Abstract
In this paper, we show that every super weakly compact convex subset of a Banach space is an absolute uniform retract, and that it also admits the uniform compact approximation property. These can be regarded as extensions of Lindenstrauss and Kalton’s corresponding results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Beauzamy, B.: Opérateurs uniformément convexifiants. Studia Math., 57(2), 103–139 (1976)
Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis, Vol. 1. American Mathematical Society Colloquium Publications, Vol. 48. American Mathematical Society, Providence, RI, 2000
Cepedello-Boiso, M.: Approximation of Lipschitz functions by Δ-convex functions in Banach spaces. Israel J. Math., 106, 269–284 (1998)
Cheng, L., Cheng, Q., Luo, S., et al.: On super weak compactness of subsets and its equivalences in Banach spaces. J. Convex Anal., 25(3), 899–926 (2018)
Cheng, L., Cheng, Q., Wang, B., et al.: On super-weakly compact sets and uniform convexifiable sets. Studia. Math., 199(2), 145–169 (2010)
Cheng, L., Cheng, Q., Zhang, J.: On super fixed point property and super weak compactness of convex subsets in Banach spaces. J. Math. Anal. Appl., 428(2), 1209–1224 (2015)
Cheng, L., Luo, Z., Zhou, Y.: On super weakly compact convex sets and representation of the dual of the normed semigroup they generate. Canad. Math. Bull., 56(2), 272–282 (2013)
Cheng, L., Wu, C., Xue, X., et al.: Convex functions, subdifferentiability and renormings. Acta Math. Sin., Engl. Ser., 14(1), 47–56 (1998)
Cheng, Q., Wang, B., Wang, C.: On uniform convexity of Banach spaces. Acta Math. Sin., Engl. Ser., 27(3), 587–594 (2011)
Clarkson, J. A.: Uniform convex spaces. Trans. Amer. Math. Soc., 40(3), 396–414 (1936)
Enflo, P.: Banach spaces which can be given an equivalent uniform convex norm. Israel J. Math., 13, 281–288 (1972)
Fabian, M., Montesinos, V., Zizler, V.: Sigma-finite dual dentability indices. J. Math. Anal. Appl., 350(2), 498–507 (2009)
Isbell, J. R.: On finite-dimensional uniform spaces. Pacific J. Math., 9, 107–121 (1959)
Isbell, J. R.: Uniform neighborhood retracts. Pacific J. Math., 11, 609–648 (1961)
James, R. C.: Super-reflexive Banach spaces. Canad J. Math., 24, 896–904 (1972)
Kalton, N. J.: Spaces of Lipschitz and Hölder functions and their applications. Collect. Math., 55(2), 171–217 (2004)
Kalton, N. J.: The uniform structure of Banach spaces. Math. Ann., 354(4), 1247–1288 (2012)
Lindenstrauss, J.: On nonlinear projections in Banach spaces. Michigan Math. J., 11, 263–287 (1964)
Mazur, S.: Une remarque sur l’homéomorphie des champs fonctionnels. Studia. Math., 1(1), 83–85 (1929)
Raja, M.: Finitely dentable functions, operators and sets. J. Convex Anal., 15(2), 219–233 (2008)
Raja, M.: Super WCG Banach spaces. J. Math. Anal. Appl., 439(1), 183–196 (2016)
Acknowledgements
We thank the referees for their time and comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Support by National Natural Science Foundation of China (Grant Nos. 11731010, 12071389)
Rights and permissions
About this article
Cite this article
Cheng, L.X., Cheng, Q.J. & Wang, J.J. On Absolute Uniform Retracts, Uniform Approximation Property and Super Weakly Compact Sets of Banach Spaces. Acta. Math. Sin.-English Ser. 37, 731–739 (2021). https://doi.org/10.1007/s10114-021-0273-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-021-0273-1