Linear mappings approximately preserving orthogonality in real normed spaces

Abstract

In a normed space we introduce an exact and approximate orthogonality relation. We consider classes of linear mappings approximately preserving this kind of orthogonality. We show that, in particular, the property that a linear mapping approximately preserves the $B$-orthogonality is equivalent to that it approximately preserves the $\rho,\rho_+$-orthogonality (although these orthogonalities need not be equivalent). Moreover, we show that every approximately orthogonality preserving linear mapping is necessarily a scalar multiple of an almost isometry.