An alternative proof for a characterization of inner product spaces

The most geometric properties of inner product spaces like strict convexity and smoothness my fail to hold in a general normed linear spaces. Also, some main properties of the orthogonality in inner product spaces do not always carry over to generalized orthogonalities. Taking these into account different types of orthogonality relations provide a good frame for studying the geometric properties of normed linear spaces. In this paper, we give a characterization of inner product spaces using the notion of Hermite–Hadamard type of Carlsson’s orthogonality in normed linear spaces. First, we provide some more results about the existence property of this orthogonality. Next, we prove that Hermite-Hadamard type of Carlsson’s orthogonality is additive in a normed linear space X if and only if X is an inner product space. Our approach to prove this fact is using the relationship between Birkhoff-James orthogonality and the Gateaux differentiability of the norm of normed linear spaces.