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The convergence space of minimal usco mappings

R. Anguelov;  O. F. K. Kalenda — 2009

Czechoslovak Mathematical Journal

A convergence structure generalizing the order convergence structure on the set of Hausdorff continuous interval functions is defined on the set of minimal usco maps. The properties of the obtained convergence space are investigated and essential links with the pointwise convergence and the order convergence are revealed. The convergence structure can be extended to a uniform convergence structure so that the convergence space is complete. The important issue of the denseness of the subset of all...

Quotients of Continuous Convex Functions on Nonreflexive Banach Spaces

P. Holický;  O. F. K. Kalenda;  L. Veselý;  L. Zajíček — 2007

Bulletin of the Polish Academy of Sciences. Mathematics

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 also gives a stronger version of Klee's result concerning renormings of nonreflexive spaces and non-norm-attaining functionals.

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