Problems Related To Orthogonalities

In this paper characterizations of inner product spaces are obtained by studying properties of generalized orthogonalities, a quantitative characterization of the difference between Birkhoff orthogonality and Isosceles orthogonality is given, the definition of metric ellipse is introduced and basic properties of metric ellipses are studied.A large number of fundamental results have been obtained by several researchers during their studying properties of generalized orthogonalities and the relationship between different kinds of generalized orthogonalities. However, their research interest mainly focus on two fields. For the first, they mainly focus on the properties of generalized orthogonalities on the entire space, and their impact on the properties of the entire space. Properties of generalized orthogonalities at some special points of the underlying space, and their impact on the pointwise properties of the space, even properties of the entire space, was neglected. And few was done in the study of the pointwise properties of generalized orthogonalities. The second, the results on the relationship between different orthogonalities that have been obtained are mostly qualitative, there was a lack of quantitative characterizations.Based on all that have mentioned above, we first proved that, a Minkowski plane is an inner product space if the dual mapping at some special points is linear or Birkhoff orthogonality implies Pythagorean orthogonality at some special points, a new proof of the result that if Birkhoff orthogonality implies Pythagorean orthogonality then the underlying plane is an inner product space is given. We also present a new proof of the result that, if Birkhoff orthogonality implies Isosceles orthogonality then the underlying space is an inner product space. Our proof is much shorter than the one mentioned in the famous book of D. Amir.Then, to present a quantitative characterization of the difference between Birkhoff orthogonality and Isosceles orthogonality, we introduced a new coefficient D(X). The lower and upper bounds of this coefficient are obtained. Further more, a necessary and sufficient condition for the D(X) of the underlying space to attain the lower and upper...