YET ON LINEAR STRUCTURES OF NORM-ATTAINING FUNCTIONALS ON ASPLUND SPACES

doi:10.1016/S0252-9602(17)30122-4

数学物理学报(英文版) ›› 2018, Vol. 38 ›› Issue (1): 151-156.doi: 10.1016/S0252-9602(17)30122-4

YET ON LINEAR STRUCTURES OF NORM-ATTAINING FUNCTIONALS ON ASPLUND SPACES

程立新, 罗思捷

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

收稿日期:2016-12-12 修回日期:2017-03-13 出版日期:2018-02-25 发布日期:2018-02-25
通讯作者: Sijie LUO E-mail:winbestlsj@163.com
作者简介:Lixin CHENG,E-mail:lxcheng@xmu.edu.cn
基金资助:
This work is partially supported by NSFC, grant 11371296, and by PhD Programs Foundation of MEC, Grant 20130121110032.

YET ON LINEAR STRUCTURES OF NORM-ATTAINING FUNCTIONALS ON ASPLUND SPACES

Lixin CHENG, Sijie LUO

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Received:2016-12-12 Revised:2017-03-13 Online:2018-02-25 Published:2018-02-25
Contact: Sijie LUO E-mail:winbestlsj@163.com
Supported by:
This work is partially supported by NSFC, grant 11371296, and by PhD Programs Foundation of MEC, Grant 20130121110032.

摘要/Abstract

摘要：

In this paper, we show that if an Asplund space X is either a Banach lattice or a quotient space of C(K), then it can be equivalently renormed so that the set of normattaining functionals contains an infinite dimensional closed subspace of X^* if and only if X^* contains an infinite dimensional reflexive subspace, which gives a partial answer to a question of Bandyopadhyay and Godefroy.

关键词: norm-attaining functional, Asplund space, Banach lattice, reflexive subspace, Banach space

Abstract:

Key words: norm-attaining functional, Asplund space, Banach lattice, reflexive subspace, Banach space

程立新, 罗思捷. YET ON LINEAR STRUCTURES OF NORM-ATTAINING FUNCTIONALS ON ASPLUND SPACES[J]. 数学物理学报(英文版), 2018, 38(1): 151-156.

Lixin CHENG, Sijie LUO. YET ON LINEAR STRUCTURES OF NORM-ATTAINING FUNCTIONALS ON ASPLUND SPACES[J]. Acta mathematica scientia,Series B, 2018, 38(1): 151-156.

参考文献

[1] Acosta M D, Aron R M, García D, Maestre M. The Bishop-Phelps-Bollobás theorem for operators. J Funct Anal, 2008, 254(11):2780-2799
[2] Bandyopadhyay P, Godefroy G. Linear structures in the set of norm-attaining functionals on a Banach space. J Convex Anal, 2006, 13(3/4):489-497
[3] Bessaga C, Pe lczyński A. On bases and unconditional convergence of series in Banach spaces. Studia Math, 1958, 17(2):151-164
[4] Bishop E, Phelps R R. A proof that every Banach space is subreflexive. Bull Amer Math Soc, 1961, 67:97-98
[5] Bishop E, Phelps R R. The support functionals of a convex set//Proc Sympos Pure Math, Vol VⅡ. Providence, RI:Amer Math Soc, 1963:27-35
[6] Bollobás B. An extension to the theorem of Bishop and Phelps. Bull London Math Soc, 1970, 2:181-182
[7] Borwein J M. A note on ε subgradients and maximal monotonicity. Pacific J Math, 1982, 103(2):307-314
[8] Bourgain J. On dentability and the Bishop-Phelps property. Israel J Math, 1977, 28(4):265-271
[9] Brøndsted A, Rockafellar R T. On the subdifferentiability of convex functions. Proc Amer Math Soc, 1965, 16:605-611
[10] Cheng L, Dai D, Dong Y. A sharp operator version of the Bishop-Phelps theorem for operators from ℓ1 to CL-spaces. Proc Amer Math Soc, 2013, 141(3):867-872
[11] Ekeland I. On the variational principle. J Math Anal Appl, 1974, 47:324-353
[12] Ekeland I. Nonconvex minimization problems. Bull Amer Math Soc (NS), 1979, 1(3):443-474
[13] Fabian M. Each weakly countably determined Asplund space admits a Fréchet differentiable norm. Bull Aust Math Soc, 1987, 36(3):367-374
[14] Fabian M, Habala P, Hájek P, et al. Functional Analysis and Infinite-Dimensional Geometry. Springer Science & Business Media, 2013
[15] García-Pacheco F J. Three Non-Linear Problems on Normed Spaces. Kent State University, 2007
[16] García-Pacheco F J, Puglisi D. Lineability of functionals and operators. Studia Math, 2010, 201(1):37-47
[17] García-Pacheco F J, Puglisi D. Renormings concerning the lineability of the norm-attaining functionals. J Math Anal Appl, 2017, 445(2):1321-1327
[18] James R. Characterizations of reflexivity. Studia Math, 1964, 23(3):205-216
[19] Johnson W, Rosenthal H. On w^*-basic sequences and their applications to the study of Banach spaces. Studia Math, 1972, 43(1):77-92
[20] Lin P K. Köthe-Bochner Function Spaces. Boston:Birkhäuser, 2004
[21] Lindenstrauss J. On operators which attain their norm. Israel J Math, 1963, 1:139-148
[22] Lindenstrauss J, Tzafriri L. Classical Banach Spaces Ⅱ:Function Spaces. Springer Science & Business Media, 2013
[23] Meyer-Nieberg P. Banach Lattices. Springer Science & Business Media, 2012
[24] Pe lczyński A. Projections in certain Banach spaces. Studia Math, 1960, 19(2):209-228
[25] Phelps R R. Convex Functions, Monotone Operators and Differentiability. Springer, 2009
[26] Phelps R R. The Bishop-Phelps Theorem, Ten Mathematical Essays on Approximation in Analysis and Topology. Amsterdam:Elsevier BV, 2005:235-244
[27] Qiu J H, Li B, He F. Vectorial Ekelaud's variational principle with a w-distance and its equivalent theorems. Acta Math Sci, 2012, 32(6):2221-2236
[28] Rosenthal H P. On subspaces of L_p. Ann Math, 1973, 97(2):344-373

YET ON LINEAR STRUCTURES OF NORM-ATTAINING FUNCTIONALS ON ASPLUND SPACES

YET ON LINEAR STRUCTURES OF NORM-ATTAINING FUNCTIONALS ON ASPLUND SPACES

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