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## On the sum of simultaneously proximinal sets

#### Longfa SUN [1] , Yuqi SUN [2] , Wen ZHANG [3] , Zheming ZHENG [4]

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.
simultaneous strong proximinality, simultaneous approximative compactness, compact sets, Banach space
• [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.
Primary Language en Mathematics Mathematics Orcid: 0000-0001-9576-5610Author: Longfa SUNInstitution: North China Electric Power UniversityCountry: China Orcid: 0000-0001-9576-5014Author: Yuqi SUNInstitution: XIAMEN UNIVERSITYCountry: China Orcid: 0000-0001-9576-5644Author: Wen ZHANG (Primary Author)Institution: XIAMEN UNIVERSITYCountry: China Orcid: 0000-0001-9576-5611Author: Zheming ZHENGInstitution: XIAMEN UNIVERSITYCountry: China Fundamental Research Funds for the Central Universities, National Natural Science Foundation of China. 2019MS121, 11731010. The authors were supported by the Fundamental Research Funds for the Central Universities 2019MS121 and the National Natural Science Foundation of China (no. 11731010 and 12071388). Publication Date : June 7, 2021
 Bibtex @research article { hujms696407, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2021}, volume = {50}, pages = {668 - 677}, doi = {10.15672/hujms.696407}, title = {On the sum of simultaneously proximinal sets}, key = {cite}, author = {Sun, Longfa and Sun, Yuqi and Zhang, Wen and Zheng, Zheming} } APA Sun, L , Sun, Y , Zhang, W , Zheng, Z . (2021). On the sum of simultaneously proximinal sets . Hacettepe Journal of Mathematics and Statistics , 50 (3) , 668-677 . DOI: 10.15672/hujms.696407 MLA Sun, L , Sun, Y , Zhang, W , Zheng, Z . "On the sum of simultaneously proximinal sets" . Hacettepe Journal of Mathematics and Statistics 50 (2021 ): 668-677 Chicago Sun, L , Sun, Y , Zhang, W , Zheng, Z . "On the sum of simultaneously proximinal sets". Hacettepe Journal of Mathematics and Statistics 50 (2021 ): 668-677 RIS TY - JOUR T1 - On the sum of simultaneously proximinal sets AU - Longfa Sun , Yuqi Sun , Wen Zhang , Zheming Zheng Y1 - 2021 PY - 2021 N1 - doi: 10.15672/hujms.696407 DO - 10.15672/hujms.696407 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 668 EP - 677 VL - 50 IS - 3 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.696407 UR - https://doi.org/10.15672/hujms.696407 Y2 - 2020 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics On the sum of simultaneously proximinal sets %A Longfa Sun , Yuqi Sun , Wen Zhang , Zheming Zheng %T On the sum of simultaneously proximinal sets %D 2021 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 50 %N 3 %R doi: 10.15672/hujms.696407 %U 10.15672/hujms.696407 ISNAD Sun, Longfa , Sun, Yuqi , Zhang, Wen , Zheng, Zheming . "On the sum of simultaneously proximinal sets". Hacettepe Journal of Mathematics and Statistics 50 / 3 (June 2021): 668-677 . https://doi.org/10.15672/hujms.696407 AMA Sun L , Sun Y , Zhang W , Zheng Z . On the sum of simultaneously proximinal sets. Hacettepe Journal of Mathematics and Statistics. 2021; 50(3): 668-677. Vancouver Sun L , Sun Y , Zhang W , Zheng Z . On the sum of simultaneously proximinal sets. Hacettepe Journal of Mathematics and Statistics. 2021; 50(3): 668-677. IEEE L. Sun , Y. Sun , W. Zhang and Z. Zheng , "On the sum of simultaneously proximinal sets", Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 3, pp. 668-677, Jun. 2021, doi:10.15672/hujms.696407

Authors of the Article
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[4]