Longfa SUN, Yuqi SUN, Wen ZHANG, Zheming ZHENG

On the sum of simultaneously proximinal sets

In this paper, we show that the sum of a compact convex subset and a simultaneously $\tau$-strongly proximinal convex subset (resp. simultaneously approximatively $\tau$-compact convex subset) of a Banach space X is simultaneously $\tau$-strongly proximinal (resp. simultaneously approximatively $\tau$-compact ), and the sum of a weakly compact convex subset and a simultaneously approximatively weakly compact convex subset of X is still simultaneously approximatively weakly compact, where $\tau$ is the norm or the weak topology. Moreover, some related results on the sum of simultaneously proximinal subspaces are presented.

Keywords:

simultaneous strong proximinality, simultaneous approximative compactness, compact sets, Banach space,

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[1] P. Bandyopadhyay, Y. Li, B. Lin and D. Narayana, Proximinility in Banach spaces, J. Math. Anal. Appl. 341, 309–317, 2008.
[2] E.W. Cheney and D.E. Wulbert, The existence and unicity of best approximation, Math. Scand. 24, 113–140, 1969.
[3] L.X. Cheng, Q.J. Cheng and Z.H. Luo, On some new characterizations of weakly compact sets in Banach spaces, Studia Math. 201, 155–166, 2010.
[4] W. Deeb and R. Khalil, The sum of proximinal subspaces, Soochow J. Math. 18, 163–167, 1992.
[5] S. Dutta and P. Shunmugaraj, Strong proximinality of closed convex sets, J. Approx. Theory 163, 547–553, 2011.
[6] M. Feder, On the sum of proximinal subspaces, J. Approx. Theory 49, 144–148, 1987.
[7] S. Gupta and T.D. Narang, Simultaneous strong proximinality in Banach spaces, Turkish J. Math. 41, 725–732, 2017.
[8] R.C. James, Weak compactness and reflexivity, Israel J. Math. 2, 101–119, 1964.
[9] P.K. Lin, A remark on the sum of proximinal subspces, J. Approx. Theory 58, 55–57, 1989.
[10] J. Mach, Best simultaneous approximation of bounded functions with values in certain Banach spaces, Math. Ann. 240, 157–164, 1979.
[11] Q.F. Meng, Z.H. Luo and H.A. Shi, A remark on the sum of simultaneously proximina subspaces (in Chinese), J. Xiamen Univ. Nat. Sci. 56 (4), 551–554, 2017.
[12] T.D. Narang, Simultaneous approximation and Chebyshev centers in metric spaces, Matematicki Vesnik. 51, 61–68, 1999.
[13] I.A. Pyatyshev, Operations on approximatively compact sets, J. Math. Notes 82, 653– 659, 2007.
[14] T.S.S.R.K. Rao, Simultaneously proximinal subspaces, J. Appl. Anal. 22 (2), 115–120, 2016.
[15] T.S.S.R.K. Rao, Points of strong subdifferentiability in dual spaces, Houston J. Math. 44 (4), 1221–1226, 2018.
[16] M. Rawashdeh, S. Al-Sharif and W.B. Domi, On the sum of best simultaneously proximinal subspaces, Hacet. J. Math. Stat. 43, 595–602, 2014.
[17] W. Rudin, Functional Analysis, 2nd ed. New York, McGraw-Hill Inc, 1991.
[18] F. Saidi, D. Hussein and R. Khalil, Best simultaneous approximation in $L^P(I,X)$, J. Approx. Theory 116, 369–379, 2002.

Hacettepe Journal of Mathematics and Statistics-Cover

Yayın Aralığı: Yılda 6 Sayı
Başlangıç: 2002
Yayıncı: Hacettepe Üniversitesi Fen Fakultesi

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