Characterizations of smooth spaces by approximate orthogonalities

Since the monograph by Amir that appeared in 1986, a lot of attention has been given to the problem of characterizing, by means of properties of the norms, when a Banach space is indeed a Hilbert space, i.e., when the norm derives from an inner product. In this paper, similar investigations will be carried out for smooth spaces instead of inner product spaces. We consider the approximate orthogonalities in real normed spaces. We show that the relations approximate semi-orthogonality and approximate ρ + -orthogonality are generally incomparable (unless the normed space is smooth). As a result, we give a characterization of smooth spaces in terms of those approximate orthogonalities.