43 research outputs found
Quotients of continuous convex functions on nonreflexive Banach spaces
On each nonreflexive Banach space X there exists a positive continuous convex
function f such that 1/f is not a d.c. function (i.e., a difference of two
continuous convex functions). This result together with known ones implies that
X is reflexive if and only if each everywhere defined quotient of two
continuous convex functions is a d.c. function. Our construction gives also a
stronger version of Klee's result concerning renormings of nonreflexive spaces
and non-norm-attaining functionals.Comment: 5 page
Dedekind order completion of C(X) by Hausdorff continuous functions
The concept of Hausdorff continuous interval valued functions, developed
within the theory of Hausdorff approximations and originaly defined for
interval valued functions of one real variable is extended to interval valued
functions defined on a topological space X. The main result is that the set of
all finite Hausdorff continuous functions on any topological space X is
Dedekind order complete. Hence it contains the Dedekind order completion of the
set C(X) of all continuous real functions defined on X as well as the Dedekind
order completion of the set C_b(X) of all bounded continuous functions on X.
Under some general assumptions about the topological space X the Dedekind order
completions of both C(X) and C_b(X) are characterised as subsets of the set of
all Hausdorff continuous functions. This solves a long outstanding open problem
about the Dedekind order completion of C(X). In addition, it has major
applications to the regularity of solutions of large classes of nonlinear PDEs
Compactness in Banach space theory - selected problems
We list a number of problems in several topics related to compactness in
nonseparable Banach spaces. Namely, about the Hilbertian ball in its weak
topology, spaces of continuous functions on Eberlein compacta, WCG Banach
spaces, Valdivia compacta and Radon-Nikod\'{y}m compacta