In this paper we introduce two mappings associated with the lower and upper semiinner 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.