investigate the strength of set existence axioms needed for separable Banach space theory. We show that a very strong axiom, Π1 comprehension, is needed to prove such basic facts as the existence of the weak-∗ closure of any norm-closed subspace of `1 = c0. This is in contrast to earlier work [6, 4, 7, 23, 22] in which theorems of separable Banach space theory were proved in very weak subsys...

In this paper, we introduce a new geometric property (UÃ2) and we show that if a separable Banach space has this property, then both X and its dual X∗ have the weak fixed point property. We also prove that a uniformly Gateaux differentiable Banach space has property (UÃ2) and that if X∗ has property (UÃ2), then X has the (UKK)-property. Criteria for Orlicz spaces to have the properties (UA2), (...

We show convergence of the weak dual greedy algorithm in wide class of Banach spaces, extending our previous result where it was shown to converge in subspaces of quotients of Lp (for 1 < p < ∞). In particular, we show it converges in the Schatten ideals Sp when 1 < p < ∞ and in any Banach lattice which is p-convex and q-concave with constants one, where 1 < p < q < ∞. We also discuss convergen...

A class of Lagrange-Newton-SQP methods is investigated for optimal control problems governed by semilinear parabolic initial-boundary value problems. Distributed and boundary controls are given, restricted by pointwise upper and lower bounds. The convergence of the method is discussed in appropriate Banach spaces. Based on a weak second order suucient optimality condition for the reference solu...

Based upon an external approximation scheme for the underlying Banach space, a nonlinear operator equation is approximated by a sequence of coercive problems. The equation is supposed to be governed by the sum of two nonlinear operators acting between a reflexive Banach space and its dual. Under suitable stability assumptions and if the underlying operators can be approximated consistently, wea...

In this paper, we study approximate amenability of matrix algebras. We show that every derivation from Mn(A) into Mn(E) is the sum of an inner derivation and a derivation induced by a derivation from A into E(m), where A is a Banach algebra and E is a Banach A-bimodule. By using this, we provide many results in approximate and permanent weak amenability of these algebras.

We study the equivalence between the weak and strong laws of large numbers for arrays of row-wise independent random elements with values in a Banach space . The conditions under which this equivalence holds are of the Chung or Chung-Teicher types. These conditions are expressed in terms of convergence of specific series and o(1) requirements on specific weighted row-wise sums. Moreover, there ...

In this paper, we study a class of Banach spaces, called φspaces. In a natural way, we associate a measure of weak compactness in such spaces and prove an analogue of Sadovskii fixed point theorem for weakly sequentially continuous maps. A counter-example is given to justify our requirement. As an application, we establish an existence result for a Hammerstein integral equation in a Banach space.

Following Davie’s example of a Banach space failing the approximation property ([D]), we show how to construct a Banach space E which is asymptotically Hilbertian and fails the approximation property. Moreover, the space E is shown to be a subspace of a space with an unconditional basis which is “almost” a weak Hilbert space and which can be written as the direct sum of two subspaces all of who...