Mappings approximately preserving orthogonality in normed spaces

Nonlinear Analysis (Impact Factor: 1.64). 12/2010; 73(12):3821-3831. DOI: 10.1016/

ABSTRACT We answer many open questions regarding approximately orthogonality preserving mappings (in Birkhoff–James sense) in normed spaces. In particular, we show that every approximately orthogonality preserving linear mapping (in Chmieliński sense) is necessarily a scalar multiple of an ε-isometry. Thus, whenever ε-isometries are close to isometries we obtain stability. An example is given showing that approximately orthogonality preserving mappings are in general far from scalar multiples of isometries, that is, stability does not hold.

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    ABSTRACT: In a normed space we introduce an exact and approximate orthogonality relation connected with “norm derivatives” r¢±{\rho^{\prime}_{\pm}} . We also consider classes of linear mappings preserving (exactly and approximately) this kind of orthogonality. Mathematics Subject Classification (2010)Primary 46B20-46C50-Secondary 39B82 KeywordsOrthogonality-approximate orthogonality-orthogonality preserving mappings-norm derivative
    Aequationes Mathematicae 01/2010; 80(1):45-55. · 0.42 Impact Factor
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    ABSTRACT: We survey mainly recent results on the two most important orthogonality types in normed linear spaces, namely on Birkhoff orthogonality and on isosceles (or James) orthogonality. We lay special emphasis on their fundamental properties, on their differences and connections, and on geometric results and problems inspired by the respective theoretical framework. At the beginning we also present other interesting types of orthogonality. This survey can also be taken as an update of existing related representations.
    Aequationes Mathematicae 01/2012; 83(1-2). · 0.42 Impact Factor


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May 31, 2014