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lower and upper semi-inner product; semi-inner products; Schwarz inequality; smooth normed spaces; Birkhoff orthogonality; best approximants
In this paper we introduce two mappings associated with the lower and upper semi-inner product $(\cdot ,\cdot )_i$ and $(\cdot ,\cdot )_s$ and with semi-inner products $[\cdot ,\cdot ]$ (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.
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