Joram Lindenstrauss

Geometric nonlinear functional analysis

Introduction Retractions, extensions and selections Retractions, extensions and selections (special topics) Fixed points Differentiation of convex functions The Radon-Nikodym property Negligible sets and Gateaux differentiability Lipschitz classification of Banach spaces Uniform embeddings into Hilbert space Uniform classification of spheres Uniform classification of Banach spaces Nonlinear quotient maps Oscillation of uniformly continuous functions on unit spheres of finite-dimensional subspaces Oscillation of uniformly continuous functions on unit spheres of infinite-dimensional subspaces Perturbations of local isometries Perturbations of global isometries Twisted sums Group structure on Banach spaces Appendices Bibliography Index.

On orlicz sequence spaces

It is proved that every Orlicz sequence space contains a subspace isomorphic to somelp. The question of uniqueness of symmetric bases in Orlicz sequence spaces is investigated.

An example concerning fixed points

An example is given of a contractionT defined on a bounded closed convex subset of Hilbert space for which ((I+T)/2)n does not converge.

On operators which attain their norm

The following problem is considered. LetX andY be Banach spaces. Are those operators fromX toY which attain their norm on the unit cell ofX, norm dense in the space of all operators fromX toY? It is proved that this is always the case ifX is reflexive. In general the answer is negative and it depends on some convexity and smoothness properties of the unit cells inX andY. As an application a refinement of the Krein-Milman theorem and Mazur’s theorem concerning the density of smooth points, in the case of weakly compact sets in a separable space, is obtained.

The ℒ p spaces

The ℒp spaces which were introduced by A. Pełczyński and the first named author are studied. It is proved, e.g., that (i)X is an ℒp space if and only ifX* is and ℒq space (p−1+q−1=1). (ii) A complemented subspace of an ℒp space is either an ℒp or an ℒ2 space. (iii) The ℒp spaces have sufficiently many Boolean algebras of projections. These results are applied to show thatX is an ℒ∞ (resp. ℒ1) space if and only ifX admits extensions (resp. liftings) of compact operators havingX as a domain or range space. We also prove a theorem on the “local reflexivity” of an arbitrary Banach space.

On the complemented subspaces problem

A Banach space is isomorphic to a Hilbert space provided every closed subspace is complemented. A conditionally σ-complete Banach lattice is isomorphic to anLp-space (1≤p<∞) or toc0(Γ) if every closed sublattice is complemented.

Contributions to the theory of the classical Banach spaces

The paper contains several results on the linear topological structure of the spaces C(K), K compact metric, and Lp(0, 1), 1 ⩽ p < ∞. The topics which are studied include: complemented subspaces, special Schauder bases, and equivalent norms in these spaces.

Extensions of lipschitz maps into Banach spaces

AbstractIt is proved that ifY ⊂X are metric spaces withY havingn≧2 points then any mapf fromY into a Banach spaceZ can be extended to a map

Banach spaces determined by their uniform structures

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