Journal of Convex Analysis 23 (2016), No. 4, 1161--1183
Copyright Heldermann Verlag 2016
Spaces of d.c. Mappings on Arbitrary Intervals
Libor Veselý
Universitŕ degli Studi di Milano, Dip. di Matematica "F. Enriques", Via C. Saldini 50, 20133 Milano, Italy
libor.vesely@unimi.it
Ludek Zajícek
Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675 Praha 8, Czech Republic
zajicek@karlin.mff.cuni.cz
[Abstract-pdf]
\newcommand{\R}{\mathbf{R}} Let $X$ be a Banach space. Using derivatives in the sense of vector distributions, we show that the space $DC([0,1],X)$ of all d.c.\ mappings from $[0,1]$ into $X$, in a natural norm, is isomorphic to the space $M_{bv}([0,1], X)$ of all vector measures with bounded variation. The same is proved for the space $BDC_b((0,\infty), X)$ of all bounded d.c.\ mappings with a bounded control function. The result for the space $DC([0,1], \R)$ of all continuous d.c.\ functions was (essentially) proved by M. Zippin [The space of differences of convex functions on $[0,1]$, Serdica Math. J. 26 (2000) 331--352] by a quite different method. The space $BDC_b((0,\infty), \R)$ consists of all differences of two bounded convex functions. Internal characterizations of its members were given by O. B{\"o}hme [On functions which are the difference of two bounded convex functions on $(0,\infty)$, Math. Nachr. 122 (1985) 45--58], but our characterization of its Banach structure is new.
Keywords: D.c. function, d.c. mapping, Banach space, vector measure, vector distribution.
MSC: 47H99, 26A51
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