Orthogonalities And Abstract Orthogonality In Normed Spaces

Along with the development of functional analysis especially the developmentof geometric theory of Banach spaces, different types of orthogonality relation havebeen introduced into general normed linear spaces. Some orthogonalities, such asisosceles orthogonality, Birkhoff orthogonality, Pythagorean orthogonality and so on,have the specific expressions in their definitions, while others are the abstract binaryrelations, such as the abstract orthogonality introduced by R. W. Freese and C. R.Diminnie and Partington orthogonality. Compared with other orthogonalities,Partington orthogonality and Pythagorean orthogonality have got less attention, andrelated results are not rich.The main result of this thesis consists of two parts.First, we study Pythagorean orthogonality based on existing results. It is provedthat a real normed linear space, whose dimension is at least3, satisfying theimplication x, y∈S_X, x⊥I y（?）‖x+y‖=2^1/2 is an inner product space. On the otherhand, it is shown that a normed linear space with dimension at least3satisfyingimplication x, y∈S X, x⊥P y（?）x⊥P（-y）is an inner product space.Second, we study the relationship between Partington orthogonality and otherorthogonality types including Roberts orthogonality, isosceles orthogonality,Pythagorean orthogonality, Singer orthogonality and Birkhoff orthogonality, andaccordingly we get some characterizations of inner product spaces.