In this paper, we discuss the relation between $\tau$-strongly Chebyshev, approximatively $\tau$-compact $k$-Chebyshev, approximatively $\tau$-compact, $\tau$-strongly proximinal and proximinal sets, where $\tau$ is the norm or the weak topology. We give some equivalent conditions regarding the above proximinality. Furthermore, we also propose the necessary and sufficient conditions that a half-space is $\tau$-strongly proximinal, approximatively $\tau$-compact and $\tau$-strongly Chebyshev.
Zihou ZHANG
,
Yu ZHOU
,
Chunyan LIU
,
Jing ZHOU
. SOME EQUIVALENT CONDITIONS OF PROXIMINALITY IN NONREFLEXIVE BANACH SPACES[J]. Acta mathematica scientia, Series B, 2022
, 42(4)
: 1621
-1630
.
DOI: 10.1007/s10473-022-0419-y
[1] Bandyopadhyay P, Li Y, Lin, B L, et al. Proximinality in Banach spaces. J Math Anal Appl, 2008, 341(1):309-317
[2] Bandyopadhyay P, Huang D, Lin B L, Troyanski S T. Some generalization of locally uniform rotundity. J Math Anal Appl, 2000, 252:906-916
[3] Cabrera I J, Sadarangani K B. Weak near convexity and smoothness of Banach spaces. Arch Math (Basel), 2002, 78(2):126-134
[4] Dutta S, Narayana D. Strongly proximinal subspaces in Banach spaces. Contemp. Math Amer Math Soc, 2007, 435:143-152
[5] Dutta S, Shunmugaraj P. Strong proximinality of closed convex sets. J Approx Theory, 2011, 163(4):547-553
[6] Fang X N, Wang, J H. Convexity and the continuity of metric projections. Math Appl (Wuhan), 2001, 14(1):47-51
[7] Godefroy G, Indumathi V. Strong proximinality and polyhedral spaces. Rev Ma Complu, 2001, 14(1):105-125
[8] Godefroy G, Indumathi V. Proximinality and renormings:some new examples. J Approx Theory, 2013, 176:118-128
[9] Godefroy G, Indumathi V, Lust-Piquard F. Strong subdifferentiability of convex functionals and proximinality. J Approx Theory, 2002, 116(2):397-415
[10] Guirao A J, Montesinos V. A note in approximative compactness and continuity of metric projections in Banach spaces. J Convex Anal, 2011, 18(2):397-401
[11] Saidi F B. On the proximinality of the unit ball of proximinal subspaces in Banach spaces:a counterexample Proc Amer Math Soc, 2005, 133(9):2697-2703
[12] Sullivan F. A generalization of uniformly rotund Banach space. Canad J Math, 197931:628-646
[13] Zhang Z H, Liu C Y. Some generalizations of locally and weakly locally uniformly convex space. Nonlinear Anal, 2011, 74(12):3896-3902
[14] Wang J H, Zhang Z H. Characterization of the property (C-K). Acta Math Sci, 1997, 17A(3):280-284(in Chinese)
[15] Zhang Z H, Liu, C Y, Zhou Y. Banach space geometry and best approximation in Banach spaces. Beijing:Science Press, 2016(in Chinese)
[16] Zhang Z H, Liu C Y. The representations and continuity of the metric projections on two classes of halfspaces in Banach spaces. Abstr Appl Anal, 2014, 2014. Art ID 908676, 5 pages
[17] Zhang Z H, Liu C Y, Zhou Y. Some examples concerning proximinality in Banach spaces. J Approx Theory, 2015, 200:136-143
[18] Zhang Z H, Zhou Y, Liu C Y. Near convexity, near smoothness and approximative compactness of half spaces in Banach spaces. Acta Math Sin (Engl Ser), 2016, 32(5):599-606
[19] Liu C Y, Zhang Z H, Zhou Y. A note in approximative compactness and midpoint locally k-uniform rotundity in Banach spaces. Acta Math Sci, 2018, 38B(2):643-650
