An orthogonality relation in complex normed spaces based on norm derivatives

Abstract

Let X be a complex normed space. Based on the right norm derivative ρ + , we define a mapping ρ ∞ byThe mapping ρ ∞ has a good response to some geometrical properties of X. For instance, we prove that ρ ∞ (x, y) = ρ ∞ (y, x) for all x, y ∈ X if and only if X is an inner product space. In addition, we define a ρ ∞ -orthogonality in X and show that a linear mapping preserving ρ ∞orthogonality has to be a scalar multiple of an isometry. A number of challenging problems in the geometry of complex normed spaces are also discussed.