Decomposition of Banach Space into a Direct Sum of Separable and Reflexive Subspaces and Borel Maps

* This paper was supported in part by the Bulgarian Ministry of Education, Science and Technologies under contract MM-506/95.

A New Hereditarily l^2 Banach Space

2000 Mathematics Subject Classification: 46B20, 46B26.

The JNR Property and the Borel Structure of a Banach Space

Research partially supported by a grant of Caja de Ahorros del Mediterraneo.

Fragmentability of the Dual of a Banach Space with Smooth Bump

We prove that if a Banach space X admits a Lipschitz β-smooth bump function, then (X ∗ , weak ∗ ) is fragmented by a metric, generating a topology, which is stronger than the τβ -topology. We also use this to prove that if X ∗ admits a Lipschitz Gateaux-smooth bump function, then X is sigma-fragmentable.

Representable Banach Spaces and Uniformly Gateaux-Smooth Norms

It is proved that a representable non-separable Banach space does not admit uniformly Gâteaux-smooth norms. This is true in particular for C(K) spaces where K is a separable non-metrizable Rosenthal compact space.

A Dichotomy for Ordinary Differential Equations in a Banach Space

A dichotomysimilar property for a class of homogeneous differential equations in an arbitrary Banach space is introduced. By help of them, existence of quasi bounded solutions of the appropriate nonhomogeneous equation is proved.

A Characterization of Weakly Lindelöf Determined Banach Spaces

2000 Mathematics Subject Classification: 46B26, 46B03, 46B04.

Every N-Dimensional Separable Metric Space Contains a Totally Disconnected (n-1)-Dimensional Subset with no True Quasi-Components

Владимир Тодоров, Петър Стоев - Тази бележка съдържа елементарна конструкция на множество с указаните в заглавието свойства. Да отбележим в допълнение, че така полученото множество остава напълно несвързано дори и след като се допълни с краен брой елементи.

Linear Colligations and Dynamic System Corresponding to Operators in the Banach Space

2000 Mathematics Subject Classification: Primary 47A48, 93B28, 47A65; Secondary 34C94.

A Remark On S.M. Bates' Theorem

In his paper [1], Bates investigates the existence of nonlinear, but highly smooth, surjective operators between various classes of Banach spaces. Modifying his basic method, he obtains the following striking results.

The Point of Continuity Property: Descriptive Complexity and Ordinal Index

∗ Supported by D.G.I.C.Y.T. Project No. PB93-1142

Approximation of Lipschitz Mappings

2000 Mathematics Subject Classification: 46B03

More lr saturated L∞ spaces

2000 Mathematics Subject Classification: 05D10, 46B03.

The Space of Differences of Convex Functions on [0, 1]

∗Participant in Workshop in Linear Analysis and Probability, Texas A & M University, College Station, Texas, 2000. Research partially supported by the Edmund Landau Center for Research in Mathematical Analysis and related areas, sponsored by Minerva Foundation (Germany).

Descriptive Sets and the Topology of Nonseparable Banach Spaces

This paper was extensively circulated in manuscript form beginning in the Summer of 1989. It is being published here for the first time in its original form except for minor corrections, updated references and some concluding comments.