arXiv subjects

Fouad Chaatit

Publications and source records attributed to Fouad Chaatit.

5 recordsLinked to original sources

arXiv1999

On Differences of Semi-Continuous Functions

Extrinsic and intrinsic characterizations are given for the class DSC(K) of differences of semi-continuous functions on a Polish space K, and also decomposition characterizations of DSC(K) and the class PS(K) of pointwise stabilizing functions on K are obtained in terms of behavior restricted to ambiguous sets. The main, extrinsic characterization is given in terms of behavior restricted to some subsets of second category in any closed subset of K. The concept of a strong continuity point is introduced, using the transfinite oscillations osc$_αf$ of a function $f$ previously defined by the second named author. The main intrinsic characterization yields the following DSC analogue of Baire's characterization of first Baire class functions: a function belongs to DSC(K) iff its restriction to any closed non-empty set L has a strong continuity point. The characterizations yield as a corollary that a locally uniformly converging series $\sum ϕ_j$ of DSC functions on K converges to a DSC function provided $\sum{osc}_αϕ_j$ converges locally uniformly for all countable ordinals $α$.

math.FA↗

arXiv1996

A Representation of Stable Banach Spaces

We show that any separable stable Banach space can be represented as a group of isometries on a separable reflexive Banach space, which extends a result of S. Guerre and M. Levy. As a consequence, we can then represent homeomorphically its space of types.

math.FA↗

arXiv1994

Uniform Kadec-Klee Property in Banach lattices

We prove that a Banach lattice $X$ which does not contain the $l^n_{\infty}$-uniformly has an equivalent norm which is uniformly Kadec-Klee for a natural topology $τ$ on $X$. In case the Banach lattice is purely atomic, the topology $τ$ is the coordinatewise convergence topology.

math.FA↗

arXiv1993

On Functions of Finite Baire Index

It is proved that every function of finite Baire index on a separable metric space $K$ is a $D$-function, i.e., a difference of bounded semi-continuous functions on $K$. In fact it is a strong $D$-function, meaning it can be approximated arbitrarily closely in $D$-norm, by simple $D$-functions. It is shown that if the $n^{th}$ derived set of $K$ is non-empty for all finite $n$, there exist $D$-functions on $K$ which are not strong $D$-functions. Further structural results for the classes of finite index functions and strong $D$-functions are also given.

math.FA↗

arXiv1992

On Uniform Homeomorphisms of the Unit Spheres of Certain Banach Lattices

We prove that if X is an infinite dimensional Banach lattice with a weak unit then there exists a probability space (Omega, Sigma,mu) so that the unit sphere S(L_1(Omega, Sigma, mu) is uniformly homeomorphic to the unit sphere S(X) if and only if X does not contain l_{infty}^n's uniformly.

math.FA↗