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Two mappings related to semi-inner products and their applications in geometry of normed linear spaces. (English) Zbl 0996.46007
Two mappings associated with the lower and upper semi-inner products \((\cdot ,\cdot)_i\) and \((\cdot ,\cdot)_s\) and with semi-inner product \([\cdot ,\cdot ]\) (in the sense of Lumer) are introduced. The mappings generate the norm of a real normed linear space. Monotonicity and boundedness of these mappings is studied and it is given 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.

MSC:
46B20 Geometry and structure of normed linear spaces
46C50 Generalizations of inner products (semi-inner products, partial inner products, etc.)
41A50 Best approximation, Chebyshev systems
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