Abstract

Let XX be a Banach space. Using derivatives in the sense of vector distributions, we show that the space DC([0,1],X)DC([0,1],X) of all d.c. mappings from [0,1][0,1] into XX, in a natural norm, is isomorphic to the space Mbv([0,1],X)M_{bv}([0,1], X) of all vector measures with bounded variation. The same is proved for the space BDCb((0,∞),X)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)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][0,1], Serdica Math. J. 26 (2000) 331–352] by a quite different method. The space BDCb((0,∞),R)BDC_b((0,\infty), \R) consists of all differences of two bounded convex functions. Internal characterizations of its members were given by O. B{ö}hme [On functions which are the difference of two bounded convex functions on (0,∞)(0,\infty), Math. Nachr. 122 (1985) 45–58], but our characterization of its Banach structure is new.

Author information

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Libor Veselý

Università degli Studi di Milano, Dip. di Matematica "F. Enriques", Via C. Saldini 50, 20133 Milano, Italy

libor.vesely@unimi.it

Luděk Zajíček

Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675 Praha 8, Czech Republic

zajicek@karlin.mff.cuni.cz

Keywords

  • D.c. function
  • d.c. mapping
  • Banach space
  • vector measure
  • vector distribution

Mathematics Subject Classification

47H99, 26A51

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Suggested citation

L. Veselý, L. Zajíček. “Spaces of d.c. Mappings on Arbitrary Intervals.” Journal of Convex Analysis 23 (2016), No. 4, 1161–1183. https://doi.org/10.68381/jca23043

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Published by Heldermann Verlag, 2016. Rights now held by Banach Press LLC.