The generalized Day norm. Part I. Properties

Monika Budzyńska, Aleksandra Grzesik, Mariola Kot


In this paper we introduce a modification of the Day norm in \(c_0(\Gamma)\) and investigate properties  of this norm.


Asymptotic normal structure; Day norm; local uniform convexity; normal structure; Opial property; strict convexity; uniform convexity in every direction

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

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

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

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

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

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.

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.

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.

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

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

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

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

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

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

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

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.

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

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

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

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

Data publikacji: 2017-12-18 20:31:32
Data złożenia artykułu: 2017-12-16 22:48:57


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