×

zbMATH — the first resource for mathematics

Maps with the Radon-Nikodým property. (English) Zbl 1396.46013
Let \(X\) 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.
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bačák, M; Borwein, JM, On difference convexity of locally Lipschitz functions, Optimization, 60, 961-978, (2011) · Zbl 1237.46007
[2] Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48. American Mathematical Society, Providence, RI (2000) · Zbl 0946.46002
[3] Borwein, J.M., Vanderwerff, J.D.: Convex Functions: Constructions, Characterizations and Counterexamples. Encyclopedia of Mathematics and its Applications, vol. 109. Cambridge University Press, Cambridge (2010)
[4] Bourgain, J, Dentability and finite-dimensional decompositions, Stud. Math., 67, 135-148, (1980) · Zbl 0434.46017
[5] Bourgin, R.D.: Geometric Aspects of Convex Sets with the Radon-Nikodým Property. Lecture Notes in Mathematics, vol. 993. Springer, Berlin (1983)
[6] Cepedello Boiso, M, Approximation of Lipschitz functions by ##δ##-convex functions in Banach spaces, Israel J. Math., 106, 269-284, (1998) · Zbl 0920.46010
[7] Cheng, L; Zhou, Y, Approximation by DC functions and application to representation of a normed semigroup, J. Convex Anal., 21, 651-661, (2014) · Zbl 1315.46050
[8] Diestel, J.: Geometry of Banach Spaces—Selected Topics. Lecture Notes in Mathematics, vol. 485. Springer, Berlin (1975) · Zbl 0307.46009
[9] Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach Space Theory. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2011). The basis for linear and nonlinear analysis · Zbl 1229.46001
[10] Frontisi, J, Smooth partitions of unity in Banach spaces, Rocky Mt. J. Math., 25, 1295-1304, (1995) · Zbl 0853.46015
[11] García, F; Oncina, L; Orihuela, J; Troyanski, S, Kuratowski’s index of non-compactness and renorming in Banach spaces, J. Convex Anal., 11, 477-494, (2004) · Zbl 1067.46002
[12] Ghoussoub, N; Maurey, B, Plurisubharmonic martingales and barriers in complex quasi-Banach spaces, Ann. Inst. Fourier (Grenoble), 39, 1007-1060, (1989) · Zbl 0678.46013
[13] Hiriart-Urruty, JB, Generalized differentiability, duality and optimization for problems dealing with differences of convex functions, 37-70, (1985), Berlin
[14] Huff, RE; Morris, PD, Geometric characterizations of the Radon-nikodým property in Banach spaces, Stud. Math., 56, 157-164, (1976) · Zbl 0351.46011
[15] Jayne, JE; Orihuela, J; Pallarés, AJ; Vera, G, Σ-fragmentability of multivalued maps and selection theorems, J. Funct. Anal., 117, 243-273, (1993) · Zbl 0822.54018
[16] Lancien, G, Dentability indices and locally uniformly convex renormings, Rocky Mt. J. Math., 23, 635-647, (1993) · Zbl 0801.46010
[17] Linde, W, An operator ideal in connection with the Radon-Nikodym property of Banach spaces, Math. Nachr., 71, 65-73, (1976) · Zbl 0328.47011
[18] Moltó, A., Orihuela, J., Troyanski, S., Valdivia, M.: A Nonlinear Transfer Technique for Renorming. Lecture Notes in Mathematics, vol. 1951. Springer, Berlin (2009) · Zbl 1182.46001
[19] Raja, M, First Borel class sets in Banach spaces and the asymptotic-norming property, Israel J. Math., 138, 253-270, (2003) · Zbl 1065.46010
[20] Raja, M, Finitely dentable functions, operators and sets, J. Convex Anal., 15, 219-233, (2008) · Zbl 1183.46018
[21] Reı̆nov, OI, RN type operators in Banach spaces, Dokl. Akad. Nauk SSSR, 220, 528-531, (1975)
[22] Reı̆nov, OI, Geometric characterization of RN-operators, Mat. Zametki, 22, 189-202, (1977)
[23] Rolewicz, S.: Metric Linear Spaces, Mathematics and its Applications (East European Series), 2nd edn., vol. 20. D. Reidel Publishing Co., Dordrecht; PWN—Polish Scientific Publishers, Warsaw (1985)
[24] Schachermayer, W, The sum of two Radon-nikodým-sets need not be a Radon-nikodým-set, Proc. Amer. Math. Soc., 95, 51-57, (1985) · Zbl 0593.46021
[25] Tuy, H, D.C. optimization: theory, methods and algorithms, 149-216, (1995), Dordrecht · Zbl 0832.90111
[26] Veselý, L; Zajíček, L, Delta-convex mappings between Banach spaces and applications, Diss. Math. (Rozprawy Mat.), 289, 52, (1989) · Zbl 0685.46027
[27] Veselý, L; Zajíček, L, Spaces of d.c. mappings on arbitrary intervals, J. Convex Anal., 23, 1161-1183, (2016) · Zbl 1373.46018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.