| The usual orthogonality plays a very important role in inner product spaces. Fora much deeper study of the geometrical properties of normed linear spaces, someconcepts and related properties of orthogonality types are introduced. Although manyscholars have done some work in this field, but many problems are still not fullystudied. In this paper, we study relations between s-orthogonality and some otherorthogonality types and related characterizations of inner product spaces.We review concepts of Birkhoff orthogonality, isosceles orthogonality andPythagorean orthogonality and related properties. As well as the relevant knowledgeof the lower and upper semi-inner-product, semi-inner-product, semi-inner-productgenerating the norm of the underlying space, Lumer orthogonality. All these basictheoretical knowledge lay a good foundation for the research carried out in this thesis.As the first part of our main result, we recall the definition and basic propertiesof sub-inner-product, get some relevant conclusions which use the definition of thesub-inner-product and present the relation of sub-inner-product to lower and uppersemi-inner-product so that the concept of sub-inner-product is connected to conceptsof lower and upper semi-inner-product, semi-inner-product, semi-inner-productgenerating the norm of the underlying space, Lumer orthogonality and Birkhofforthogonality.Next, we recall the definition and basic properties of s-orthogonality, study therelations of s-orthogonality to isosceles orthogonality, Pythagorean orthogonality andBirkhoff orthogonality and characterizations of inner product space, discuss therelation of s-orthogonality to Birkhoff orthogonality and characterizations of Radonspace. Several related examples are also presented. |
