Ball Proximinal and Strongly Ball Proximinal Spaces

Pei-Kee Lin; Wen Zhang; Bentuo Zheng

doi:10.68381/jca22034

Abstract

Let

Y

be an

E

-proximinal (respectively, a strongly proximinal) subspace of

X

. We prove that

Y

is (strongly) ball proximinal in

X

if and only if for any

x\in X

with

(x+Y)\cap B_X\ne\emptyset

,

(x+Y)\cap B_X

is (strongly) proximinal in

x+Y

. Using this characterization and a smart construction, we obtain three Banach spaces

Z\subset Y\subset X

such that

Z

is ball proximinal in

X

and

Y/Z

is ball proximinal in

X/Z

, but

Y

is not ball proximinal in

X

. This solves a problem raised by P. Bandyopadhyay, Bor-Luh Lin and T.S.S.R.K. Rao [Ball proximinality in Banach spaces, in: Banach Spaces and Their Applications in Analysis (Oxford/USA, 2006) B. Randrianantoanina et al (eds.) Proceedings in Mathematics, de Gruyter, Berlin (2007) 251–264].

Author information

Contact details are reproduced from the original publication and may be historical.

Pei-Kee Lin

Dept. of Mathematics, University of Memphis, Memphis, TN 38152, U.S.A.

pklin@memphis.edu

Wen Zhang

School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China

wenzhang@xmu.edu.cn

Bentuo Zheng

Dept. of Mathematics, University of Memphis, Memphis, TN 38152, U.S.A.

bzheng@memphis.edu

Keywords

Ball proximinal
strongly ball proximinal
E-proximinal

Mathematics Subject Classification

46B20, 41A50

References

P. Bandyopadhyay, Bor-Luh Lin, T. S. S. R. K. Rao: Ball proximinality in Banach spaces, in: Banach Spaces and Their Applications in Analysis (Oxford/USA, 2006), B. Randrianantoanina et al. (ed.), de Gruyter, Berlin (2007) 251–264.Google Scholar
V. Barbu, T. Precupanu: Convexity and Optimization in Banach Spaces, 4th Ed., Springer Monographs in Mathematics, Springer, Dordrecht (2012).https://doi.org/10.1007/978-94-007-2247-7 Google Scholar
G. Godefroy, V. Indumathi: Proximinality and renormings: some new examples, J. Approx. Theory 176 (2013) 118–128.https://doi.org/10.1016/j.jat.2013.09.004 Google Scholar
V. Indumathi, S. Lalithambigai: Ball proximinal spaces, J. Convex Analysis 18 (2011) 353–366.https://doi.org/10.68381/jca18019 Google Scholar
V. Indumathi, N. Prakash: Ball proximinal and strongly ball proximinal hyperplanes, J. Approx. Theory 172 (2013) 37–46.https://doi.org/10.1016/j.jat.2013.04.006 Google Scholar
F. B. Saidi: On the proximinality of the unit ball of proximinal subspaces in Banach spaces: a counter example, Proc. Amer. Math. Soc. 133 (2005) 2697–2703.https://doi.org/10.1090/S0002-9939-05-08152-9 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2015. Rights now held by Banach Press.

[1] P. Bandyopadhyay, Bor-Luh Lin, T. S. S. R. K. Rao: Ball proximinality in Banach spaces, in: Banach Spaces and Their Applications in Analysis (Oxford/USA, 2006), B. Randrianantoanina et al. (ed.), de Gruyter, Berlin (2007) 251–264.Google Scholar

[2] V. Barbu, T. Precupanu: Convexity and Optimization in Banach Spaces, 4th Ed., Springer Monographs in Mathematics, Springer, Dordrecht (2012).https://doi.org/10.1007/978-94-007-2247-7 Google Scholar

[3] G. Godefroy, V. Indumathi: Proximinality and renormings: some new examples, J. Approx. Theory 176 (2013) 118–128.https://doi.org/10.1016/j.jat.2013.09.004 Google Scholar

[4] V. Indumathi, S. Lalithambigai: Ball proximinal spaces, J. Convex Analysis 18 (2011) 353–366.https://doi.org/10.68381/jca18019 Google Scholar

[5] V. Indumathi, N. Prakash: Ball proximinal and strongly ball proximinal hyperplanes, J. Approx. Theory 172 (2013) 37–46.https://doi.org/10.1016/j.jat.2013.04.006 Google Scholar

[6] F. B. Saidi: On the proximinality of the unit ball of proximinal subspaces in Banach spaces: a counter example, Proc. Amer. Math. Soc. 133 (2005) 2697–2703.https://doi.org/10.1090/S0002-9939-05-08152-9 Google Scholar