Complete sets and completion of sets in Banach spaces

In this paper we study properties of complete sets and of completions of sets in Banach spaces. We consider the family of completions of a given set and its size; we also study in detail the relationships concerning diameters, radii, and centers. The results are illustrated by several examples.     

Null sets and games in Banach spaces

The notion of Aronszajn-null sets generalizes the notion of Lebesgue measure zero in the Euclidean space to infinite dimensional Banach spaces. We present a game-theoretic approach to Aronszajn-null sets, establish its basic properties, and discuss some ensuing open problems.     

Null sets and games in Banach spaces

The notion of Aronszajn-null sets generalizes the notion of Lebesgue measure zero in the Euclidean space to infinite dimensional Banach spaces. We present a game-theoretic approach to Aronszajn-null sets, establish its basic properties, and discuss some ensuing open problems.     

Proper holomorphic discs avoiding closed convex sets

Denote by the open unit disc in . Let C be a closed convex subset of . We prove that for each there is a proper holomorphic map such that and if and only if either C is a complex line or C does not contain any complex line. Received: 17 July 2001; in final form: 22 November 2001 / Published online: 5 September 2002     

Almost limited sets in Banach lattices

We introduce and study the class of almost limited sets in Banach lattices, that is, sets on which every disjoint weak^?${\mathrm{weak}}^{?}$ null sequence of functionals converges uniformly to zero. It is established that a Banach lattice has order continuous norm if and only if almost limited sets and L  -weakly compact sets coincide. In particular, in terms of almost Dunford–Pettis operators into _c0$c_0$, we give an operator characterization of those σ-Dedekind complete Banach lattices whose relatively weakly compact sets are almost limited, that is, for a σ-Dedekind Banach lattice E, every relatively weakly compact set in E   is almost limited if and only if every continuous linear operator T:E→_c0$T:E\to c_0$ is an almost Dunford–Pettis operator.     

Strongly antiproximinal sets in Banach spaces

Translated from Matematicheskie Zametki, Vol. 47, No. 2, pp. 130–136, February, 1990.     

Banach spaces that admit support sets

It is shown that the existence of a closed convex set all of whose points are properly supported in a Banach space is equivalent to the existence of a certain type of uncountable ordered one-sided biorthogonal system. Under the continuum hypothesis, we deduce that this notion is weaker than the existence of an uncountable biorthogonal system.

     

Accretive sets and differential equations in Banach spaces

This paper is concerned with global solutions of the initial value problem (1)du/dt +Au∋0,u(0)=x whereA is a (nonlinear) accretive set in a Banach spaceX. We show that various approximation processes converge to the solution (whenever it exists). In particular we obtain an exponential formula for the solutions of (1). AssumingX* is uniformly convex, we also prove the existence of a solution under weaker assumptions ofA than those made by previous authors (F. Browder, T. Kato). Results obtained at the Courant Institute of Mathematical Sciences, New York University, with the National Science Foundation, Grant NSF-GP-11600.     

Determinacy and weakly Ramsey sets in Banach spaces

We give a sufficient condition for a set of block subspaces in an infinite-dimensional Banach space to be weakly Ramsey. Using this condition we prove that in the Levy-collapse of a Mahlo cardinal, every projective set is weakly Ramsey. This, together with a construction of W. H. Woodin, is used to show that the Axiom of Projective Determinacy implies that every projective set is weakly Ramsey. In the case of we prove similar results for a stronger Ramsey property. And for hereditarily indecomposable spaces we show that the Axiom of Determinacy plus the Axiom of Dependent Choices imply that every set is weakly Ramsey. These results are the generalizations to the class of projective sets of some theorems from W. T. Gowers, and our paper ``Weakly Ramsey sets in Banach spaces.'

     

On compact connected sets in Banach spaces

Let be a separable strictly convex Banach space of dimension at least 2. It is shown that there exists a nonempty compact connected set such that the nearest point mapping is not single valued on a set of points dense in . Furthermore, it is proved that most (in the sense of the Baire category) nonempty compact connected sets have the above property. Similar results hold for the furthest point mapping.