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Two mappings related to semi-inner products and their applications in geometry of normed linear spaces

Sever Silvestru Dragomir, Jaromír J. Koliha (2000)

Applications of Mathematics

In this paper we introduce two mappings associated with the lower and upper semi-inner 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.

What is a disk?

Kari Hag (1999)

Banach Center Publications

This paper should be considered as a companion report to F.W. Gehring’s survey lectures “Characterizations of quasidisks” given at this Summer School [7]. Notation, definitions and background results are given in that paper. In particular, D is a simply connected proper subdomain of R 2 unless otherwise stated and D* denotes the exterior of D in R ¯ 2 . Many of the characterizations of quasidisks have been motivated by looking at properties of euclidean disks. It is therefore natural to go back and ask...

ρ-Orthogonality and its preservation - revisited

Jacek Chmieliński, Paweł Wójcik (2013)

Banach Center Publications

The aim of the paper is to present results concerning the ρ-orthogonality and its preservation (both accurate and approximate) by linear operators. We survey on the results presented in [11] and [23], as well as give some new and more general ones.

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