On some Banach space properties sufficient for weak normal structure and their permanence properties

Sims, Brailey; Smyth, Michael

doi:10.1090/S0002-9947-99-01862-0

On some Banach space properties sufficient for weak normal structure and their permanence properties
HTML articles powered by AMS MathViewer

by Brailey Sims and Michael A. Smyth PDF

Trans. Amer. Math. Soc. 351 (1999), 497-513 Request permission

Abstract:

We consider Banach space properties that lie between conditions introduced by Bynum and Landes. These properties depend on the metric behavior of weakly convergent sequences. We also investigate the permanence properties of these conditions.

References

M. A. Akcoglu and L. Sucheston, On uniform monotonicity of norms and ergodic theorems in function spaces, Proceedings of the conference commemorating the 1st centennial of the Circolo Matematico di Palermo (Italian) (Palermo, 1984), 1985, pp. 325–335. MR 881411
Handbook of mathematical logic, Studies in Logic and the Foundations of Mathematics, vol. 90, North-Holland Publishing Co., Amsterdam, 1977. With the cooperation of H. J. Keisler, K. Kunen, Y. N. Moschovakis and A. S. Troelstra. MR 457132
L. P. Belluce, W. A. Kirk, and E. F. Steiner, Normal structure in Banach spaces, Pacific J. Math. 26 (1968), 433–440. MR 233178
Tomás Domínguez Benavides, Some properties of the set and ball measures of noncompactness and applications, J. London Math. Soc. (2) 34 (1986), no. 1, 120–128. MR 859153, DOI 10.1112/jlms/s2-34.1.120
Tomás Domínguez Benavides, Weak uniform normal structure in direct sum spaces, Studia Math. 103 (1992), no. 3, 283–290. MR 1202012, DOI 10.4064/sm-103-3-283-290
W. L. Bynum, Normal structure coefficients for Banach spaces, Pacific J. Math. 86 (1980), no. 2, 427–436. MR 590555
D. van Dulst, Reflexive and superreflexive Banach spaces, Mathematical Centre Tracts, vol. 102, Mathematisch Centrum, Amsterdam, 1978. MR 513590
D. van Dulst and Brailey Sims, Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK), Banach space theory and its applications (Bucharest, 1981) Lecture Notes in Math., vol. 991, Springer, Berlin-New York, 1983, pp. 35–43. MR 714171
D. van Dulst and V. de Valk, (KK)-properties, normal structure and fixed points of nonexpansive mappings in Orlicz sequence spaces, Canad. J. Math. 38 (1986), no. 3, 728–750. MR 845675, DOI 10.4153/CJM-1986-038-4
Kazimierz Goebel and W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge, 1990. MR 1074005, DOI 10.1017/CBO9780511526152
R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980), no. 4, 743–749. MR 595102, DOI 10.1216/RMJ-1980-10-4-743
A. Jiménez-Melado, Stability of weak normal structure in James quasi reflexive space, Bull. Austral. Math. Soc. 46 (1992), no. 3, 367–372. MR 1190338, DOI 10.1017/S0004972700012016
Denka Kutzarova and Thomas Landes, $k$-NUC of infinite direct sums, Boll. Un. Mat. Ital. A (7) 9 (1995), no. 3, 527–534 (English, with Italian summary). MR 1363553
D. Kutzarova, E. Maluta, and S. Prus, Property $(\beta )$ implies normal structure of the dual space, Rend. Circ. Mat. Palermo (2) 41 (1992), no. 3, 353–368. MR 1230583, DOI 10.1007/BF02848940
Thomas Landes, Permanence properties of normal structure, Pacific J. Math. 110 (1984), no. 1, 125–143. MR 722744
Thomas R. Landes, Normal structure and the sum-property, Pacific J. Math. 123 (1986), no. 1, 127–147. MR 834143
Zdzisław Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591–597. MR 211301, DOI 10.1090/S0002-9904-1967-11761-0
Stanisław Prus, On Bynum’s fixed point theorem, Atti Sem. Mat. Fis. Univ. Modena 38 (1990), no. 2, 535–545. MR 1076471
Brailey Sims and M. Smyth, On non-uniform conditions giving weak normal structure, Quaestiones Math. 18 (1995), no. 1-3, 9–19. First International Conference in Abstract Algebra (Kruger Park, 1993). MR 1340469
Timothy G. Feeman, Nest algebras of operators and the Dunford-Pettis property, Canad. Math. Bull. 34 (1991), no. 2, 208–214. MR 1113298, DOI 10.4153/CMB-1991-033-7
Daryl Tingley, The normal structure of James quasireflexive space, Bull. Austral. Math. Soc. 42 (1990), no. 1, 95–100. MR 1066363, DOI 10.1017/S0004972700028185
Guang Lu Zhang, Weakly convergent sequence coefficient of product space, Proc. Amer. Math. Soc. 117 (1993), no. 3, 637–643. MR 1152993, DOI 10.1090/S0002-9939-1993-1152993-1
Weiyu Zhao, Remarks on various measures of noncompactness, J. Math. Anal. Appl. 174 (1993), no. 1, 290–297. MR 1212933, DOI 10.1006/jmaa.1993.1117