On best approximations in Banach spaces from the perspective of orthogonality

Abstract

We study best approximations in Banach spaces via Birkhoff-James orthogonality of functionals. To exhibit the usefulness of Birkhoff-James orthogonality techniques in the study of best approximation problems, some algorithms and distance formulae are presented. As an application of our study, we obtain some crucial inequalities, which also strengthen the classical Hölder's inequality. The relevance of the algorithms and the inequalities are discussed through concrete examples.2010 Mathematics Subject Classification. Primary 46B28, Secondary 46B20. Key words and phrases. Birkhoff-James orthogonality; weak * continuous functionals; best approximations; inequalities.The research of Dr. Debmalya Sain is sponsored by DST-SERB N-PDF Fellowship under the mentorship of Professor Apoorva Khare. Dr. Sain feels grateful to have the opportunity to acknowledge the colossal contribution of Mr. Swapan Bandyopahyay, an extraordinarily devoted teacher, towards rightly shaping the philosophies of so many young learners like himself. The research of Mr. Saikat Roy is supported by CSIR MHRD in form of Senior Research Fellowship under the supervision of Professor Satya Bagchi.