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3 Bulletin of the Polish Academy of Sciences. Mathematics

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Quotients of continuous convex functions on nonreflexive Banach spaces

Holicky P., Kalenda O. F., Vesely L., Zajícek L.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

Vol. 55, no 3

211-217

On each nonreflexive Banach space X there exists a positive continuous convex function ƒ such that 1/ƒ 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.

Extensions of Borel measurable maps and ranges of Borel bimeasurable maps

Holicky P.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 2

151-167

We prove an abstract version of the Kuratowski extension theorem for Borel measurable maps of a given class. It enables us to deduce and improve its nonseparable version due to Hansell. We also study the ranges of not necessarily injective Borel bimea-surable maps f and show that some control on the relative classes of preimages and images of Borel sets under f enables one to get a bound on the absolute class of the range of f. This seems to be of some interest even within separable spaces.

Descriptive properties of spaces of measures

Holicky P., Kalenda O.

Bulletin of the Polish Academy of Sciences. Mathematics

1999

Vol. 47, no 1

37-51

The spaces of Borel probabilities on a topological space X inherit a number of topological properties of X. We show in particular that the space of tight probabilities on a Cech-analytic space is Cech-analytic. Analogical results are shown for several other classes of generalized analytic and complete topological spaces.