S. S. Dragomir, School of Communications and Informatics, Victoria University of Technology, PO Box 14428, MCMC, Melbourne VIC 8001, Australia, e-mail: sever@matilda.vut.edu.au; J. J. Koliha, Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3052, Australia, e-mail: j.koliha@ms.unimelb.edu.au
Abstract: In this paper we introduce two mappings associated with the lower and upper \sip{} $(\cdot,\cdot)_i$ and $(\cdot,\cdot)_s$ and with \sip s $[\cdot,\cdot]$ (in the sense of Lumer) which generate the norm of a real normed linear space, and study properties of monotonicity and boundedness of these mappings. We give a refinement of the Schwarz inequality, applications to the Birkhoff orthogonality, to smoothness of normed linear spaces as well as to the characterization of best approximants.
Keywords: lower and upper \sip, \sip s, Schwarz inequality, smooth normed spaces, Birkhoff orthogonality, best approximants
Classification (MSC 1991): 46B20, 46B99, 46C99, 41A50
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