Zhang, Z. H.; Montesinos, Vicente; Liu, C. Y.; Gong, W. Z. Geometric properties and continuity of the pre-duality mapping in Banach space. (English) Zbl 1337.46013 Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 109, No. 2, 407-416 (2015). Summary: We use the preduality mapping in proving characterizations of some geometric properties of Banach spaces. In particular, those include nearly strongly convexity, nearly uniform convexity – a property introduced by K. Goebel and T. Sekowski – and nearly very convexity. Cited in 2 Documents MSC: 46B20 Geometry and structure of normed linear spaces 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) Keywords:duality mapping; pre-duality mapping; \(\alpha\)-upper semi-continuity; usco mapping; nearly strongly convex space; nearly uniformly convex space; nearly very convex space PDFBibTeX XMLCite \textit{Z. H. Zhang} et al., Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 109, No. 2, 407--416 (2015; Zbl 1337.46013) Full Text: DOI Link References: [1] Bandyopadhyay, P., Huang, D., Lin, B.L., Troyanski, S.L.: Some generalizations of local uniform rotundity. J. Math. Anal. Appl. 252, 906-916 (2000) · Zbl 0978.46004 [2] Bandyopadhyay, P., Li, Y., Lin, B., Narayana, D.: Proximinality in Banach spaces. J. Math. Anal. Appl. 341, 309-317 (2008) · Zbl 1138.46008 [3] Diestel, J.: Geometry of Banach Spaces. Selected Topics, LNM, vol. 485. Springer, Berlin (1975) · Zbl 0307.46009 [4] Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach Space Theory. The Basis for Linear and Nonlinear Analysis, CMS Books in Mathematics. Springer, Berlin (2011) · Zbl 1229.46001 [5] Giles, J.R., Gregory, D.A., Sims, B.: Geometrical implications of upper semi-continuity of the duality mapping on a Banach space. Pacific J. Math. 79(1), 99-109 (1978) · Zbl 0399.46012 [6] Goebel, K., Sekowski, T.: The modulus of non-compact convexity. Ann. Univ. M. Curie-Sklodowska, Sect. A 38, 41-48 (1984) · Zbl 0607.46011 [7] Guirao, A.J., Montesinos, V.: A note in approximative compactness and continuity of metric projections in Banach spaces. J. Convex Anal. 18, 397-401 (2011) · Zbl 1219.46019 [8] Huff, R.: Banach spaces which are nearly uniformly convex. Rocky Mountain J. Math. 10(4), 743-749 (1980) · Zbl 0505.46011 [9] Kutzarova, D., Rolewicz, S.: On nearly uniformly convex sets. Arch. Math. 57, 385-394 (1991) · Zbl 0756.52004 [10] Kutzarova, D., Lin, B.L., Zhang, W.: Some geometrical properties of Banach spaces related to nearly uniform convexity. Contemp. Math. 144, 165-171 (1993) · Zbl 0804.46025 [11] Kutzarova, D., Prus, S.: Operators which factor through nearly uniformly convex spaces. Boll. Un. Mat. Ital. B (7) 9, 2, 479-494 (1995) · Zbl 0934.47011 [12] Montesinos, V.: Drop property equals reflexivity. Studia Math. 87, 93-100 (1987) · Zbl 0652.46009 [13] Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability, LNM, vol. 1364, 2nd edn. Springer, Berlin (1993) · Zbl 0921.46039 [14] Rolewicz, S.: On drop property. Studia Math. 85, 27-37 (1986) · Zbl 0642.46011 [15] Rolewicz, S.: On \[\Delta\] Δ-uniform convexity and drop property. Studia Math. 87, 181-191 (1987) · Zbl 0652.46010 [16] Wu, C.X., Li, Y.J.: Strong convexity in Banach spaces. J. Math. Wuhan Univ. 13(1), 105-108 (1993) · Zbl 0802.46026 [17] Wang, J.H., Nan, C.X.: The continuity of subdifferential mapping. J. Math. Anal. Appl. 210, 206-214 (1997) · Zbl 0912.46046 [18] Wang, J.H., Zhang, Z.H.: Characterization of the property (C-K). Acta Math. Sci. Ser. A Chin. Ed. 17(A)(3), 280-284 (1997) · Zbl 0917.46013 [19] Zhang, Z.H., Liu, C.Y.: Some generalization of locally and weakly locally uniformly convex space. Nonlinear Anal. 74(12), 3896-3902 (2011) · Zbl 1232.46016 [20] Zhang, Z.H., Liu, C.Y.: Convexity and proximinality in Banach spaces. J. Funct. Spaces Appl. 2012, 11 (2012). doi:10.1155/2012/724120. Article ID 724120 · Zbl 1241.46011 [21] Zhang, Z.H., Liu, C.Y.: Convexity and existence of the farthest point. Abstract Appl. Anal. 2011, 9 (2011). doi:10.1155/2011/139597. Article ID 139597 · Zbl 1241.46010 [22] Zhang, Z.H., Shi, Z.R.: Convexities and approximative compactness and continuity of the metric projection in Banach spaces. J. Approx. Theory 161(2), 802-812 (2009) · Zbl 1190.46018 [23] Zhang, Z.H., Zhang, C.J.: On very rotund Banach spaces. Appl. Math. Mech. (English Ed.) 21(8), 965-970 (2000) · Zbl 0967.46017 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.