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.


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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