Maps with the Radon-Nikodým property.

*(English)*Zbl 1396.46013Let \(X\) be a real Banach space with dual \(X^*\), \(C\) a closed convex subset of \(X\) and \((M,d)\) a metric space. A function \(f:C\to M\) is called dentable if, for every nonempty bounded subset \(A\) of \(C\) and every \(\varepsilon>0\), there exists an open half-space \(H\) of \(X\) such that \(A\cap H\neq\emptyset\) and diam\((f(A\cap H))<\varepsilon\). Denote by \(\mathcal D(C,M)\) the set of all dentable mappings from \(C\) to \(M\) and by \(\mathcal D_U(C,M)\) its subset formed by all dentable mappings uniformly continuous on bounded subsets of \(C\).

The notion is related to the Radon-Nikodým (RN) property: the set \(C\) has the RN property iff the identity mapping \(I:C\to(C,\|\cdot\|)\) is dentable. Also, a continuous linear operator from \(X\) to another Banach space \(Y\) is dentable iff it is an RN operator in the sense of O. I. Reinov [Sov. Math., Dokl. 16, 119–123 (1975; Zbl 0317.47022); translation from Dokl. Akad. Nauk SSSR 220, 528–531 (1975)], and so the study of dentable mappings is, in some sense, a nonlinear extension of the RN property. The authors show that the set \(C\) has the RN property iff every Lipschitz mapping \(f:C\to M\) is dentable. If \(M\) is a Banach space (Banach algebra, Banach lattice), then \(\mathcal D_U(C,M)\) is also a Banach space (Banach algebra, Banach lattice, respectively) with respect to the norm of uniform convergence on \(C\).

It is known that the strongly exposing functionals on a closed convex set with the RN property form a dense \(G_\delta\) subset of \(X^*\). The authors extend this result to this frame by replacing strongly exposing functionals by a class of functionals called strongly slicing. The possibility of uniform approximation of a uniformly continuous function \(f\) by DC (difference of convex) functions is also studied. It turns out that this happens iff the function \(f\) is finitely dentable in the sense defined by M. Raja [J. Convex Anal. 15, No. 2, 219–233 (2008; Zbl 1183.46018)]. Other results as, for instance, Stegall’s variational principle, are no longer true beyond the usual hypotheses, sending back to the classical case.

The notion is related to the Radon-Nikodým (RN) property: the set \(C\) has the RN property iff the identity mapping \(I:C\to(C,\|\cdot\|)\) is dentable. Also, a continuous linear operator from \(X\) to another Banach space \(Y\) is dentable iff it is an RN operator in the sense of O. I. Reinov [Sov. Math., Dokl. 16, 119–123 (1975; Zbl 0317.47022); translation from Dokl. Akad. Nauk SSSR 220, 528–531 (1975)], and so the study of dentable mappings is, in some sense, a nonlinear extension of the RN property. The authors show that the set \(C\) has the RN property iff every Lipschitz mapping \(f:C\to M\) is dentable. If \(M\) is a Banach space (Banach algebra, Banach lattice), then \(\mathcal D_U(C,M)\) is also a Banach space (Banach algebra, Banach lattice, respectively) with respect to the norm of uniform convergence on \(C\).

It is known that the strongly exposing functionals on a closed convex set with the RN property form a dense \(G_\delta\) subset of \(X^*\). The authors extend this result to this frame by replacing strongly exposing functionals by a class of functionals called strongly slicing. The possibility of uniform approximation of a uniformly continuous function \(f\) by DC (difference of convex) functions is also studied. It turns out that this happens iff the function \(f\) is finitely dentable in the sense defined by M. Raja [J. Convex Anal. 15, No. 2, 219–233 (2008; Zbl 1183.46018)]. Other results as, for instance, Stegall’s variational principle, are no longer true beyond the usual hypotheses, sending back to the classical case.

Reviewer: Stefan Cobzaş (Cluj-Napoca)

##### MSC:

46B22 | Radon-Nikodým, Kreĭn-Milman and related properties |

46E40 | Spaces of vector- and operator-valued functions |

41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |

46T20 | Continuous and differentiable maps in nonlinear functional analysis |

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\textit{L. García-Lirola} and \textit{M. Raja}, Set-Valued Var. Anal. 26, No. 1, 77--93 (2018; Zbl 1396.46013)

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