Two mappings related to semi-inner products and their applications in geometry of normed linear spaces

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In this paper we introduce two mappings associated with the lower and upper semi-inner product (·, ·)i and (·, ·)S and with semi-inner products [·, ·] (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.

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