Geodesic
The generalized Day norm. Part I. Properties
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 71 (2017) no. 2.

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In this paper we introduce a modification of the Day norm in c_0(Γ) and investigate properties  of this norm.
Keywords: Asymptotic normal structure, Day norm, local uniform convexity, normal structure, Opial property, strict convexity, uniform convexity in every direction 
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Budzyńska, Monika; Grzesik, Aleksandra; Kot, Mariola. The generalized Day norm. Part I. Properties. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 71 (2017) no. 2. http://geodesic.mathdoc.fr/item/AUM_2017_71_2_a6/

[1] Boas, R. P., Jr., Some uniformly convex spaces, Bull. Amer. Math. Soc. 46 (1940), 304-311.

[2] Baillon, J.-B., Schoneberg, R., Asymptotic normal structure and fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 81 (1981), 257-264.

[3] Brodskii, M. S., Mil’man, D. P., On the center of a convex set, Doklady Akad. Nauk SSSR (N.S.) 59 (1948), 837-840.

[4] Clarkson, J. A., Unifomly convex spaces, Trans. Amer. Math. Soc. 78 (1936), 396-414.

[5] Day, M. M., Strict convexity and smoothness of normed spaces, Trans. Amer. Math. Soc. 78 (1955), 516-528.

[6] Day, M. M., James, R. C., Swaminathan, S., Normed linear spaces that are uniformly convex in every direction, Canad. J. Math. 23 (1971), 1051-1059.

[7] Dodds, P. G., Dodds, T. K., Sedaev, A. A., Sukochev, F. A., Local uniform convexity and Kadec-Klee type properties in K-interpolation spaces. I: General theory, J. Funct. Spaces Appl. 2 (2004), 125-173.

[8] Garkavi, A. L., On the optimal net and best cross-section of a set in a normed space (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962), 87-106.

[9] Goebel, K., Kirk, W. A., Topics in Metric Fixed Point Theory, Cambridge University Press, 1990.

[10] Goebel, K., Reich, S., Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, 1984.

[11] Hanner, O., On the uniform convexity of \(L^p\) and \(l^p\), Ark. Mat. 3 (1956), 239-244.

[12] Holmes, R. B., Geometric Functional Analysis and Its Applications, Springer, 1975.

[13] Kirk, W. A., A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006.

[14] Lovaglia, A. R., Locally uniformly convex Banach spaces, Trans. Amer Math. Soc. 78 (1955), 225-238.

[15] Maluta, E., A diametrically complete set with empty interior in a reflexive LUR space, J. Nonlinear Conv. Anal. 18 (2017),105-111.

[16] Mariadoss, S. A., Soardi, P. M., A remark on asymptotic normal structure in Banach spaces, Rend. Sem. Mat. Univ. Politec. Torino 44 (1986), 393-395.

[17] Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597.

[18] Rainwater, J., Local uniform convexity of Day’s norm on \(c_0(\Gamma)\), Proc. Amer. Math. Soc. 22 (1969), 335-339.

[19] Smith, M. A., Some examples concerning rotundity in Banach spaces, Math. Ann. 233 (1978), 155-161.

[20] Smith, M. A., Turett, B., A reflexive LUR Banach spaces that lacks normal structure, Canad. Math. Bull. 28 (1985), 492-494.