Geodesic
Two mappings related to semi-inner products and their applications in geometry of normed linear spaces
Applications of Mathematics, Tome 45 (2000) no. 5, pp. 337-355.

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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.
DOI : 10.1023/A:1022268627299
Classification : 41A50, 46B20, 46B99, 46C50, 46C99
Keywords: lower and upper semi-inner product; semi-inner products; Schwarz inequality; smooth normed spaces; Birkhoff orthogonality; best approximants 
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Dragomir, S. S.; Koliha, J. J. Two mappings related to semi-inner products and their applications in geometry of normed linear spaces. Applications of Mathematics, Tome 45 (2000) no. 5, pp. 337-355. doi : 10.1023/A:1022268627299. http://geodesic.mathdoc.fr/articles/10.1023/A:1022268627299/

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